Paper Title: Coordinating Decisions via Quantum Telepathy.




1]Dawei Dingdding@alumni.stanford.edu
[1]Independent
2]Liang Jiang
[2]Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637

Coordinating Decisions via Quantum Telepathy
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§ ABSTRACT 
Quantum telepathy, or pseudotelepathy, is the phenomenon where two non-communicating parties can exhibit correlated behaviors that are impossible to achieve using classical mechanics. This is also known as Bell inequality violation and is made possible by quantum entanglement. In this work, we present a conceptual framework for applying quantum telepathy to real-world problems. In general, the problems involve coordinating decisions given a set of observations without being able to communicate. 
We argue this inability is actually quite prevalent in the modern era where the decision-making timescales of computer processors are so short that speed of light delay is actually quite appreciable in comparison. We highlight the example of high-frequency trading (HFT), where trades are made at microsecond timescales, but the speed of light delay between different exchanges can range from the order of 10 microseconds to 10 milliseconds. Due to the maturity of Bell inequality violation experiments, experimental realization of quantum telepathy schemes that can attain a quantum advantage for real-world problems is already almost immediately possible.

We demonstrate this by conducting a case study for a concrete HFT scenario that gives rise to a generalization of the CHSH game and evaluate different possible physical implementations for achieving a quantum advantage. It is well known that Bell inequality violation is a rigorous mathematical proof of a quantum advantage over any classical strategy and does not need any complexity-theoretic assumptions such as BQP≠BPP. Moreover, fault tolerance is not necessary to realize a quantum advantage: for example, violating the CHSH inequality only requires single-qubit gates applied on two entangled qubits.






§ INTRODUCTION

Quantum entanglement is a unique testament to the strangeness of quantum mechanics. As stated in the celebrated result of John Bell <cit.>, no local hidden variable theory can reproduce the correlated behaviors of entangled particles. His prediction, which is known as Bell inequality violation, and its subsequent experimental verification <cit.>, completely overturned traditional assumptions of how nature works and made mathematically explicit the departure quantum mechanics makes from the familiar classical world. This phenomenon was also described by Einstein, Podolsky, and Rosen as “spooky action at a distance” <cit.>: although they are not communicating, entangled particles can display behaviors that seem impossible, hence the term quantum telepathy, or pseudotelepathy <cit.>. At a high level, quantum telepathy can be described as locally programmable nonlocal correlations. That is, global properties of a system of entangled particles can be altered solely via actions only on individual particles and without communication. Why this phenomenon is possible can be tied to a particularly counterintuitive property of quantum entanglement. Stated plainly, just as relativity defies the ancient intuition that time is absolute, quantum entanglement defies the ancient intuition that a description of a combination of objects is the combination of their descriptions. Borrowing the words of Aristotle, for a system of entangled particles, “the whole is greater than the sum of its parts” in an information-theoretic sense. 

Quantum telepathy is not a just topic of fundamental physics or philosophy. A series of patents <cit.> pointed out that quantum telepathy can actually have real-world applications. 
To better elucidate how Bell inequalities can appear in real-world settings, the patents propose a new, slightly different, concept called “coordinating decisions between non-communicating parties” (CDNP) problems, which we will henceforth refer to more simply as tacit coordination (TC) problems. We also introduce this terminology because we will make some generalizations and even deviations from the usual theory of Bell inequalities and the equivalent computer science concept of nonlocal games <cit.>. In short, a TC problem arises when there are multiple non-communicating parties who each make a local observation and a local decision. The aim is to optimize a global “utility” of their collective decisions based on their collective observations.[See <ref> for a more detailed description and see <ref> for a technical definition. ] Note that for consistency, we will use this terminology throughout the paper.
One very natural real-world setting where TC problems can appear is high-frequency trading (HFT). In this case, each party is a colocated server at a stock exchange engaged in HFT for the same firm. Each server has access to local information, such as stock price fluctuations at their respective exchange. Now, these stock exchanges are spatially separated by distances ranging from tens of kilometers to thousands of kilometers. The speed of light-delay thus ranges from hundreds of microseconds to tens of milliseconds. However, modern HFT is conducted on timescales as short as microseconds and in the future may even shorten to nanoseconds <cit.>. No matter how much effort is put into shortening latencies, with these numbers it is physically impossible for different servers to communicate before making a trade decision. However, there could be a nontrivial globally optimal pair of trades given a pair of observations to maximize expected returns or some other monetary value. See <ref> for an example. Thus, HFT is naturally a TC problem.

Now, we emphasize Bell inequality violation is irrefutable proof of a quantum advantage. It does not require any complexity-theoretic assumptions such as BQP≠BPP and is actually a straightforward mathematical argument. In fact, Bell inequality violation is so convincing that it can be used to prove quantum advantage in other settings, such as shallow quantum circuits <cit.>. Furthermore, to achieve quantum advantage we do not necessarily need complicated schemes such as fault-tolerant quantum computing. For example, the CHSH inequality <cit.> only requires two qubits and a single-qubit rotation prior to measurement. Indeed, the number of qubits and fidelities necessary to violate the inequality have already been achieved half a century ago <cit.>. Even for Bell inequalities that require high-dimensional quantum systems to achieve the maximum violation, it is sufficient to find (if it exists) a low-dimensional scheme that also violates the inequality, just not maximally, to attain a quantum advantage. 




The main contribution of this paper is to point out that TC problems are actually quite prevalent in the real world and that a quantum advantage can be attained with currently available or near-future technologies. 
The core insight is that speed of light delays for distances on the order of 10 km is already 10 to 100 μs, 
which is a relatively long period of time in our modern world satiated with classical processors that have GHz clock speeds.
In particular, we conduct a case study of a concrete HFT scenario that gives rise to a generalization of the CHSH game and assess different possible physical implementations of quantum telepathy schemes.



Our paper will be organized as follows. In <ref>, we first give an informal definition of TC problems. Then, in <ref>, we present the concrete HFT scenario and compute what quantum advantages can be attained. In <ref>, we go into the detailed physical implementations of quantum telepathy schemes. In particular, we evaluate their ability to achieve a quantum advantage for the said HFT scenario, considering practical issues such as photon loss, entanglement generation rates, and robustness to noise. We end with a discussion in <ref>, where we consider other settings where TC problems may arise such as distributed computing and computer architecture. <ref> gives technical definitions and proves facts about TC problems that we use in our analyses, and <ref> presents a numerical optimizer for computing the quantum advantage attainable for general TC problems.



  
Related work Quantum telepathy is not a new concept. There are multiple papers discussing applications of quantum telepathy <cit.>, but none point out that the speed of light delay is a fundamental barrier for communication in many real-world settings. Furthermore, many of the examples given are artificial and do not immediately appear relevant for real-world problems. The concept of using quantum telepathy for HFT was first presented in <cit.>, but no realistic example was given and the distance scales considered were thousands of kilometers. Distributing entanglement across such a distance would require a quantum satellite <cit.>, which is not always practical. These patents also do not go into much detail about physical implementations nor quantitatively evaluate their ability to achieve a quantum advantage. We note the ideas of this paper were included in a new patent <cit.>.



§ DEFINITION OF TC PROBLEMS


We give an informal definition of TC problems. For a technical exposition, see <ref>. A TC problem involves multiple parties that each make an observation followed by a decision so as to maximize a global utility. In <ref>, we give an illustration of a TC problem. For simplicity, we will assume there are two parties. 

A utility array describes what is the utility of a set of decisions given a set of observations. For simplicity, suppose each party can only have two possible observations {o,o'} and two possible decisions {d,d'}. The utility array can be written in matrix form:

    𝒰 =
            d,d     d,d'    d',d     d',d' 
        o,o     u_11    u_12    u_13    u_14
        o,o'     u_21    u_22    u_23    u_24
        o',o     u_31    u_32    u_33    u_34
        o',o'    u_41    u_42    u_43    u_44∈ℝ^4× 4,

where the rows are labeled by possible pairs of observations and the columns are labeled by possible pairs of decisions. The matrix element is then the utility of that pair of decisions given that pair of observations. The parties are to each make a decision given their observation that maximizes the expected value of the utility, given an input distribution, that is, a probability distribution over the observations. The catch is that the parties are cannot communicate during this decision-making process. The reason for this could be various: 

    
  * The parties may be spatially separated and have to make decisions faster than the communication latency or even the speed of light delay.
    
  * The parties may have lost their communication link due to a technical fault or other problem.

The former case can be quite prevalent in modern settings: take the parties to be computers that perform tasks with GHz clock speeds. As long as the spatial separation between the computers is more than 30 cm, communication is physically impossible.

Without communication, how do the parties maximize their utilities? If they're smart, the parties would have coordinated a strategy beforehand given knowledge of the utility array. The implementation of the strategy leads in general to a probability distribution of decisions conditioned on observations, which can also be expressed as a matrix:

    p(d| o) = 
            d,d     d,d'    d',d     d',d' 
        o,o     p_11    p_12    p_13    p_14
        o,o'     p_21    p_22    p_23    p_24
        o',o     p_31    p_32    p_33    p_34
        o',o'    p_41    p_42    p_43    p_44∈ℝ_≥ 0^4× 4.

Because these are probabilities, this is a right stochastic matrix, i.e. the rows sum to 1. We will call this conditional probability distribution the behavior of the parties. The expected utility is then given by multiplying these probabilities with the entries of the utility array and some input distribution p_O(o).

In a deterministic strategy, each individual party simply makes a decision based on their local observation:

    
  * Before starting, the parties prepare respective functions f_1, f_2 : {o,o'}→{d,d'} based on the utility matrix.
    
  * After starting, the parties compute their respective functions by taking their respective local observation as input and use the corresponding output as their decision.

In general, a classical strategy, also known as a local hidden variable theory, can involve randomness: both local randomness for each party and shared randomness establishing correlations between different parties. However, the expected utility of any classical strategy is always a convex combination of the expected utilities of deterministic strategies. Hence, to find the maximum possible expected utility function of all classical strategies, it is sufficient to only consider deterministic strategies. 

Now, the phenomenon of Bell inequality violation in the language of TC problems is this: for certain TC problems, we can devise strategies using quantum mechanics that achieve higher expected utilities than that of any strategy we can devise classically. Such quantum strategies are of the following form:

    
  * Before starting, the parties share entangled particles. The precise state of the entangled particles is specifically designed for the utility matrix.
    
  * After starting, the parties apply a quantum measurement to their particle, the type of measurement being based on their observation.
    
    
  * The parties make decisions based on the measurement result.

Note here that no communication is conducted at any step. However, the parties need to be able to share entangled particles. 

In general, we will be interested in the quantum value and classical value, which are defined as the expected utilities optimized over quantum and classical strategies, respectively. We will refer to the difference between the quantum and classical value as the gap, or quantum advantage. For detailed technical definitions of the concepts in this section, readers can refer to <ref>. 






§ LATENCY TC PROBLEMS


We currently live in a world where the speed of light delay between different parties can be appreciable compared to the timescales in which decisions need to be made. Classical processors have GHz clock speeds, while light in vacuum can only travel 30 cm in 1 ns. Hence, TC problems can easily arise in real-world scenarios where communication is not possible due to latency. In general, we give the following criteria for a latency TC problem:

    
  * Multiple parties are involved that each make local observations and decisions.
    
  * There is a global utility associated with a set of decisions given a set of observations.
    
  * The parties do not have enough time to communicate their observations with each other before having to make a decision.

The last criterion could be due to fundamental physics constraints such as speed of light (in vacuum) delay or other factors such as having only an optical fiber connection through which light has to travel farther than the displacement between the parties and travels more slowly than in vacuum.


































 §.§ High Frequency Trading































In this paper we will conduct a detailed case study involving a high frequency trading scenario where quantum entanglement can provide an advantage. We will attempt to make the scenario as detailed as possible, but some simplifications will be necessary due to lack of real market data and to keep things conceptually straightforward. The TC problem we present should therefore be interpreted as a toy model. In general, to obtain quantitative predictions of quantum advantage in concrete monetary units, real historical HFT data is needed.  We hope that this HFT scenario and the quantum advantage that can be attained will inspire other interesting examples and attract further research into this area.


Consider the HFT setup in <ref>. Suppose a market making trader named Zhuo operates in NYSE and NASDAQ on a cross-listed instrument. In layman's terms, Zhuo is a supplier: he provides trades (both buying and selling) for a stock listed in both NYSE and NASDAQ near current market prices so that trading can proceed. Zhuo has two colocated servers, one at each exchange. Each server receives market data from its local exchange (observations) and modifies its HFT algorithm accordingly (decision). Thus the first criterion for latency TC problems is satisfied. The pair of decisions given the observations will determine how favorable Zhuo's position is given the market information (utility). Thus the second criterion is fulfilled. Furthermore, the data centers of these two stock exchanges are around 35 miles <cit.>, or 56.3 km, apart which means even a line of sight connection in vacuum would entail a speed of light delay of at least about 188 μs. However, HFT is sometimes conducted on timescales as short as microseconds <cit.>, which means in general the two colocated servers do not have enough time to communicate before they have to make a trade decision. Thus the third criterion is satisfied.

We now provide the full details of this particular TC problem. As Zhuo is a market maker, each server conventionally issues orders in pairs where one order is a bid and the other an ask. However, the pair of orders is issued sequentially,[In general this will depend on the order types available at the exchanges. For example, some order types send both the ask and bid orders in one message, but even then the server has to sequentially decide on their respective price values. A possible exception to this is if the order type allows for specifying a spread. ] and the first order is in general more likely to be filled. This is due to the first order arriving earlier and having being made with slightly more up-to-date information. Thus, if a bid order is always issued first in the pair, then there will be a slight bias toward successful bid orders over ask orders. Which order to issue first corresponds to the decision to be made in the TC problem for each server. In general, if Zhuo lets one server issue a bid order first and the other server issue an ask order first, this will effectively lead to hedging. On the other hand, if both servers issue an ask order first or both servers issue a bid order first, then Zhuo will effectively take a directional position. 

Under normal market conditions, Zhuo wants to hedge so as to reduce the variance of his position. However, each server will also look for a technical indicator of an anti-toxic flow,[This is the opposite of toxic flow, which is described for example in <cit.>. ]  that is, a future overreaction. This corresponds to the observation in the TC problem that each server makes. In finance terms, it is an indicator that the next market move, regardless of direction, will tend to be reverting. In layman's terms, it is an indicator the parties Zhuo is trading with are going to be wrong (for example the other party buys when the price will go down). In this case, Zhuo wants to take a directional position, that is, make a net change in the quantity of stocks held. 
Now, both exchanges are looking for this indicator. If both see the indicator, then Zhuo should definitely take a directional position. If only one sees an indicator, there can be slight preference for taking a directional over hedging position or vice versa.

At a high level, we see that in this HFT scenario the utility (favorability of Zhuo's position) is only dependent on whether the servers make the same decision or different decisions. Hence, it is an instance of an XOR game <cit.>. Here, the XOR of the decisions of the two servers has a natural finance interpretation as effectively taking a directional position or a hedging position. We next wish to explicitly write out the utility array. Assuming that only one indicator of an anti-toxic flow is not sufficient to prefer a directional position over a hedging position, the utility array is actually that of the opposite winning conditions of the CHSH game:

    𝒰 =
            A,A     A,B    B,A     B,B 
        N,N     0     1     1     0 
        N,I     0     1     1     0 
        I,N     0     1     1     0 
        I,I     1     0     0     1
        ,

where the observations N,I correspond to “No indicator” versus “Indicator” of anti-toxic flow, and the decisions A,B correspond to “Ask-first” or “Bid-first” order issuing, respectively. We will choose the order of parties as the NYSE server followed by the NASDAQ server. This establishes that a TC problem such as the CHSH game can manifest in real-world scenarios.  Now, to quantitatively evaluate a quantum advantage, we also need to consider the input distribution. Suppose the input distribution is independent Bernoulli distributed with parameter p for both servers, where with probability p the indicator is observed (I). We analytically solve this TC problem in <ref> where we find that a quantum advantage exists iff p ∈ (1 - 1/√(2), 1/√(2)).

We can also consider the natural extension where one server observing the indicator leads to a partial preference of a directional or hedging position. In this case we can introduce a parameter β∈ [0,1] and the utility array is given by

    𝒰 =
            A,A     A,B    B,A     B,B 
        N,N     0     1     1     0 
        N,I    β    1-β    1-β   β
        I,N    β    1-β    1-β   β
        I,I     1     0     0     1
        .

Assuming again an independent Bernoulli distribution for the input distribution with parameter p, we can numerically evaluate the quantum advantage. The result is shown in <ref> for a grid of values for p, β, as well as different cross-sections. These and other numerical results are obtained via a general-purpose numerical optimizer applicable to any TC problem outlined in <ref>.

We see that when β=0, we obtain the previous results where there is a quantum advantage for p ∈ (1-1/√(2), 1/√(2)). Interestingly, as β increases to 0.5, the range of p for which there is a quantum advantage increases, although the value of the quantum advantage decreases. At β =0.5 there is no quantum advantage. For β > 0.5 we observe a mirror image by taking β↦ 1-β. This is because the corresponding utility array can be obtained by appropriately relabeling the observations and decisions, which will lead to the same gap as per <ref> in <ref>. Hence, we again see a variety of real-world scenarios where quantum entanglement can provide an indubitable advantage. For convenience of reference, we will refer to this TC problem with the utility array given in <ref> and with an independent Bernoulli distributed input with parameter p as the hedge-or-not problem.



§ PHYSICAL IMPLEMENTATION


In this section we explore the concrete details of physically implementing a quantum strategy for a latency TC problem. We will in particular take the hedge-or-not problem as a case study. Our results suggest that there is a tantalizing possibility for quantum entanglement to bring practical gains to real-life TC problems such as those encountered in HFT with current or near-future quantum technologies. Moreover, we have a concrete and realistic roadmap to realize said near-future quantum technologies, such as vacuum beam guides <cit.>.

In general, we will classify possible physical implementations into two types: direct photonic connection (Type I) and quantum memory (Type II). 



 §.§ Direct Photonic Connection


In a Type I implementation, the two parties directly receive entangled photons from an intermediate source node. This is exhibited pictorially in <ref>.


We will focus on two key specs for this setup to successfully implement a quantum strategy for a latency TC problem:

    
  * Entangled photon generation rate
    
  * Photon loss.

A latency TC problem arises due to the extremely short time window in which decisions have to be made. Furthermore, for simplicity we will assume the same TC problem repeats itself with high frequency. This is sensible for many HFT scenarios: for instance, market makers such as Zhuo would be solving the same TC problem over and over again because their method of getting returns or taking a favorable position is always the same. The hedge-or-not problem is one such example.

For cases where the utility matrix changes with time, the parties and entangled photon source could synchronize changing the quantum strategy over time.
Nevertheless, this leads to the first requirement for the physical setup: the entangled photons are generated at a sufficiently high frequency so that a quantum strategy can be executed within the time window. In HFT, these time windows are on the order of microseconds, which means entangled photons need to be generated at MHz rates. This is already possible with current technologies using a continuous-wave laser and a nonlinear crystal to produce photon pairs maximally entangled in polarization via parametric down-conversion. These pairs can be produced at MHz rates (see for instance <cit.>). It is possible to accommodate even shorter time windows by using pumped pulses, thereby achieving generation rates up to 50 GHz <cit.>. We can also simply increase the number of entangled photon sources. However, note that it is not strictly necessary that the entanglement generation rate be sufficiently high to attain a quantum advantage overall. The parties can simply execute a deterministic strategy during times when entanglement is unavailable, and execute a quantum strategy when it is.

The second spec is photon loss. This is a key difficulty for a Type I implementation that could even preclude a quantum advantage <cit.>. Because of photon loss, not only are photons sometimes unavailable, each party does not know whether the other party received their photon.[There is also the problem of dark counts where one party erroneously detects a photon. This scenario can be captured with the same mathematical framework. ] One could imagine that if they both knew that one of the photons was lost, they could revert to the best classical strategy and thereby effectively achieve a nontrivial convex combination of the classical and quantum values. This would retain a quantum advantage. However, because they each do not know whether a photon was lost for the other party, they only have probabilistic control over what strategy they implement.  We explicitly define in <ref> how exactly loss affects the resulting behavior of a quantum strategy. In general, the resulting behavior will be a convex combination of behaviors where the weight is the probability of entanglement loss and the parties that experience loss fall back to a deterministic strategy. As the probability of loss increases, the quantum behavior eventually becomes a classical behavior, thereby leading to a loss of a quantum advantage.

Given a TC problem, assume each party successfully obtains their entangled particle with probability η, which we call the efficiency. We can then define a threshold efficiency η^* as the minimal value such that for η > η^*, we can attain a quantum advantage. If the TC problem had no quantum advantage even without loss, we define η^* := 1. Consider for example the hedge-or-not problem. We use the numerical optimizer outlined in <ref> to compute η^* for different possible values of p and β. The results are shown in <ref>.

At a qualitative level, we see the same pattern as in <ref>. This is intuitive as lowering the efficiency will continuously lower the achievable expected utility for quantum strategies with loss until it reaches the classical value. 

It is known that for TC problems with two parties where each party has two possible observations, the threshold efficiency is at least 2/3 <cit.>. This is also apparent in <ref>. If we use optical fiber to distribute the photons, this efficiency is already difficult to achieve for the distances involved for HFT. In general, η exponentially decreases with the length of the channel l:

    η = 10^-0.1 α l,

where α is the attenuation rate of the physical medium used for the channel. Now, the lowest attenuation rate achievable by optical fiber for the optimal wavelength of 1550 nm is 0.17 dB/km <cit.>. Hence, to achieve an efficiency of at least 2/3, the maximum length l allowed is around 10.4 km. Note that this is the distance from the source to each party is therefore half the distance between the two servers. The efficiency computed should also include detector efficiency, inter-component coupling efficiency, and other possible sources of photon loss. This places an even stronger bound on length. We see η^* values considerably higher than 2/3 in <ref>, which shows that using optical fiber in a type I physical implementation is insufficient for the hedge-or-not problem when the two servers are separated by larger distances, such as in the case of NYSE to NASDAQ. There are other physical medium to consider, however. The authors of <cit.> consider using vacuum beam guides to distribute entangled photons with which an attenuation rate as low as 5 × 10^-5 dB/km can be achieved according to numerical simulations. In this case, an efficiency of 2/3 is achievable for distances less than about 35,000 km, which is about the circumference of the Earth. Thus this would allow for quantum strategies to be executed at continental distance scales, for which speed of light delays can be significant. For example, the geodesic distance between the NYSE data center and the HKEX data centre is about 12.9 × 10^3 km, which is a delay of about 43.1 ms. This is very long compared to the time scale of HFT. For the case of NYSE and NASDAQ, using vacuum beam guides we can achieve 1-η of about 3.24 × 10^-4, which is sufficient for the majority of the points in <ref>. Note that optical fiber can still be useful for TC problems with shorter distances, or with a higher number of parties <cit.> or possible observations <cit.>.

We next go into the fine details of strategies for the hedge-or-not problem setting p = 0.3, β =0.3 for concreteness. In this case the threshold efficiency is computed to be η^* ≈ 0.941. So that we attain a noticeable quantum advantage, we set η = 0.95. 
We run the numerical optimizer and find that the classical value is

    c^* = 0.79,

where an optimal deterministic strategy is for the NYSE server to always bid first and the NASDAQ server to ask first when it sees no indicator and to bid first when it sees an indicator. The highest expected utility for a quantum strategy with efficiency η = 0.95 is

    q^*(η = 0.95) ≈ 0.792.

We prove in <ref> and <ref> that when there are only two possible observations and decisions for each party, qubit systems are sufficient for achieving the quantum value, even in the presence of loss. Furthermore, we can parameterize the measurement basis with only one parameter. Define the states

    |ψ(θ)⟩    := cosθ| 0 ⟩  - sinθ| 1⟩
    |ψ^⊥(θ)⟩    := sinθ| 0 ⟩  + cosθ| 1⟩.

One optimal quantum strategy uses the following measurement operators. When neither server sees an indicator, both servers measure in the computational basis[Since we only care about the largest eigenvalue of the Bell operator (see <ref>), there is a local unitary degree of freedom and therefore we can without loss of generality always assume the measurement for the first observation is in the computational basis. ]

    {| 0 ⟩⟨ 0|, | 1 ⟩⟨ 1 |}.

When either server sees an indicator, it instead uses the measurement 

    {|ψ(-0.590) ⟩⟨ψ(-0.590) |, |ψ^⊥(-0.590) ⟩⟨ψ^⊥(-0.590) |},

where the angle is in radians and three significant figures are kept. The shared entangled state is

    |Φ⟩ = 0.0401 | 00 ⟩ - 0.902 | 01 ⟩ - 0.428 | 10 ⟩ - 0.0401 | 11⟩,

keeping three significant figures for the coefficients. Note that since the measurement operators have all real elements, the shared entangled state also has all real elements. 
To realize this quantum physically, we perform a Schmidt decomposition:

    |Φ⟩ = 0.903 | u_0 v_0 ⟩ + 0.429 | u_1 v_1 ⟩,

where

    | u_0 ⟩ := 0.9995 | 0 ⟩ + 0.0301 | 1 ⟩, | u_1 ⟩ := 0.0301 | 0 ⟩ - 0.9995 | 1⟩

and

    | v_0 ⟩ := 0.0301 | 0 ⟩ + 0.9995 | 1 ⟩, | v_1 ⟩ := -0.9995 | 0⟩ + 0.0301 | 1 ⟩.

The state in <ref> can for example be realized by letting | u_i ⟩, | v_i⟩ be the vertical-horizontal polarization basis and using the technique of <cit.>. The single-qubit rotations needed before the measurement to switch back to the computational basis can be realized using linear optics. Lastly, the optimal fallback deterministic strategy is for the NYSE server to always ask first and the NASDAQ server to always bid first. 
















 §.§ Using Quantum Memory







The main shortcoming of Type I implementations is that the magnitude of photon loss in optical fiber, the predominant physical medium for photonic communication in industrial applications, is too high for many TC problems. One solution for this is for the parties to each have a quantum memory, which could be a simple (few qubits) quantum computer with a long coherence time. The memories are entangled via photons in a heralded entanglement scheme. This constitutes a Type II implementation. The setup is shown in <ref>.
 
Because the entanglement is heralded, we can avoid the problem in Type I implementations where the parties do not know if entanglement was successfully distributed. Furthermore, by using quantum memories we can realize more complicated and higher dimensional entangled states that may be difficult to realize with photons. 
Furthermore, we know for XOR problems (TC problems with two possible decisions {0,1} whose utility only depends on the XOR of the decisions) it is sufficient to use maximally entangled states <cit.> to achieve all possible XOR behaviors (probability of XOR = 0 or 1). Without having to worry about loss, for Type II implementations all we have to do is to realize the maximally entangled state in this case.

The main specs we need to consider for Type II implementations are

    
  * Effective entanglement generation rate r_e
    
  * Fidelity of quantum operations F.

The effective entanglement generation rate is determined by the following parameters:

    
  * Entanglement attempt time t_a
    
  * Heralded entanglement success probability p_s
    
  * Quantum memories multiplicity M

Here, t_a is the time needed to generate entanglement for each quantum memory. This includes the time for the physical operation to create memory-photon entanglement, the time for the photons to travel to the intermediate node, and the time for the heralded entanglement information to be received. Usually t_a is dominated by the traversal times. Next, p_s includes the probability of successfully creating memory-photon entanglement, the probability of the photons reaching the intermediate node, and the probability of correctly projecting the photons into the desired state. Lastly, M is the number of quantum memories available to each party. This increases the effective entanglement generation rate by a multiplicative factor. In summary, 

    r_e = M p_s/t_a.

We would like a Type II implementation to have an effective generation rate sufficiently high for the TC problem's demand for entanglement (although again like Type I implementations this isn't necessary to get an overall quantum advantage).

We compute the effective entanglement generation rate for HFT between NYSE and NASDAQ as a function of M. Assuming t_a is dominated by photon traversal time, we split it into two terms. The entangled photon traversal we will assume is through fiber with velocity v_f, while to minimize time for heralding we can use free space transmission with velocity v_s. Then, 

    t_a = 56.3/2 km÷ v_f + 56.3/2 km÷ v_s ≈ 230  μs,

where we take 

    v_f = 2/3· 3 × 10^8  m/s = 2 × 10^8  m/s

and

    v_s = 3 × 10^8  m/s.

Next, we assume p_s is dominated by photon loss and the probability of successful photon state projection which is usually 1/2. Thus, using the optical fiber attenuation rate of 0.17 dB/km,

    p_s = 1/2· (10^-0.1 · 0.17  dB/km· 56.3/2  km)^2 ≈ 0.055.

We therefore have

    r_e ≈ M · 240  Hz.

As an example, to achieve 1 MHz rates needed for HFT, we require M ≈ 4300. This is entirely feasible, for not only are multi-thousand-qubit processors already within reach <cit.>, but what is needed here is not a 4300-qubit processor, but simply 4300 copies of single-qubit processors, where each copy has a good photonic interface for establishing long-range entanglement. The technology to achieve this has already been demonstrated: see for instance <cit.>. Note that we are assuming that the TC problem only requires qubit systems as in the case of the hedge-or-not problem. In general, letting d be the distance between the parties, with the above assumptions on t_a and p_s, the effective entanglement rate is 
    r_e = M ·10^-0.1 α d/2(d/2v_f + d/2v_s ) = M ·10^-0.1 α d/d(1/v_f + 1/v_s ).


For Type II implementations, fidelity is also an important specification. Entanglement generation involves noisy memory-photon entanglement, imperfect photon detection, as well as decoherence of the memories during photon traversal. Measurement of the memories themselves can also be noisy. We will evaluate how much the quantum advantage computed for the hedge-or-not problem in <ref> is robust to noise. For simplicity, we will assume depolarizing noise on the entangled state, but in realistic scenarios, a detailed physics simulation should be conducted. We define the robustness ν^* as how large of a depolarizing error can be tolerated before the quantum advantage disappears. 

Now, the effect of depolarizing noise

    ρ↦ (1-ν) ρ + νπ,

where π is the maximally mixed state and ν is the magnitude of the noise, on the behavior of a quantum strategy for the hedge-or-not problem is the following:

    p(d | o) ↦ (1-ν) p(d | o) + ν1/4,


where we assume the quantum strategy achieves the quantum value and uses only qubits. Using higher dimensional systems could affect the robustness, but at the cost of also incurring larger error rates for the physical implementation. For a derivation of these results, see <ref>. Also see <cit.> for a general study on noise tolerance.  Using <ref> and the utility array <ref>, we can directly compute the effect of depolarizing noise on the expected utility:

    u̅   ↦ (1-ν) u̅ + ν1/4× [(1-p)^2 (1+1) + 2 p(1-p) (β + β + 1- β +1-β) + p^2 (1+1)] 
        = (1-ν) u̅ + ν1/2.

Conveniently, this transformation does not depend on p or β. We observe the second term in <ref> corresponds to a constant classical behavior where each party independently chooses one of the possible decisions A,B with probability 1/2 regardless of the observation, so the utility of 1/2 achieved by such a behavior as computed in <ref> is always at most the quantum value. Thus, we can simply compute the robustness as

    ν^* = q^* - c^*/q^* - 1/2.

We plot the robustness for the hedge-or-not problem as a function of p and β in <ref>.

In the range of p and β considered, other than some extreme values, we observe a robustness to depolarizing noise of about 10^-3 to 10^-1. We even see robustness values of up to 0.293. That is, optimal quantum strategies using qubits can tolerate depolarizing noise up to (not including) that magnitude and a quantum advantage would remain. For fixed β, we also observe some regions with constant robustness, which implies in those regions the quantum value q^* is a fixed affine function of the classical value c^*. In general, noise robustness can be interpreted as some form of normalized maximal Bell inequality violation <cit.>. We also plot the quantum advantage for different values of p and β under various noise levels in <ref>. The range of p and β for which there is a quantum advantage shrinks as the noise level increases.




§ DISCUSSION



Quantum telepathy is a potential new application of quantum technologies. In our modern era of digital computing where timescales are measured in microseconds or even nanoseconds, the speed of light delay is actually appreciable in many scenarios. Coordinating decisions at such timescales between space-like separated parties could benefit from using quantum entanglement. High frequency trading is a notable example. 

Aside from HFT, other possible applications include distributed computing and (classical) computer architecture. While HFT is somewhat specialized, these latter two scenarios are much more common, and any interesting usage of quantum telepathy would have a greater impact. We consider the first application. Distributed computing may involve computers with a spatial separation of around 100 meters, the square root of the area of an average data center in 2024 <cit.>. At this distance, any computational processes faster than 0.1 microseconds would be space-like separated. Current entangled photon sources can achieve generation rates at this timescale <cit.>, and optical fiber can achieve an efficiency of η≈ 0.998 at this distance scale.[Note in this case, the dominating source of loss would instead be from coupling. With current technologies, coupling efficiencies of 90% or higher can be achieved <cit.>. ]
Hence this is the best regime to make use of current technologies. As for the latter application of computer architecture, distance scales may be on the order of 1 cm or shorter in a CPU (for example an Intel i7 CPU core package size is 50× 25 mm <cit.>). Although it would be difficult to find any processes faster than the speed of light delay at such a distance scale, in general signal delays in a microprocessor cache can be dominated by resistive-capacitive time delays in long on-chip interconnects <cit.>. Furthermore, modern computer memories can have latencies significantly longer than the speed of light delay: DDR memories can have latencies on the order of 10 ns <cit.>, while SSD memories can have latencies of 10 to 100 μs <cit.>. Needing to coordinate processes in different parts of the memory at these timescales may therefore engender a TC problem. However, for such timescales, we may need to use photon sources with very high generation rates such as <cit.> as well as faster electronics for measurement. Now, in the case of computer architecture, for integration purposes it is more practical to use optical waveguides to distribute photons. Compared to optical fiber, waveguides have a much higher attenuation rate of about 0.2 dB/cm <cit.>, which implies an efficiency of η≈ 0.955 for a 1 cm distance. Note that the speed of light in a waveguide is around the same as that of optical fiber: 2/3 of that in vacuum. Taking a step back, whether the application is HFT, distributed computing, or computer architecture, the crucial next step for assessing the practical usefulness of quantum telepathy is to analyze a TC problem encountered in an industrial setting with a utility array based on real data, not just a toy model such as the hedge-or-not problem, and to comprehensively analyze achievable efficiencies, fidelities, and entanglement generation rates to evaluate how big of a quantum advantage can be attained. We leave this for future research.

Indeed, as a potential important application of quantum technologies, we anticipate quantum telepathy will inspire new lines of theoretical and experimental research to make the technology practically viable. On the theory side, a key research direction is to figure out techniques to compute the quantum value and a quantum strategy that realizes it in an efficient way for general TC problems, or special classes of TC problems encountered in real-life scenarios. Such techniques should ideally also be able to consider entanglement loss. In particular, stronger results regarding the minimum number of continuous variables needed in the general-purpose optimizer outlined in <ref> would be very useful. It would also be interesting to theoretically take into account the multi-round nature <cit.> of TC problems encountered in practice. On the experimental side, a research direction of crucial importance is to devise ways to share programmable entangled states between the different parties. 
For Type I implementations, one possible approach is for the entanglement source to use quantum computers such as atoms <cit.> or superconducting circuits <cit.> to produce optical or microwave entangled photons with a programmable quantum state.
For Type II implementations, we may need to devise heralded entanglement schemes that produce more general entangled states and not just the maximally entangled state or a state of the type in <ref>.
Regardless of implementation type, the authors of <cit.> mention it would also be particularly interesting to physically realize quantum embezzling states <cit.> which can be used to produce any bipartite entangled state without communication. Such an entanglement source can accommodate all possible TC problems, which is very convenient for standardization purposes. It could also be used in more complicated scenarios where the utility array is changing over time.







  
Acknowledgment We would like to thank Jianxin Chen, Dalu Ding, Yong Ding, Jack Gidding, Patrick Hayden, Zhengfeng Ji, Rongzhi Jiang, Matt Meinel, Anand Natarajan, Mario Szegedy, and Zhaoyou Wang for helpful discussions. DD would like to thank God for all of His provisions. LJ acknowledges support from the Packard Foundation (2020-71479). 

unsrt






§ GENERAL TC PROBLEMS: DEFINITIONS AND FACTS



In this section we present the theory of TC problems. Many of the concepts before <ref> are well known in the literature on Bell inequalities (see for example <cit.>), although we do make some generalizations and even deviations <cit.> to capture essential elements of a real-world TC problem and give a convenient framework to prove some results. We will make such deviations explicit. 

We first define a general TC problem.

    Let n ≥ 2 be an integer. An n-party TC problem is a tuple 
    
    ({𝒪_i}_i=1^n, {𝒟_i}_i=1^n, p_𝒪(o), u),

    where 𝒪_i, 𝒟_i are non-empty finite[Theoretically, we could consider infinite sets, but this is not feasible in practical experimental settings. ] sets, p_𝒪(o) is a probability distribution over 𝒪 := 𝒪_1 ×𝒪_2 ×⋯×𝒪_n, and u is a multidimensional array indexed by observations and decisions: u_o_1, o_2, ⋯, o_n^d_1, d_2, ⋯ d_n∈ℝ, o_i ∈𝒪_i and d_i ∈𝒟_i. We also define 𝒟 := 𝒟_1 ×𝒟_2 ×⋯×𝒟_n.

𝒪_i and 𝒟_i are the sets of observations and decisions for party i. p_𝒪(o) is the input distribution, a probability distribution over the set of possible observation tuples across all parties. The marginal distributions of observations for each party in general may not be independent. This is an additional element that is present in nonlocal games but not general Bell expressions as presented in <cit.>. Lastly, u is the utility array, where[For convenience here and in other places we will often use the shorthand u_o^d:= u_o_1, o_2,⋯, o_n^d_1, d_2, ⋯ d_n. ] u_o^d ∈ℝ is the utility of making a combination of decisions d ∈𝒟 given the combination of observations o ∈𝒪. We deviate from the theory in <cit.> by proposing a multidimensional array structure instead of a vector structure for their equivalent notion of a Bell expression. This is more natural since for one we should differentiate the observation indices from the decision indices and each party's index from each other. We also introduce the terminology that when each party has the same number of observations m and the same number of decisions Δ, we have an (n,m,Δ) problem.

For any i ∈ [n], where [n]:={1, 2, ⋯, n}, let o_i ∈𝒪_i, d_i ∈𝒟_i. We define a behavior as a multidimensional array p_o_1, o_2, ⋯ o_n^d_1, d_2, ⋯ d_n as conditional probabilities p(d | o). The expected utility of a behavior is given by 

    ∑_o ∈𝒪 p_𝒪(o) 
        ∑_d ∈𝒟 p_o^d u_o^d.
 

The conditional probability distribution p(d| o) describes how parties make decisions given the observations. Again, we deviate from <cit.> by choosing to describe this as a multidimensional array instead of a vector. Now, in a TC problem, the parties are not in communication. Hence, their behavior has to satisfy a no-signaling condition, that is, for all i ∈ [n], o_1, o_2, ⋯, o_i-1, o_i+1, ⋯, o_n, d_1, d_2, ⋯, d_i-1, d_i+1, ⋯, d_n in their respective alphabets,

    ∑_d_i ∈𝒟_i p^d_1, d_2, ⋯, d_i, ⋯ d_n_o_1, o_2, ⋯, o_i, ⋯, o_n

is the same for all o_i ∈𝒪_i.
In words, the marginal distribution over decisions of all parties other than party i is independent of the observation of party i. Here, our choice using a array structure for the behavior gives the no-signaling condition a more natural interpretation as a form of multidimensional array symmetry.
Note that the TC problem formalism generalizes nonlocal games that usually only have only binary outcomes of winning or losing. The formalism can capture this notion as a special case by letting u_o^d=1 if (o,d) satisfies the winning conditions and equal 0 otherwise. Then, the expected utility of a behavior is the probability of winning the nonlocal game.

We next define what behaviors are possible for parties with classical and quantum resources. This part is very similar to the concepts in <cit.>, so we will be brief.

    A deterministic strategy is given by {f_i}_i=1^n where f_i : 𝒪_i →𝒟_i. 

In a deterministic strategy, each player has a local function f_i : 𝒪_i →𝒟_i which they use to make decision d_i given observation o_i. The corresponding deterministic behavior is given by

    p_o^d := ∏_i=1^n δ_d_i, f_i(o_i),

where o := (o_1, o_2, ⋯, o_n), d := (d_1, d_2, ⋯, d_n), and δ is the Kronecker delta function. The expected utility of a deterministic behavior is given by

    ∑_o ∈𝒪 p_𝒪(o) u_o^f(o),

where f:𝒪→𝒟 is given by f((o_1, o_2, ⋯, o_n)) := (f_1(o_1), f_2(o_2), ⋯, f_n(o_n)). More generally, a classical behavior is defined by convex combinations of deterministic behaviors, which are attained by classical strategies: probabilistic mixtures of deterministic strategies where shared randomness can be used.


    A quantum strategy is a tuple ({q_i}_i=1^n, {M_i}_i=1^n, |ψ⟩), where q_i≥|𝒟_i| is a positive integer, M_i :𝒪_i → S_M(q_i, |𝒟_i|), S_M(q_i, |𝒟_i|) being the set of all projective measurements on a q_i-dimensional Hilbert space consisting of |𝒟_i| projectors, and |ψ⟩ is a ∏_i=1^n q_i-dimensional quantum state. 

In a quantum strategy, the players share a global quantum state |ψ⟩, where each of their shares is of dimension q_i. They apply a projective measurement M_i(o_i) on their share when they make the observation o_i. There are |𝒟_i| possible measurement outcomes, which correspond to the possible decisions each party can make. We can more conveniently denote the projectors of M_i(o_i) as Π_i^(d_i | o_i), indexed by d_i ∈𝒟_i. The corresponding quantum behavior is then given by
    
    p_o^d := ⟨ψ|⊗_i=1^n Π_i^(d_i| o_i)|ψ⟩.

We can therefore express the expected utility as

    ⟨ψ|∑_o ∈𝒪 p_𝒪(o) ∑_d ∈𝒟⊗_i=1^n Π_i^(d_i| o_i) u_o^d |ψ⟩.

We will call the operator 

    ∑_o ∈𝒪 p_𝒪(o) ∑_d ∈𝒟⊗_i=1^n Π_i^(d_i| o_i) u_o^d

the Bell operator of a quantum strategy. For the purposes of computing the quantum value it is sufficient to compute the largest eigenvalue of the Bell operator. Note in general we can also allow density matrices and POVM elements, but such strategies can always be considered to be a quantum strategy according to <ref> but in a higher dimensional space via Naimark's dilation theorem.



 §.§ Quantum Advantage


Now, we want to optimize the expected utility over different strategies. We can see from <ref> that for such a purpose the key object of interest is the weighted utility array w where w_o^d := p_𝒪(o) u_o^d.




Indeed, we can express the expected utility of a behavior p(d| o) as simply

    ∑_o ∈𝒪, d ∈𝒟 p_o^d w_o^d.


We denote the set of all possible deterministic behaviors as 𝐃 and quantum behaviors as 𝐐. 
Then, we define the classical value and quantum value of w as 

    c^*(w)     := max_p_o^d ∈𝐃∑_o ∈𝒪, d ∈𝒟 p_o^d w_o^d, 
    
        q^*(w)    := max_p_o^d ∈𝐐∑_o ∈𝒪, d ∈𝒟 p_o^d w_o^d.


We will call g(w) := q^*(w) - c^*(w) the gap. 

Now, since deterministic strategies form a subset of quantum strategies, g(w) ≥ 0. We will particularly be interested in cases for which g(w) > 0, in which case we call w gapped. Otherwise, g(w) = 0 and we call it gapless. We denote the set of all gapped weighted utility arrays by 𝒢. We establish some basic properties of 𝒢. We in particular explore transformations of an array that preserves the gapped property.

    𝒢 is a cone, that is, it is closed under positive scalar multiplication.


Let α>0. Then, let w ∈𝒢. It is clear that

    c^*(α· w) = max_p(d| o) ∈𝐃∑_o ∈𝒪, d ∈𝒟 p(d| o) α w(o,d) = αmax_p(d | o) ∈𝐃∑_o ∈𝒪, d ∈𝒟 p(d| o) w(o,d) = α c^*(w).

Similarly, q^*(α·  w) = α q^*(w). Thus, g(α· w) = α g(w) >0 and so α· w ∈𝒢.

We also define the constant array e ∈𝒢 where e_o^d = 1. It is clear translation by a multiple e also preserves 𝒢:

    Let w be a weighted utility array. Then, g(w + x e) = g(w) for x ∈ℝ. In particular, if w ∈𝒢, then w + xe ∈𝒢.


    We have trivially
    
    c^*(w+η e)    = c^*(w) + x |𝒪|
    
            q^*(w+η e)   = q^*(w) + x |𝒪|.

    Hence, g(w+x e) = g(w). The second statement is immediate.




    
    
        
    






 


    



Another observation is that the ordering of observations and decision is arbitrary. Hence, the following holds.


    The gap of a weighted utility array is invariant under permutations of observations and decisions. That is, letting w be a weighted utility array, define the multidimensional array v whose elements are
    
    v_o_1, o_2, ⋯, o_n^d_1, d_2, ⋯, d_n := w_π_1(o_1), π_2(o_2), ⋯, π_n(o_n)^σ_1(d_1), σ_2(d_2), ⋯, σ_n(d_n),

    where π_i, σ_i are permutations of 𝒪_i, 𝒟_i, respectively. Then, g(v) = g(w). In particular, if w ∈𝒢, then v ∈𝒢.


    This is immediate via simply relabeling observations and decisions according to π_i and σ_i, respectively, for all strategies.


We also state properties that do not hold regarding 𝒢. 


    𝒢 is not closed under addition.

That is, 𝒢 is a cone but not a convex cone. To see this, consider the weighted utility matrix w_CHSH corresponding to the CHSH game <cit.>:

    4 (w_CHSH)_o_1, o_2^d_1, d_2 := (o_1  o_2) ⊕ (d_1 ⊕ d_2) ⊕ 1.

Here, we let o_i, d_i ∈{0,1} so that we can use logical and bitwise addition operators.  Intuitively, the two parties have to make opposite decisions when both observations are 1 and have to make the same decision otherwise. It is well known that w_CHSH∈𝒢. Then, define the “anti-CHSH game” weighted utility matrix w̅_CHSH as

    4 (w̅_CHSH)_o_1, o_2^d_1, d_2 := (o_1  o_2) ⊕ (d_1 ⊕ d_2).

That is, it has the opposite winning condition. By a similar argument as that of the CHSH game, w̅_CHSH∈𝒢. However, 

    w_CHSH + w̅_CHSH= 1/4 e ∉𝒢.

This establishes <ref>.

Furthermore, counter to intuition, scaling the weighted utility by a different positive constant for each set of observations does not always preserved gappedness. This kind of scaling can be interpreted as changing the input distribution.


    ∃ w ∈𝒢, α_o > 0 such that the array w', defined as (w')_o^d := α_o w_o^d, is not in 𝒢.

The example is again CHSH, but this time we will make use of the correlation form:
 
    1/4⟨ A_0 B_0 ⟩ + 1/4⟨ A_0 B_1 ⟩ + 1/4⟨ A_1 B_0 ⟩ - 1/4⟨ A_1 B_1 ⟩ .


































































































































































    
    
    
    
    

























































































The scaling we choose is a natural scenario in which the input distribution for each party is i.i.d. according to a Bernoulli distribution with parameter p ∈ [0,1] instead of the uniform distribution as is usually assumed for the CHSH game. In this case, we want to instead compute

    (1-p)^2 ⟨ A_0 B_0 ⟩ + p(1-p) ⟨ A_0 B_1 ⟩ + p(1-p) ⟨ A_1 B_0 ⟩ -p^2⟨ A_1 B_1 ⟩ .

For conciseness, we will denote the correlations with a, b, c, d ∈ [-1,1], respectively. Then, the quantum value is given by <cit.>

    max_a,b,c,d ∈ [-1,1] (1-p)^2 a + p(1-p) b + p(1-p) c - p^2 d





subject to 

    | arcsin a + arcsin b +arcsin c - arcsin d| ≤π.

We will compute this in full generality. 
First, we see that if <ref> is strictly satisfied, we can always increase a,b,c and decrease d until

    arcsin a + arcsin b + arcsin c - arcsin d = π

and the expression in <ref> can only increase. Thus, it is sufficient to consider the equality condition <ref>.








    

















































We make use of the method of Lagrange multipliers:

    ℒ = (1-p)^2 a + p(1-p)b + p(1-p) c - p^2 d - λ (arcsin a + arcsin b + arcsin c - arcsin d - π).

We solve for the stationary points:

    ∂ℒ/∂ a = (1-p)^2 -  λ1/√(1-a^2) = 0,

so

    a = ±√(1- λ^2/(1-p)^4).

Similarly, 

    b = ±√(1  - λ^2/p^2(1-p)^2),


    c = ±√(1  - λ^2/p^2(1-p)^2),

and

    d = ±√(1  - λ^2/p^4).

In particular, we see that b  = ± c. If b = -c, the correlation expression becomes

    (1-p)^2 a - p^2 d ≤ (1-p)^2 + p^2.

The RHS can be obtained by the classical behavior where a = 1, b = 1, c = -1, d = -1, which also does not attain the classical value (It is clear that depending on the value of p, we either want to set either the first or last term in <ref> to be negative.). Thus, WOLOG, we will set c to equal b.
Thus, we have the simpler optimization

    max_a,b,d ∈ [-1,1] C,

where

    C:=(1-p)^2 a + 2 p (1-p)b - p^2 d

and

    arcsin a + 2 arcsin b - arcsin d = π.

Let
 
    (x) =
         
             -1     x< 0
    
             0     x= 0
    
             1     x>0

 be the sign function.
 We consider the following cases
 
     
  * (a) (d) ≥ 0: Then, taking the cosine of both sides of
     
    arcsin a -arcsin d = π - 2arcsin b,

     we obtain
     
    cosarcsin√(1- λ^2/(1-p)^4)cosarcsin√(1- λ^2/p^4) + √(1- λ^2/(1-p)^4)√(1- λ^2/p^4)
        = - cos 2arcsin√(1-λ^2/p^2(1-p)^2)
 
     since cosine is an even function. We simplify this equation to get
     
    λ/(1-p)^2λ/p^2 + √(1- λ^2/(1-p)^4)√(1- λ^2/p^4) = 2 (1 - λ^2/p^2(1-p)^2) - 1.

     We plug this into Mathematica <cit.> and obtain
     
    λ = 0 , ±1/2√(2)√((2p^2-1)(2p^2-4p+1)) =: ±λ^*.

     The λ =0 solutions are classically attainable solutions, which might be optimal for certain values of p. 
     
     
  * (a) (d) < 0: 
     We follow similar steps to get
     
    λ = ±λ^*.

     
 
We need to check what conditions on p ∈ [0,1] guarantee that λ^* ∈ℝ. We see that this is true if










    p ∈ [1- 1/√(2), 1/√(2)].

We can easily check that (λ^*)^2 ≤ p^4, (1-p)^4 and so is ≤ p^2(1-p)^2, their geometric mean. This ensures that a,b,d ∈ℝ, which means when λ^* ∈ℝ, we obtain feasible solutions.

Now, we observe that to satisfy <ref>, we must have at least one of a,b ≥ 0. Furthermore, at least one of b,d ≥ 0. We therefore have the following 5 cases:
 
     
  * a,b,d ≥ 0: Then (a) (d) ≥ 0. Hence, 
     
    C|_λ = ±λ^* =  1/2√(2)(|2p^2-6p+3|+ 4p^2 - 4p +2  - |2p^2+2p-1|),

     which is feasible when p∈ [1-1/√(2), -1+√(3)/2] by checking if <ref> is satisfied. Note that the three-term expression is written in the same order as <ref>. In this range for p, we can further simplify this to
     
    C |_λ = ±λ^* = √(2) [1-2p(1-p)].

     There is also the classical solution which is possible for all p ∈ [0,1]:
     
    C_λ = 0 = -2p^2+1.

     
  * a,b ≥ 0, d <0: 
     
    C |_λ = ±λ^* = √(2) [1-2p(1-p)],

     feasible when p ∈ [-1+√(3)/2, 3-√(3)/2].
     
  * a,d ≥ 0, b <0: One can check the λ^* solution is not feasible for any p ∈ [1-1/√(2),1/√(2)]. The classical solution in this case is
     
    C|_λ=0 = 2p^2-4p+1.

     
  * b,d≥ 0, a<0: One can check the λ^* solution is not feasible for any p ∈ [1-1/√(2),1/√(2)].
     
  * b ≥ 0, a,d<0: 
     
    C |_λ = ±λ^* = √(2) [1-2p(1-p)],

     feasible when p ∈ [3-√(3)/2,1/√(2)]. The classical solution is
     
    C_λ=0 = -2p^2+4p-1.





















 
 











Since the CHSH inequality with an independent Bernoulli distributed input is a natural generalization of the original inequality, we summarize our result as a theorem.

    Consider the utility array w_CHSH,p for the CHSH game with each input distribution being an independent Bernoulli random variable with probability p. Then, the classical value is given by
    
    c^* =
        
            1-p^2     0 ≤ p ≤1/2
    
            -p^2 +2p    1/2≤ p ≤ 1.

    while the quantum value is given by
    
    q^* =
            
                1-p^2     0 ≤ p ≤ 1 -1/√(2)
    1/√(2)[1-2p(1-p)] + 1/2    1- 1/√(2)≤ p ≤1/√(2)
    
                
                
                -p^2 +2p    1/√(2)≤ p ≤ 1.

    In particular, w_CHSH,p∈𝒢 when p ∈ (1-1/√(2), 1/√(2)) and ∉𝒢 otherwise.

Hence, for p ∉(1- 1/√(2), 1/√(2)), we obtain <ref>. For convenience, we also provide the numerical values for the bounds: 

    q^* =
            
                1-p^2     0 ≤ p ≤ 0.293…
    1/√(2)[1-2p(1-p)] + 1/2    0.293…≤ p ≤ 0.707…
    
                
                
                -p^2 +2p     0.707…≤ p ≤ 1.

We can check that indeed for p = 1/2,

    q^* = 1/√(2)(1-1/2)+1/2 = 1+√(2)/2√(2) = cos^2(π/8)

as expected. 





 §.§ XOR Arrays

An interesting class of TC problems have utilities that only depend on the XOR of the decisions. We make the following definition.

    Suppose we have a TC problem where ∀ i, 𝒟_i = {0,1}. An array m indexed by o ∈𝒪, d ∈𝒟 is an XOR array if 
    
    m_o^d = f(o, ⊕_i=1^n d_i) ∈ℝ.


We will call TC problems whose utility array is an XOR array an XOR problem.  Note that this implies the weighted utility array is also an XOR array.
Such problems are equivalent to correlation expressions <cit.> and the quantum values can be explicitly computed via a semidefinite program (SDP) <cit.>. 

An interesting observation is that we can conclude g(w̅_CHSH) = g(w_CHSH) via <ref> applied to w_CHSH and setting π_1 be a bit flip while letting all else be equal. We can generalize this observation by making the following definition.


    Let m be an XOR array, where m_o^d = f(o, ⊕_i=1^n d_i). Then, define m̅ as the anti-array that has elements
    
    m̅_o^d := f(o, ¬⊕_i=1^n d_i),

    where ¬ x denotes the bit flip of x.

Intuitively, if w is a nonlocal XOR game, then w̅ is the same game with the opposite winning conditions. We can now easy establish the following

    Suppose w is an XOR array. Then, g(w̅) =g(w). 


    Let π_1 be the bit flip permutation on {0,1}. Then,
    
    w̅_o^d = f(o, ¬⊕_i=1^n d_i) = f(o, π_1(d_1) ⊕⊕_i=2^n d_i) = w_o^π_1(d_1), d_2, d_3, ⋯, d_n.

    Hence, by <ref>, the conclusion follows.

Intuitively, for XOR problems, the procedure of reversing the winning conditions is equivalent to local index permutation of the utility array, thereby preserving the gap. 



A central result for XOR problems with two parties is the following theorem by Tsirelson.


    The following four conditions for real numbers c_kl, k = 1,⋯ ,m, l = 1, ⋯, n are equivalent.
    
        
  * There are a C^* algebra 𝒜 with identity I, Hermitian A_1,⋯, A_m,B_1,⋯,B_n ∈𝒜, and a state f on 𝒜 such that, for every k,l,
        
    A_k B_l = B_l A_k;    -I ≤ A_k ≤ I;    -I ≤ B_l ≤ I;    f(A_k B_l) = c_kl.

        
  * There are Hermitian operators A_1, ⋯, A_m, B_1, ⋯ B_n and a density matrix ρ in a Hilbert space ℋ such that, for every k,l,
        
    A_k B_l = B_l A_k;   Spec(A_k) ⊆ [-1,+1];   Spec(B_l)⊆ [-1,+1];   [A_k B_l ρ] = c_kl.

        
  * The same as 2 and in addition for every k,l; and ℋ = ℋ_1 ⊗ℋ_2, A_k = A_k^(1)⊗ I_2 and B_l = I_1 ⊗ B_l^(2) act on ℋ_1 and ℋ_2, respectively; furthermore
        
    (A_k^(1))^2 = I_1, (B_l^(2))^2 = I_2, [(A_k^(1)⊗ I_2) ρ]=0, [(I_1 ⊗ B_l^(2)) ρ]=0;

        besides that 
        
    {A_k^(1), A_k'^(1)}∝ I_1, {B_l^(2), B_l'^(2)}∝ I_2;
 
        ℋ_1, ℋ_2 are finite dimensional, obeying
        
    dim(ℋ_1) ≤
                    2^m/2    m  even
    
                    2^m+1/2    m  odd

        and similarly for ℋ_2.
        
  * There are unit vectors x_1,⋯,x_m,y_1,⋯,y_n ∈ℝ^m+n such that, for every k,l,
        
    ⟨ x_k, y_l⟩ = c_kl.

    

The set {c_kl}_k,l is called a quantum correlation matrix. In terms of the TC problem formalism, we can relate a quantum correlation matrix c_kl corresponding to the quantum behavior p_o^d via

    c(p_o^d)_kl := (p_(k,l)^(0,0) + p_(k,l)^(1,1)) - (p_(k,l)^(0,1) + p_(k,l)^(1,0)) = 2 p( XOR=0 | (k,l)) - 1.

Note for XOR problems the expected utility of a quantum behavior p_o^d only depends on the quantities p(XOR=0| (k,l)). Furthermore, we can relate the operators A_k^(1), B_l^(2) with the measurement operators Π_i^(d_i | o_i) by

    A_k^(1) = 2 Π_1^(0 | k) - I_1, B_l^(2) = 2 Π_2^(0 | l) - I_2.


We make the following definition.

    Define a projection operator Π as trivial if Π is the zero or identity operator. We define a quantum strategy to be degenerate if one party uses a trivial measurement operator. Otherwise, we call it non-degenerate.

Note that if a party with only two possible decisions uses a trivial measurement operator, they are locally implementing a deterministic strategy. We can thereby obtain the following corollary.


    Consider an XOR problem with two parties. Then, any possible quantum correlation matrix c_kl can be realized using a non-degenerate quantum strategy. Moreover, such a strategy can attain the quantum value.


    
    Let 𝒮_q be a quantum strategy and the c_kl be the quantum correlation matrix realized. If one of the parties uses a trivial measurement operator, WOLOG the first party, then for some k, Π_1^(0 | k) is the zero or identity operator. Then, A_k = ± I. But this implies [A_k ρ] ≠ 0 for any quantum state ρ. Thus, measurement operators obeying condition 3 of <ref> must all be nontrivial measurement operators. The first conclusion follows. Since c_kl are all that is involved in the expected utility of an XOR problem, the second conclusion follows.

Results like these can help reduce the search space for numerical optimizers that search over all possible quantum strategies. 

















 §.§ Behaviors with Loss


Here we take a look from a theoretical standpoint the phenomenon of entanglement loss when physically implementing a quantum strategy. As mentioned in <ref>, such loss is common when using photons to distribute entanglement <cit.>. In general, when loss occurs for a certain party, that party can fall back to a predetermined local deterministic strategy. We can rigorize this concept using the following definition.[Note that instead of defining a semiclassical strategy as a special case of a quantum strategy, we could define it as a generalization of a classical strategy, which would then allow for the choice of trivial measurement operators to be based on observations and shared randomness. This is more general than what we define but does not lead to higher expected utilities. Furthermore, “semiclassical” more often implies starting from quantum and then taking the classical limit. ]

    Let S ⊆ [n]. Then, we define an S-semiclassical strategy as a quantum strategy where the measurement operators of the parties in S are all trivial for all possible observations. That is, Π_i^(d_i | o_i) for i ∈ S are either the zero or identity operator. 

In other words, the parties in S are each implementing a local deterministic strategy. An S-semiclassical behavior is the corresponding behavior of an S-semiclassical strategy. Now, in general entanglement loss is stochastic, therefore leading to the following definition.

    Let 𝒮_q be a quantum strategy, 𝒮_d a deterministic strategy, and S ⊆ [n]. Then, we use 𝒮_q ⊔_S 𝒮_d to denote the S-semiclassical strategy obtained by modifying the quantum strategy 𝒮_q so that the each party in S locally implements the deterministic strategy 𝒮_d. 
    
    Next, let η_i ∈ [0,1] where i ∈ [n]. Then, the {η_i}_i-lossy behavior of the tuple (𝒮_q, 𝒮_d) is given by 
    
    p_(𝒮_q , 𝒮_d); {η_i}_i(d| o) := ∑_S ⊆ [n]∏_i ∈ S (1-η_i) ∏_j ∉Sη_j · p_𝒮_q ⊔_S 𝒮_d(d | o),

    where p_𝒮(d| o) denotes the behavior corresponding to a strategy 𝒮.

    Finally we define q^*({η_i}_i) as the {η_i}_i-lossy value of a TC problem the maximum expected utility with respect to all possible {η_i}_i-lossy behaviors.

It will be useful to define the {η_i}_i-lossy Bell operator for a tuple (𝒮_q, 𝒮_d)

    ∑_S ⊆ [n]∏_i ∈ S (1-η_i) ∏_j ∉Sη_j ·∑_o ∈𝒪 p_𝒪(o) ∑_d ∈𝒟⊗_i=1^n Π_i^(d_i| o_i) (𝒮_q ⊔_S 𝒮_d) u_o^d,

where Π_i(𝒮_q ⊔_S 𝒮_d) are the measurement operators used in the S-semiclassical strategy 𝒮_q ⊔_S 𝒮_d, which is itself a quantum strategy. The largest eigenvalue of this operator is highest expected utility for the choice of measurement operators over all possible shared quantum states.

It is straightforward to show that the set of all quantum behaviors 𝐐 is convex <cit.>. Since a semiclassical behavior is a quantum behavior, by the definition in <ref> we can conclude a lossy behavior also belongs to 𝐐. Thus, all possible {η_i}_i-lossy behaviors constitute a subset of 𝐐. We can interpret {η_i}_i as a “shrinking factor” of 𝐐 to 𝐃⊆𝐐: when η_i =1 for all i, we can attain all of 𝐐, whereas when η_i = 0 for all i, we can only attain 𝐃.





















We will establish the following basic result.


    Consider a (2,2,2) problem. Then, the behavior of a degenerate quantum strategy is a classical behavior. Moreover, the same is true for lossy behavior where the quantum strategy is degenerate.


    
    

    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    The set of all possible classical behaviors is given by a polytope, called the local polytope <cit.>. By <cit.>, the inequalities that define the local polytope for the case of two parties, two observations, and two decisions are positivity conditions and permutations of the four terms in the CHSH inequality. 
    
    We will explicitly show that the behavior of a degenerate quantum strategy satisfies all possible CHSH inequalities. WOLOG, suppose the first party always outputs 0 when they make the first observation. When they make the second observation, they perform some quantum measurement {Π_A, I_A - Π_A}. Let the second party's measurement operators be given by {Π_B, I_B - Π_B} and {Π_B', I_B - Π_B'}, in order of their observations. Then, we can compute
    
    p(XOR=0| (0,0))     = tr[(I_A ⊗Π_B)ρ_AB]
    
            p(XOR=0| (0,1))     = tr[(I_A ⊗Π_B')ρ_AB]
    
            p(XOR=0| (1,0))     = tr[(Π_A ⊗Π_B)ρ_AB]+ tr[((I_A-Π_A) ⊗ (I_B -Π_B))ρ_AB] 
        =  2tr[(Π_A ⊗Π_B)ρ_AB]+1-tr[(I_A ⊗Π_B)ρ_AB]-tr[(Π_A ⊗ I_B)ρ_AB]
    
            p(XOR=0| (1,1))     =  2tr[(Π_A ⊗Π_B')ρ_AB]+1-tr[(I_A ⊗Π_B')ρ_AB] -tr[(Π_A ⊗ I_B')ρ_AB].


    Now, the CHSH inequalities are usually expressed in terms of the quantum correlation matrix elements c_kl. We first consider the usual permutation 
    
    c_00+c_01+c_10-c_11
       = 2tr[(I_A ⊗Π_B)ρ_AB] -1+2tr[(I_A ⊗Π_B')ρ_AB]-1+4tr[(Π_A ⊗Π_B)ρ_AB]+2-2tr[(I_A ⊗Π_B)ρ_AB]
       -2tr[(Π_A ⊗ I_B)ρ_AB]-1-4tr[(Π_A ⊗Π_B')ρ_AB]-2+2tr[(I_A ⊗Π_B')ρ_AB]+2tr[(Π_A ⊗ I_B)ρ_AB]+1
        = -2+4tr[(I_A ⊗Π_B')ρ_AB]+4tr[(Π_A ⊗Π_B)ρ_AB]-4tr[(Π_A ⊗Π_B')ρ_AB]
       =4tr[(Π_A ⊗Π_B+ (I_A-Π_A) ⊗Π_B')ρ_AB]-2.

    Now, it is easy to see that the operator Π_A ⊗Π_B+ (I_A-Π_A) ⊗Π_B' is an orthogonal projector. Thus, its eigenvalues are either 0 or 1. Hence,
    
    | c_00+c_01+c_10-c_11|≤ 2.

    By symmetry, we only need to consider one other permutation:
    
    c_00+c_10+c_11-c_01
       = 2tr[(I_A ⊗Π_B)ρ_AB] -1+4tr[(Π_A ⊗Π_B)ρ_AB]+2-2tr[(I_A ⊗Π_B)ρ_AB]-2tr[(Π_A ⊗ I_B)ρ_AB]-1
       +4tr[(Π_A ⊗Π_B')ρ_AB]+2-2tr[(I_A ⊗Π_B')ρ_AB]-2tr[(Π_A ⊗ I_B)ρ_AB]-1-2tr[(I_A ⊗Π_B')ρ_AB]+1
        = 2+4tr[(Π_A ⊗Π_B)ρ_AB]-4tr[(Π_A ⊗ I_B)ρ_AB] +4tr[(Π_A ⊗Π_B')ρ_AB]-4tr[(I_A ⊗Π_B')ρ_AB]
        = -4tr[(Π_A ⊗ (I_B-Π_B)+(I_A-Π_A)⊗Π_B')ρ_AB]+2.

    Again, Π_A ⊗ (I_B-Π_B)+(I_A-Π_A)⊗Π_B' is an orthogonal projector. Thus,
    
    | c_00+c_10+c_11-c_01|≤ 2.

    We can conclude that the quantum behavior satisfies the inequalities of the local polytope and is therefore a classical behavior.

    Now, given a tuple (𝒮_q, 𝒮_d) where 𝒮_q is degenerate, the corresponding lossy behavior is a convex combination of quantum behaviors (possibly semiclassical) where each behavior results from a degenerate strategy. Hence, by the previous result, the lossy behavior is a convex combination of classical behaviors and is therefore itself classical.

We next prove that 2-dimensional (qubit) quantum systems are sufficient to achieve lossy values for (n,2,2) problems.


    Consider an (n,2,2) problem. Then, the {η_i}-lossy value can be attained where the optimal quantum strategy 𝒮_q uses only qubit quantum systems.


    Fix a deterministic strategy 𝒮_d. Let 𝒮_q be a quantum strategy. 
    The results in <cit.> show that the extremal points of the set of all quantum behaviors for (n,2,2) problems are realized by quantum strategies using only qubit quantum systems. Thus, in this setting every quantum behavior is a convex combination of qubit quantum behaviors. 
    We therefore have
    
    p_(𝒮_q , 𝒮_d); {η_i}_i(d| o)    = ∑_S ⊆ [n]∏_i ∈ S (1-η_i) ∏_j ∉Sη_j · p_𝒮_d(d_S | o_S) p_𝒮_q(d_S̅| o_S̅) 
        = ∑_S ⊆ [n]∏_i ∈ S (1-η_i) ∏_j ∉Sη_j · p_𝒮_d (d_S | o_S) (∑_k λ_k p_𝒮_q_k (d_S̅| o_S̅)) 
        = ∑_k λ_k ∑_S ⊆ [n]∏_i ∈ S (1-η_i) ∏_j ∉Sη_j · p_𝒮_d (d_S | o_S)  p_𝒮_q_k (d_S̅| o_S̅),

    where S̅ is the complementary set of S, p(d_S | o_S) is the marginal probability[Note that by the no-signaling property the marginal probability does not depend on the observations of S̅. ] on S ⊆ [n], λ_k is a probability vector, and 𝒮_q_k are qubit strategies. Thus, for fixed 𝒮_d, the maximum expected utility is achieved by a qubit behavior. We can then maximize over 𝒮_d to obtain the desired conclusion.
    
        
    
    
    
    
    
    
    
    
    
    




 §.§ Noisy Quantum Behavior


We give a theoretical exposition on the effect of depolarizing noise in quantum strategies. We make the following definition.

    Let 𝒮_q = ({q_i}_i=1^n, {M_i}_i=1^n, |ψ⟩) be a quantum strategy and ν∈ [0,1]. Then, the ν-noisy behavior of 𝒮_q is the behavior that corresponds to the strategy 𝒮_q except the quantum state |ψ⟩ is replaced by (1-ν)|ψ⟩⟨ψ| + νπ, where π is the maximally mixed state. That is, the ν-noisy behavior is given by
    
    p_o^d(ν) := [⊗_i=1^n Π_i^(d_i | o_i) ( (1-ν)|ψ⟩⟨ψ| + νπ)].

    The robustness ν^* of a quantum strategy whose expected utility is at least the classical value is the smallest ν such that the expected utility of the ν-noisy behavior is equal to the classical value.

We can expand out <ref>:

    p_o^d(ν) = [ ⊗_i=1^n Π_i^(d_i| o_i)  ( (1-ν)|ψ⟩⟨ψ| + νπ)] = (1-ν) p_o^d(0) + νrank[⊗_i=1^n Π_i^(d_i | o_i)]/∏_i=1^n q_i.

We see that the effect of the noise is completely determined by the ranks of the measurement operators. Now, the fraction in the second term can be interpreted as a behavior. Indeed, we observe that it factorizes into behaviors for individual parties:

    rank[⊗_i=1^n Π_i^(d_i | o_i)]/∏_i=1^n q_i = ∏_i=1^n rank[Π_i^(d_i | o_i)]/q_i.

Hence, it is a classical behavior. We can therefore interpret ν as a shrinking factor of 𝐐 to the set of all classical behaviors (with rational probabilities) that are factorizable, that is, where the behavior of each party is independent. Note that this set includes 𝐃, so we can always attain the classical value even for ν=1. Furthermore, this implies for a quantum strategy with an expected utility higher than the classical value, the expected utility for the ν-noisy behavior is monotonically decreasing with ν.

For the case of the hedge-or-not problem when there is a quantum advantage, if a quantum strategy only uses qubits and achieves the quantum value, the ranks of the measurement operators must be rank 1 according to <ref>. Hence, in this case 

    p_o^d(ν) = (1-ν) p_o^d(0) + ν1/4.


We claim that in the presence of depolarizing noise, increasing the dimensions of the quantum systems used can increase the expected utility, even for the CHSH inequality. Recall the utility array is given by

    𝒰 =
            0,0     0,1     1,0     1,1 
        0,0     1     0     0     1 
        0,1     1     0     0     1 
        1,0     1     0     0     1 
        1,1     0     1     1     0
        ,

and the inputs are uniformly distributed. Then, by <ref> and <ref>, there exists an optimal qubit strategy for the noiseless setting, call it 𝒮_q, which is non-degenerate. Thus, the factorizable behavior in <ref> must be the uniform distribution. The expected utility for the ν-noisy behavior of 𝒮_q is 

    (1-ν) cos^2π/8 + ν1/2.

Now, consider modified quantum strategy 𝒯_q where both parties use ququarts (dimension 4). Instead of the original entangled state |ψ⟩, we use |ψ⟩⊗| 00 ⟩. And for the all the projectors Π_i^(0 | o_i) for the 0 decision, we replace it with Π_i^(0 | o_i)⊗| 0 ⟩⟨ 0 | (the corresponding 1 decision projector is the orthogonal complement). This clearly preserves the expected utility of the noiseless quantum behavior. However, the ranks of the projectors are not doubled while the quantum dimensions are. It is not difficult to see that the factorizable behavior in this case is, in matrix form,

    0,0     0,1     1,0     1,1 
        0,0    1/16   3/16   3/16   9/16
        0,1    1/16   3/16   3/16   9/16
        1,0    1/16   3/16   3/16   9/16
        1,1    1/16   3/16   3/16   9/16,

which implies for 𝒯_q, the ν-noisy behavior achieves an expected utility of

    (1-ν) cos^2 π/8 + ν9/16 > (1-ν) cos^2π/8 + ν1/2

for ν > 0. This establishes the claim.



§ NUMERICAL OPTIMIZER









We give some details of the numerical optimizer used to compute classical and quantum values, as well as how some of the figures were computed. The optimizer itself is quite straightforward but can be used for arbitrary TC problems and not just special classes such as XOR problems.



 §.§ Classical value

The classical value is straightforward to compute. The optimizer proceeds via brute force by iterating over all possible deterministic strategies and tracking the largest expected utility found. Each deterministic strategy is simply a choice of decision given an observation for each party, which for a ({𝒪_i}_i=1^n, {𝒟_i}_i=1^n, p_𝒪(o), u) TC problem, would simply be parameterized by  

    ∑_i=1^n |𝒪_i |

discrete variables, the variables for party i taking |𝒟_i| different values. Aside from this brute force approach, there are also linear programming methods as detailed in <cit.>.



 §.§ Quantum value

Our numerical optimizer computes the quantum value via an optimization over explicit parameterizations of projective measurements on a q-dimensional Hilbert space ℋ_q with Δ≤ q possible outcomes. This parameterization has both continuous and discrete variables. 

The continuous variables parameterize an orthogonal basis of ℋ_q modulo nonzero scalar multiplication. We can do this via <cit.>, which gives a parameterization of a unitary matrix via q^2 real parameters λ_m,n, m, n ∈ [q]:

    U = [ ∏_m=1^q-1( ∏_n=m+1^q exp(i P_n λ_n,m) exp(i σ_m,nλ_m,n)]  ·( ∏_l=1^q exp(i P_l λ_l,l)),

where the matrix multiplication proceeds from left to right,

    P_l := | l ⟩⟨ l |,

and

    σ_m,n := -i | m ⟩⟨ n | + i | n ⟩⟨ m |.

The range for λ_m,n is [0,2π] for m ≥ n and [0, π/2] for m < n. Since an orthogonal basis modulo nonzero scalar multiplication is basically a unitary matrix modulo multiplication by a diagonal unitary, we see from <ref> that we simply can remove the rightmost term to obtain our desired parameterization. Thus, in the end we only need the q^2-q parameters λ_m,n where m ≠ n, and use the parameterization

    U = [ ∏_m=1^q-1( ∏_n=m+1^q exp(i P_n λ_n,m) exp(i σ_m,nλ_m,n)],

where the columns of U are the basis vectors that we use.































We have figured out how to parameterize a projective measurement for each party. We next must combine the parameters for each party together to parameterize a Bell operator in <ref>. Note that we do not need to parameterize the shared quantum state since we can simply compute the largest eigenvalue of the Bell operator. In fact, WOLOG we can assume each party performs the measurement in the computational basis upon the first observation, with respect to some ordering of the observations. This is because we can always conjugate the Bell operator by a tensor product unitary that rotates for each party the measurement operator used upon making the first observation to the computational basis. In sum, for a TC problem ({𝒪_i}_i=1^n, {𝒟_i}_i=1^n, p_𝒪(o), u), a quantum strategy where party i uses a quantum system of dimension q_i has a total of

    ∑_i=1^n (q_i^2-q_i)(|𝒪_i | -1)

continuous variables and 

    ∑_i=1^n |𝒪_i |

discrete variables. 

Interestingly, this is not always the minimum number of variables we need: for some cases we can further eliminate variables. In the case of (n,2,2) problems, we can obtain the following result.


    Consider an (n,2,2) problem. Then, it is sufficient to use n continuous variables to parameterize a quantum strategy that achieves the quantum value. Moreover, it is also sufficient to parameterize a quantum strategy that with the appropriate deterministic strategy achieves the lossy value.


    By <ref>, to achieve the quantum value we can assume we are only using qubit systems. By <ref>, we therefore seem to need ∑_i=1^n 2 · (2-1) = 2n continuous variables to parameterize a Bell operator up to local unitary equivalence. We can halve this. Let λ^(j)_m,n denote the continuous parameters for party j's projective measurement upon their second observation (recall measurement upon first observation is in the computational basis). Now, for q=2, <ref> gives
    
    U = 
            [               cosλ_1,2^(j)               sinλ_1,2^(j); -e^i λ_2,1^(j)sinλ_1,2^(j)      e^iλ_2,1^(j)λ_1,2^(j) ].

    Let | v⟩ denote the first column vector. Hence, considering all possible observations and partitions, the choices of a projector for party j is the following:
    
    Π_j ∈{| v⟩⟨ v |, I - | v ⟩⟨ v |, 0, I, | 0 ⟩⟨ 0 |, | 1⟩⟨ 1|}.

    Then, we observe for all possibilities, the operator
    
    Z_-λ_2,1^(j)Π_j Z_λ_2,1^(j)

    is independent of λ_2,1^(j). Here we used the Z rotation matrix
    
    Z_λ :=
            [    1    0;    0 e^iλ ].

    Thus, we see that
    
    (⊗_j=1^n Z_-λ_2,1^(j)) ∑_o ∈𝒪 p_𝒪(o) ∑_d ∈𝒟⊗_j=1^n Π_j^(d_j| o_j) u_o^d  (⊗_j=1^n Z_λ_2,1^(j)) 
        = ∑_o ∈𝒪 p_𝒪(o) ∑_d ∈𝒟(⊗_j=1^n Z_-λ_2,1^(j)Π_j^(d_j| o_j) Z_-λ_2,1^(j)) u_o^d

    is independent of λ_2,1^(j) for all j. However, this operator is local unitarily equivalent to the Bell operator. This implies that the largest eigenvalue is independent of these n parameters and so the first conclusion follows.

    In the lossy case, for any quantum strategy 𝒮_q and deterministic strategy 𝒮_d, every term in the convex combination of <ref> also satisfies the above independence property since <ref> still holds. The second conclusion therefore follows. 


As an example, a quantum strategy for a (2,2,2) problem using only qubits would have 2 continuous variables and 2+2 = 4 discrete variables. Note that by <ref> in this case a quantum strategy whose behavior is not classical uses only nontrivial measurement operators. Thus, the 4 discrete variables can be eliminated as we only need to consider the 1+1 =2 partition. 
Thus, to compute <ref>, <ref> and <ref>, we need to only optimize over 2 continuous variables. 

Optimization is conducted via two methods. The first is brute force. For the continuous variables, we choose a grid of values to evaluate over and also perform a gradient descent for each point in the grid. This is implemented via  <cit.>. The discrete variables are evaluated at every possible value. This method is clearly not scalable, so we also include another method, the CMA-ES evolutionary algorithm <cit.> with discrete variables <cit.>. We use the Python implementation of <cit.> in our optimizer. Note that CMA-ES does not guarantee a global optimum. However, in practical scenarios it may be sufficient to find a quantum strategy with a higher expected utility than the classical value. To double-check the results in <ref>, <ref>, and <ref>, we performed both brute force (with a grid linear size of 20 points) and CMA-ES optimizations and verified that we obtained the same results. Note that since the hedge-or-not problem is an XOR problem, we can also use semidefinite programming to solve for the quantum value <cit.>. We instead use our optimizer since it can handle general TC problems and use the hedge-or-not problem as an example of how to apply it.



 §.§ Lossy value

To compute the lossy value of a TC problem, we optimize over tuples (𝒮_q, 𝒮_d). This simply combines the parameterization of a deterministic strategy and that of a quantum strategy. The number of continuous variables is the same, but now the number of discrete variables is 

    2∑_i=1^n |𝒪_i |.

Note that we are now instead computing the largest eigenvalue of the lossy Bell operator in <ref>, so we again do not need to parameterize the shared quantum state. We can perform optimization via brute force where for the deterministic strategy variables we also iterate over all possible values. The CMA-ES optimization can be done by treating the deterministic strategy variables as just additional discrete variables.

By <ref> and <ref>, we can compute the lossy value of a (2,2,2) problem by parameterizing a quantum strategy that only uses qubits and nontrivial measurement operators. By <ref>, we again only need 2 continuous variables. However, we now have 4 discrete variables parameterizing the deterministic strategy which must be included in the optimization. We perform brute force optimization over both continuous (grid linear size of 20) and discrete variables for a given η to compute the lossy value, then perform a binary search over η∈ [2/3, 1] to find η^*, thereby obtaining the results in <ref>. 



 Paper Title: Information flow-enhanced precision in collisional quantum thermometry.







tmendonca@ifsc.usp.br
Instituto de Física de São Carlos,
Universidade de São Paulo, CP 369, 13560-970, São Carlos, SP, Brazil
Centre for Quantum Materials and Technologies, School of Mathematics and Physics, Queen’s University, Belfast BT7 1NN, United Kingdom


dosp@ifsc.usp.br
Instituto de Física de São Carlos,
Universidade de São Paulo, CP 369, 13560-970, São Carlos, SP, Brazil


mauro.paternostro@unipa.it
Università degli Studi di Palermo, Dipartimento di Fisica e Chimica - Emilio Segrè, via Archirafi 36, I-90123 Palermo, Italy
Centre for Quantum Materials and Technologies, School of Mathematics and Physics, Queen’s University, Belfast BT7 1NN, United Kingdom



§ ABSTRACT 

We describe and analyze a quantum thermometer based on a multi-layered collisional model. The proposed architecture provides significant sensitivity even for short interaction times between the ancillae comprised in the thermometer and the system to be probed, and a small number of information-acquiring collisions. The assessment of the flow of information taking place within the layered thermometer and between system and thermometer reveals that the tuning of the mutual backflow of information has a positive influence on the precision of thermometry, and helps unveiling the information-theoretic mechanisms behind the working principles of the proposed architecture. 


 
Pacs

Information flow-enhanced precision in collisional quantum thermometry   
    Mauro Paternostro
    August 5, 2024
=========================================================================







§ INTRODUCTION



Quantum metrology uses resources such as entanglement <cit.>, quantum coherence <cit.> and squeezing <cit.> to enhance the precision with which measurement protocols are able to estimate relevant physical quantities beyond the capabilities of classical strategies. When applied to the problem of estimating the temperature of a system, such quantum tools make quantum thermometry <cit.> effective even when operating under conditions that are classically challenging, such as when the thermometer has not yet thermalized with the system. 
This makes such framework potentially useful to probe non-equilibrium environments, which could showcase coherence and correlations. 

Energetic resource value of information is an active research area in quantum thermodynamics, they can power up, cool, or heat systems using information instead of consuming energy. Quantum thermometry will be an integral part of practical implementations of such quantum information engines and devices.
A particularly powerful way to model the process of estimating temperature leverages upon the framework of collisional models <cit.>, which has been successfully applied to describe the thermodynamics of non-equilibrium quantum processes <cit.>.

In collisional quantum thermometry, a set of probing particles acquire information on the temperature of the system of interest through a sequence of short and intense, crash-like interactions. Ref. <cit.> explored such approach showing how it allows for the estimation of the temperature of a quantum system beyond the precision constraints set by thermal fluctuations <cit.>. Additional studies have grounded the suitability of collisional quantum thermometry to pursue this important metrological task <cit.>. In this paper, we use a collision-based approach to build a thermometric sensor that measures the temperature of an environment indirectly, in a completely out-of-equilibrium regime, by resorting to measurements performed on auxiliary qubits arranged in the form of layers of chains. We go beyond state-of-the-art in collisional quantum thermometry by alllowing explicitly for inter-layer interactions: although this obviously increases the complexity of the problem at hand, it also enhances the process of information-spreading that is responsible for the performance of this metrological task. Quantitatively, we achieve a thermometric sensitivity  that is orders of magnitude larger than that of a single-layered thermometer as proposed by <cit.>. We also investigated the case where there is a complete exchange of information during the system-ancilla interaction, a model proposed by <cit.>. In this case, we showed that the same amount of information flows between the qubits, not allowing for a gain in sensitivity.

To investigate the motivating factors for such gain in sensitivity of our multi-layered thermometer, we studied the behavior of the information flow using the measure of non-Markovianity  proposed in Ref. <cit.> (dubbed here as BLP), this procedure was demonstrated in Ref.<cit.>. Finally, we compare the QFI and the BLP measurement, qubit by qubit, to investigate the path taken by information in our complete system. It was possible to determine how the accumulation of information provides an increase in the sensitivity of our thermometer, we also show that such accumulation can also be characterized through Mutual Information.





§ DESCRIPTION OF THE MODEL



We consider a collisional model where a qubit – embodying the system – is initially thermalized with an environment at temperature T. A set of ancillary chains is used to indirectly estimate the temperature of the environment, as showed in the Fig. <ref>(a). The dynamics of the overall compound comprises a sequence of collisions between the system and N chains of n ancillae each. In a schematic manner, the process unfolds according to the following steps:

(1) The system S thermalizes with the environment through the thermal map 𝒯_S=e^ℒτ_SE  generated by <cit.>

    dρ_S/dt = ℒ(ρ_S)= γ(n̅+1)𝒟[𝒮_-] + γ(n̅)𝒟[𝒮_+],

where 𝒟[L]=Lρ_SL^†-1/2{L^†L,ρ_S}, n̅=(e^βΩ_S-1)^-1 with β=ħΩ_S/k_B and k_B the Boltzmann constant. Here,  Ω_S is the characteristic frequency of the system, T is the temperature of the environment and τ_SE is a time long enough for the system qubit to be thermalized with the temperature of the environment. At the end of this step, the qubit system will be in the Gibbs  state ρ_S^th=e^-H_S/k_BT/Z, where H_S= ħΩ_S𝒮_z/2 is the Hamiltonian of the system.

The following steps involving A_k,j, i.e. the j^th element of the k^th chain   (j=1,..,n and k=1,..,N), then take place: For a set value of j=1,..,n, a collision between the system S and A_k,j, ruled by the Hamiltonian 

    H_SA_k,j = ħ g (𝒮_+α^k,j_-+ 𝒮_-α^k,j_+)


occurs, followed by the collision between A_k,j and A_k+1,j according to the generator 

    H_A_k,jA_k+1,j = ħ J (α^k,j_+α^k+1,j_-+ α^k,j_-α^k+1,j_+).

Here, g (J) is the system-ancilla (inter-ancilla) coupling strength, which we assume to be independent of j and k, while the operators 𝒮_±=(|0⟩±|1⟩)_S/√(2) and α_±^k,j=(|0⟩±|1⟩)_A_k,j/√(2) refer to system and  the relevant ancilla, respectively, with 

 {|0⟩,|1⟩} being the computational basis of our problem (we have shifted the energy of the states in such a way that 𝒪_z|0⟩=|0⟩ and 𝒪_z|1⟩=-|1⟩).

The collisions continue so as to span all the chains (i.e. for k=1,...,N), after which a thermalization process occurs again. The sequence of system-ancilla and ancilla-ancilla collisions starts again, this time involving the elements j+1 of each chain. then consider the 
After that, the process restarts for the j+1 ancillae spanning all the N chains.
Fig. <ref> provides a visual depiction of the first few steps of the process, as well as its circuit representation representing the interactions for the case of three chains only.

The sequence of operations that describes the described interaction protocol for a single repetition and considering a generic ancilla leads to the state

    ρ_S,A_1,j⋯ A_N,j=𝒰_S,A_k,j∘[◯^N-1_k=1𝒰_A_k+1,jA_k,j]∘𝒰_S,A_k,j
    ∘𝒯_S(ρ_0),

where 𝒰_S,A_k,j∘μ = V_S,A_k,j μ  V_S,A_k,j^† and 𝒰_A_k,j,A_k+1,j∘μ = V_A_k,j,A_k+1,j μ  V_A_k,j,A_k+1,j^† for a density matrix μ and  V_S,A_k,j=exp(-i H_SA_k,jτ_SA/ħ), V_A_k,j,A_k+1,j=exp(-i H_A_k,jA_k+1,jτ_A/ħ). In our analysis, we will consider values of the dimensionless system-ancilla  interaction time gτ_SA≪π/2, while taking the inter-ancilla one as Jτ_A=π/2. Under such conditions, a full exchange of information between the ancillae will be allowed, as V_A_k,jA_k+1,j is the phased SWAP gate performing the transformation |a,a⟩_A_k,jA_k+1,j→(-i)^a⊕ b|b,a⟩_A_k,jA_k+1,j (a,b=0,1 and ⊕ standing for sum modulo 2). On the other hand, the information exchange between the system and ancilla will only be partial.







§ ANALYSIS OF THE QUANTUM FISHER INFORMATION



We make use of the formal apparatus of quantum metrology to provide a bound to the precision of the estimation of the temperature of the environment, which has been encoded in the state of system. In particular, using the framework of quantum parameter estimation <cit.>, we can upper bound to said precision through the quantum Fisher Information (QFI)

    QFI_k,j(T,ρ_A) = Π_θ,ϕmax F_k,j(Π_θ,ϕ,ρ_A_k,j,T),

where Π_θ,ϕ=|ψ(θ,ϕ)⟩⟨ψ(θ,ϕ)|_A_k,j with |ψ(θ,ϕ)⟩_A_k,j=cosθ|0⟩_A_k,j+e^iϕsinθ|1⟩_A_k,j (θ∈[0,π], ϕ∈[0,2π]) are the elements of a positive operator valued measurement (POVM) on  ancilla A_k,j, and F_k,j(Π_θ,ϕ,ρ_A_k,j,T) is the classical Fisher Information of the POVM 

    F_k,j(Π_θ,ϕ,T,ρ_A)= ∬ dθ dϕ  p(θ,ϕ) (∂∂ Tlnp(θ,ϕ))^2.

Here, p(θ,ϕ)=Tr(Π_θ,ϕρ_A_k,j) is the probability to get outcome (θ,ϕ) from the measurement. The  QFI implies the maximization of its classical counterpart over all possible POVMs, and is thus optimized over the measurement strategy on the ancillae. 
The precision of a probe in thermal equilibrium with a system is limited by the Cramer-Rao bound

    (Δ T)^2 ≥1/QFI(T,ρ),

where (Δ T)^2 is the temperature variance. When evaluated over a thermal state, such lower bound to the variance of a temperature estimator (sometimes referred to as thermal Fisher information <cit.>) can be surpassed through the use of quantum resources such as entanglement and coherence.

The results of the QFI measurements for k=1,..,8 and j=1,..,30 are shown in Fig. <ref>(a), which shows the value of the QFI for measurements performed on the ancillae of each chain as we vary the temperature. Specifically, we have assumed to measure the last ancilla of each chain. As the number of layered chains increases, the QFI grows thus lowering the upper bound entailed by the Cramér-Rao bound and thus witnessing and increase in sensitivity. Fig. <ref>(b) shows the behavior of the QFI as we vary the number of ancillae in each chain, i.e. varying j, showing that, in a sufficiently layered structure, even a few ancillae will be sufficient for our collisional thermometer to achieve high sensitivity. Fig. <ref>(c) shows the maximum value of the QFI growing with the number of layers.








§ ASSESSING INFORMATION FLOW AND ITS ROLE IN THE THERMOMETRIC TASK



To understand the reason for the increased thermometric sensitivity, we will analyze the flow of information during the various operations carried out between system and ancillae. 

We consider the system as prepared as proposed in <cit.>. 
The effect of the system-ancilla interaction can be quantified, at every instant of time, by considering the distinguishability between the the time-evolved form of such initial preparations as given by the quantity

    D(ρ_S_+A_k,j,ρ_S_-A_k,j;t)= ρ_S_+A_k,j(t)-ρ_S_-A_k,j(t)_1

with ·_1 the trace distance. The lack of monotonicity of D(ρ_S_+A_k,j,ρ_S_-A_k,j;t) signals a back-flow of information into the system stemming from the joint system-ancillae dynamics. Following Refs. <cit.>, this witnesses the non-Markovian nature of the open dynamics undergone by S. 

Fig. <ref> reports the behavior of Eq. (<ref>) and its derivative σ(t)=∂_t D(ρ_S_+(A_k,j),ρ_S_-(A_k,j);t) when S interacts with only the first ancilla of each chain.  
Fig. <ref>(a) and (c) refer to the Markovianity in the system. Needless to say, a complementary picture can be taken where we characterize non-Markovianity by looking at the distinguishability between evolved states of the ancillae. The results stemming from such a perspective are shown in Fig. <ref>(b) and (d).
We observe a flow of information between system and ancillae, growing with the number of chains being considered.
Moreover, there is clearly information is flowing off the system [as σ<0 in Fig. <ref>(c)], and into the ancillae [whose corresponding σ is positive, as seen from Fig. <ref>(d)].



In order to gather insight in the dynamics of information flow during the thermometric task, in the following numerical experiments we will consider gτ_SA=Jτ_A=π/2, i.e., a full exchange of information between system-ancilla and ancilla-ancilla.
Measurements were performed on all parties involved int he processes ruled by V_S,A_k,j and V_A_k,j,A_k+1,j. However, for a clearer visualization we plot only the results for j=1 and k=1,2, i.e. 
V_S,A_1,1 (measuring S and A_1,1), V_A_1,1,A_2,1 (measuring A_1,1 and A_2,1) and V_S,A_2,1 (measuring S and A_2,1). These are the operations
illustrated in steps 2, 3 and 4 of Fig. <ref>(a). These analyses and results does not depend on the choice of j made here.

We show the behavior of the trace distance, its derivative σ and the QFI in Fig. <ref> and Fig. <ref>, respectively. We use the protocol put forward in  Ref. <cit.>, where such tools have been used to analyze the flow of information passing through a qubit environment. We take the initial states of the system to be ρ_S^(±) (0)=|±⟩⟨±|_S while the initial state of the ancillae is ρ_A_k,j(0) = ⊗^N_k=1|0⟩⟨0|_k. The initial state of the system-ancillae compound is the uncorrelated one ρ_SA^(±)(0)=ρ_S^(±)(0)⊗ρ_A_k,j(0).

Fig. <ref>(a) and Fig. <ref>(b) show the dynamics of the trace distance and its derivative, respectively, for the states described above. 
Fig. <ref>(b) shows clearly how each collision (either system-ancilla or ancilla-ancilla) results in a pouring of information that flows according to the preferential direction 
    S→chain k→chain k+1→ S
 for k=1,...,N since the positions of the ancillae such that k is odd and k+1 is even. Here we can also see that information leaves a given qubit [as witnessed by having σ<0, cf. solid line in Fig. <ref>(b)], and reaches the other qubit involved in the operation [resulting in σ>0, cf. dot-dashed, dotted, and dashed lines].





We now evaluate the QFI using the same protocol described above.
Figs. <ref>(a) and (b) show the QFI resulting from measurements performed on the qubits from which the information is coming out and coming in, respectively. 
Note that  the QFI is only defined when the information is coming in to the qubit.
Fig. <ref>(b) shows that the same QFI can be extracted from measurements on S, A_k,1 or A_k+1,1 (since k is odd and k + 1 is even), a result that is clearly stemming from the highly symmetric interaction configuration that we have chosen for this illustration. We see that the QFI measurement does not vary when gτ_SA=Jτ_A=π/2 there is no gain in sensor sensitivity, this happens because the same amount of information is being transferred in the full Swap operation [cf. Fig. <ref>(b)]. If the measured qubit is sending the information during the full exchange interaction, the QFI measurement will not have a defined result (Fig. <ref>(a)).

We now extend the analysis and return to the case of gτ_SA≪π/2, i.e. an incomplete exchange of information between system and ancillae. In Fig. <ref> we consider the results corresponding to j=1 and a compound comprising 5 chains. The QFI for the system and the ancillae are shown in Figs. <ref>(a) and (b), respectively. Fig. <ref>(c) shows the QFI for the ancilla that receives information during the ancilla-ancilla interaction. Since the interactions characterizes an full Swap operation, i.e. Jτ_A=π/2, the QFI measurements performed on the ancillae from which the information is leaving the qubit follow the result shown in Fig. <ref>(a), thus we will not repeat such results here.

As in Fig. <ref>(b), Fig. <ref>(a) shows that the same amount of information can be measured in the different parts of the compound. However,  Figs. <ref>(b) and (c) show that there is an increase in the sensitivity of our thermometer as we increase the number of ancillary chains. Therefore, this represents there is an accumulation of information caused by the short interaction time between system and ancillae τ_SA, as is evident in the characterization of the distinguishability in Fig. <ref>(d). Since we can only obtain the QFI when the information flows into an element, we can state that there is also information returning to the system.



The accumulation of information can also be seen when we calculate the mutual information between S and an ancillary chain. We compute the mutual information between the elements with j=n of the k^th chain in all interactions as

    I(ρ_S:ρ_A_k,n)= ℰ(ρ_S)+ℰ(ρ_A_k,n)-ℰ(ρ_SA_k,n).

where ℰ(ρ)= -tr(ρlogρ) is the von Neumann entropy <cit.>. Fig. <ref> shows that as we increase the number of ancilla chains, the greater the amount of information shared between the system and the ancillae.





Conclusion.–

We have built a model to improve and increase the sensitivity of a quantum thermometer based on the use of a multilayered system. Even with a short interaction time, i.e. without requesting thermalization of the system with the thermometer, we can adjust the sensitivity of the latter by increasing the number of its ancillary chains. We have unveiled a tight link between the sensitivity of the estimation of temperature as provided by the QFI and the information flow among various parts of the compound. The accumulation of information entailed by the information flow leads to better sensitivity (i.e. a larger QFI). This  can also be seen in the increase of mutual information with the number of chains comprised in the thermometer.





§ ACKNOWLEDGMENTS

The authors thank discussions with G. T. Landi.
TMM thanks the grants No. 2021/01277-2 and No. 2022/09219-4, São Paulo Research Foundation (FAPESP).
DOSP acknowledges the support by the Brazilian funding agencies CNPq (Grant No. 304891/2022-3), FAPESP (Grant No. 2017/03727-0), and the Brazilian National Institute of Science and Technology of Quantum Information (INCT/IQ). MP acknowledges the support by the European Union's Horizon Europe EIC Pathfinder project QuCoM (Grant Agreement No. 101046973), the Royal Society Wolfson Fellowship (RSWF/R3/183013), the UK EPSRC (EP/T028424/1), and the Department for the Economy Northern Ireland under the US-Ireland R&D Partnership Programme. 







 Paper Title: Engineering Rydberg-pair interactions in divalent atoms with hyperfine-split ionization thresholds.






fhummel@atom-computing.com
Atom Computing, Inc., Berkeley, California


weber@itp3.uni-stuttgart.de
Institute for Theoretical Physics III and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Germany


Institute for Theoretical Physics III and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Germany


Department of Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany


Atom Computing, Inc., Berkeley, California


Atom Computing, Inc., Berkeley, California
 

hofferberth@iap.uni-bonn.de
Institute of Applied Physics, University of Bonn, Germany


mli@atom-computing.com
Atom Computing, Inc., Berkeley, California



§ ABSTRACT 
Quantum information processing with neutral atoms relies on Rydberg excitation for entanglement generation.
While the use of heavy divalent or open-shell elements, such as strontium or ytterbium, has benefits due to their optically active core and a variety of possible qubit encodings, their Rydberg structure is generally complex.
For some isotopes in particular, hyperfine interactions are relevant even for highly excited electronic states.
We employ multi-channel quantum defect theory to infer the Rydberg structure of isotopes with non-zero nuclear spin and perform non-perturbative Rydberg-pair interaction calculations.
We find that due to the high level density and sensitivities to external fields, experimental parameters must be precisely controlled.
Specifically in , we study an intrinsic Förster resonance, unique to divalent atoms with hyperfine-split thresholds, which simultaneously provides line stability with respect to external field fluctuations and enhanced long-range interactions.
Additionally, we provide parameters for pair states that can be effectively described by single-channel Rydberg series.
The explored pair states provide exciting opportunities for applications in the blockade regime as well as for more exotic long-range interactions such as largely flat, distance-independent potentials.



Engineering Rydberg-pair interactions 
 in divalent atoms with hyperfine-split ionization thresholds
    Ming Li
    August 5, 2024
====================================================================================================







§ INTRODUCTION 


The long-range interactions between Rydberg atoms cause exotic processes in cold quantum gases and enable applications in non-linear quantum optics <cit.>.
Coherent Rydberg excitations are essential to quantum information processing using neutral atoms in order to introduce and mediate entanglement <cit.>.
Ensembles of individually trapped atoms in optical tweezer arrays have enabled incredible progress for quantum simulation and computation with defect-free assembly <cit.>, reconfigurable lattice geometries <cit.>, mid-circuit erasure conversion and readout <cit.>, continuous reloading <cit.>, and unseen connectivities by aid of rearrangement <cit.>.
Recently, a variety of atomic species has been explored for quantum gas experiments and quantum technology applications exploiting different aspects of the atomic structure <cit.>.
While alkali metals benefit from a long history of scientific research, the presence of a second valence electron in alkaline-earth and some lanthanide elements provides compelling control opportunities <cit.>, e.g. the core electron can be utilized to match polarizabilities of ground- and Rydberg states for magic wavelength trapping <cit.> and isolated core excitations can be exploited to introduce coherent light shifts and probe Rydberg states <cit.>.
Some isotopes allow for the storage of quantum information in their nuclear spin, achieving coherence times on the order of tens of seconds <cit.>.

Most schemes to entangle neutral atoms rely on strong dipolar interactions between Rydberg states, which are either coupled by near-resonant laser fields <cit.> or by dressing ground states with far off-resonant lasers <cit.>.
Yet, for the Rydberg structure, typically only one or at most a few energy levels are considered explicitly.
Non-perturbative Rydberg-pair interaction calculations are required to confirm the absence of other pair states and leakage channels in the Rydberg spectrum <cit.>.

In this article, we present calculations of non-perturbative Rydberg-pair interactions for divalent atoms with hyperfine-split ionization thresholds in order to study interaction tunabilty and the robustness of perturbative predictions.
The complex spectra of these atoms are described within multi-channel quantum defect theory <cit.>, which has been adapted to include nuclear spin using frame transformation <cit.>.
Recently,  was predicted to feature unusual properties not encountered in atoms with a simpler structure, namely, for some Rydberg states, simultaneous insensitivity to electric fields and resonant dipolar interaction in equal-state pairs <cit.>.
We find this to be a general feature caused by the hyperfine interaction between different Rydberg series when the energy splitting of adjacent Rydberg manifolds becomes comparable to the threshold splitting in the ionic core.
Specifically, the coupling of Rydberg series belonging to different ionic-core hyperfine configurations gives rise to varying electron spin character of some Rydberg states as a function of the principal quantum number <cit.> and to a corresponding energy dependence of their quantum defects.
The resulting properties are appealing for quantum technology applications and Rydberg gas experiments: 
High degrees of level stability help preparing Rydberg states that minimize amplitude and phase errors.
Intrinsic Förster resonances lead to strong dipole-dipole interactions, allowing for the engineering of exotic many-body Hamiltonians and for the implementation of two- and multi-qubit gates <cit.>.
These Förster resonances do not require careful tuning of external fields, which is often the case for atoms with a single valence electron <cit.>, but tuning of the principal quantum number is in general sufficient as we will demonstrate.
Moreover, the Förster resonances occur for two atoms being in the same state, allowing for global addressing of the atoms via a single laser beam.

This article is structured as follows:
In Sec. <ref>, we review the theory and methods, particularly the Hamiltonian for a pair of Rydberg atoms in Sec. <ref>.
After this general discussion of Rydberg interactions, we provide the methods specific to divalent Rydberg atoms in Sec. <ref>, where we outline multi-channel quantum defect theory and frame transformation.
Sec. <ref> details , with comments on exact diagonalization in Sec. <ref> and on the developed software in Sec. <ref>.
Sec. <ref> contains the results, discussing the single-atom spectral structure in Sec. <ref> and introducing field-free pair-interaction potentials for the Förster resonant Rydberg series in Sec. <ref>.
Sec. <ref> covers the response of single atoms and Rydberg pairs to magnetic fields, similarly for electric fields in Sec. <ref>.
A comparison to a different, single-channel Rydberg series is provided in Sec. <ref>.
Finally, Sec. <ref> contains the conclusion and an outlook.



§ THEORY AND METHODS 




 §.§ Rydberg-pair Hamiltonian 




We consider a pair of atoms as illustrated in Fig. <ref> and modelled by the Hamiltonian 

    H_2 = H_1 ⊗ 11 + 11 ⊗ H_1 + H_int(R⃗),

where H_1 is the single-atom Hamiltonian detailed further below and (using atomic units throughout)

    H_int(R⃗) = ∑_κ_1, κ_2=1^∞ V_κ_1 κ_2 |R⃗|^-1-κ_1-κ_2

is the multipole expansion of the electrostatic interaction between the two atoms suitable for internuclear separations exceeding the Le Roy radius and neglecting retardation effects.
Here, V_κ_1 κ_2 is proportional to the spherical multipole operators on the Rydberg electron in atom 1 and 2, respectively.
The exact form of V_κ_1 κ_2 depends on the choice of the coordinate system.
We choose the internuclear axis R⃗ to align with the z axis and restrict Eq. <ref> to dipole-dipole interactions:

    V_11 = -∑_q=-1^1 21+q r_1q^(1)r_1-q^(2),

where r_κ q^(i) is the spherical multipole operator of order κ for atom i.

We find a matrix representation of H_2 by expanding in a set of appropriately symmetrized wave functions of pairs of atoms.
In the absence of external fields, H_2 preserves reflection symmetry through a plane containing the internuclear axis.
Since we consider homonuclear pairs, H_2 also preserves inversion symmetry.
In the case of external field alignment with the internuclear axis, the total magnetic quantum number M=m_1+m_2 is a conserved quantity, where we introduce an index, indicating m_i corresponds to atom i.
Finally, since only dipole-dipole interaction are considered in the multipole expansion (Eq. <ref>), H_2 preserves permutation symmetry.
We use these symmetries to decompose H_2 into blocks that can be treated independently of each other <cit.>.

The Hamiltonian for a single Rydberg atom is given by

    H_1 = H_0 + H_s + H_z + H_d,

where H_0 is the field-free Hamiltonian with the good quantum numbers f and m corresponding to the total hyperfine spin, as well as parity.
The eigenvalues of H_0, 

    E_ν f = E_I-R_μ/ν^2,

are conveniently expressed in terms of an effective principal quantum number ν, defined with respect to the ionization threshold E_I.
Here, R_μ is the mass scaled Rydberg constant.
If an eigenstate is a superposition of multi-channel configurations as described in the following section, it is still useful to choose a specific E_I in order to label eigenstates by a global ν and, hence, indicate their ordering.
For hyperfine split-ionization thresholds, we choose E_I to be the energetically higher one, which some single-channel Rydberg series converge to.

The effect of static and homogeneous external fields is considered by the following three terms:
For the electric field F⃗, the Stark Hamiltonian is given by

    H_s = -r⃗·F⃗,

where r⃗ is the electronic coordinate (and, since the electron charge e=1 in atomic units, corresponds to the electric dipole moment).
For the magnetic field B⃗, the Zeeman Hamiltonian is given by

    H_z = -μ⃗·B⃗,

where μ⃗ is the magnetic dipole moment taking into account the electronic and nuclear spins

    μ⃗= -μ_B [ ℓ⃗_c + ℓ⃗_r + g_s ( s⃗_c + s⃗_r ) ] + μ_I I⃗,
   
where μ_B and μ_I are the Bohr magneton and the nuclear magnetic moment and g_s is the electron spin g-factor.
Here, the subscripts, c and r label core and Rydberg electron quantum numbers, while ℓ⃗ and s⃗ are orbital and electronic spin angular momentum, respectively, and I⃗ is the nuclear spin (cf. Fig. <ref>).
The diamagnetic Hamiltonian is given by

    H_d = 1/8( r⃗^2 B⃗^2 - | r⃗·B⃗|^2 ).

This term is typically considered for ground-state atoms only when the magnetic field is on the order of 1 atomic unit (∼ 10^4 T).
However, as it scales with the surface area of the atom, H_d∼ n^4, for the Rydberg states considered here, diamagnetic interaction is comparable to Zeeman interaction for fields as low as ∼ 100 G.

In practise, before dealing with the interactions between two atom, the total single-atom Hamiltonian H_1 is constructed within a restricted basis (see Sec. <ref>) and diagonalized for a specific external field configuration; a process we call prediagonalization <cit.>.
An appropriate two-atom basis is then constructed by pairwise, symmetrized combinations of the resulting eigenstates with the option to provide an energy cutoff.
This is particularly relevant in order to only include well converged single-atom states in the two-atom basis.
Finally, Rydberg pair-interaction potentials are obtained by constructing H_2 in the two-atom basis and subsequent diagonalization at the internuclear distances of interest.



 §.§ Multi-channel quantum defect theory 


In order to find the eigensystem of H_0, we follow the formalism presented by Robicheaux et al. combining multi-channel quantum defect theory (MQDT) and frame transformations to capture complex spin interactions in heavy atoms <cit.>.
Generally, the perspective of quantum defect theory is to consider an atom as a collision of an electron and an ionic core <cit.>.
In MQDT, the collision is described in terms of the real, symmetric K matrix that parameterizes the scattering between different spin configurations, so-called channels <cit.>.
The frame transformation is the process of unitarily transforming K matrices according to different coupling schemes that depend on the relevant length scale of the Rydberg electronic coordinate <cit.>.
Each frame has a different advantage: 
In the fragmentation frame, it is straight-forward to incorporate multiple ionization thresholds and account for atomic hyperfine structure (or in the case of molecules, for vibrational or rotational degrees of freedom).
The close-coupling frame, on the other hand, provides a diagonal representation of the K matrix.

From a scattering perspective, the total hyperfine spin of the Rydberg atom f is formed by coupling the asymptotically separated spins of the Rydberg electron j_r and the ionic core's hyperfine spin f_c.
However, at small separations, j_r and f_c are not meaningful and the fragmentation-frame coupling scheme has to be replaced by a more suitable close-coupling scheme, e.g. spin-orbit coupling.
This provides the additional advantage that the the K matrix in the close-coupling frame, obtained from spectroscopic data, are applicable (to a good approximation) independently of the isotope and the corresponding nuclear spin configuration.
An intuitive explanation for this is provided by comparing the energy scales of different spin couplings: 
Close to the ground-state, the hyperfine structure is only a small correction to the eigenstates dominated by spin-orbit coupling (or j_cj_r-coupling for some excited states, j_c being the total angular momentum of the core electrons) and therefore has very limited impact on the coupling parameters.

The formalism presented in <cit.> and outlined below solves for the bound states of H_0 after transforming from the closed-coupling frame into the fragmentation frame.
This is equivalent to, but leads to different equations than solving directly in the close-coupling frame, presented e.g. in <cit.>.
Let us start in the close-coupling frame, in which we assume the K matrix to be diagonal, denoted as K̅⃗̅, with elements

    K̅_α = tanπμ_α,

where μ_α are (possibly energy dependent) quantum defects and α labels the close-coupling channels.
We choose the transformation into the fragmentation frame to be expressed as a series of unitary matrices

    U_cc-fj = U_jj-fjU_LS-jjU_cc-LS,

where the subscripts label coupling schemes (detailed below), the first two matrices are solely determined by the corresponding angular momentum recouplings.
U_cc-LS accounts for corrections to pure spin-orbit (LS) coupling that is typically expressed as a series of simple rotations between specific LS channels, the (possibly energy dependent) angles of which serve as an optimization parameter in order to fit experimental data[
For light atoms, this is often not required and the K matrix is diagonal in the LS frame.
As pointed out in <cit.>, for , a coupling between the 5snd series singlet and triplet states, ^1D_2 and ^3D_2, can be accounted for by this additional rotation.].
We list here the representations of the coupling schemes that make for the frame transformation matrices analogously to references <cit.>.
When the Rydberg electron is close to the core, we write the wave function as

    |LS⟩ = |(((s_cs_r)S(l_cl_r)L)JI)fm ⟩,

where parentheses indicate which spins are coupled consecutively.
This configuration is recoupled in an intermediate step to

    |jj⟩ = |(((s_cl_c)j_c(s_rl_r)j_r)JI)fm ⟩,

before the final recoupling to the fragmentation frame

    |fj⟩ = |(((s_cl_c)j_cI)f_c(s_rl_r)j_r)fm ⟩.

This scheme can be summarized to be propagating the coupling of the valence electrons to the nuclear spin through different stages.

Finally, the K matrix in the fragmentation frame is obtained as

    K = U_cc-fjK̅⃗̅U_cc-fj^†.

Generally, the K matrix is block diagonal in f and in parity.
Thus, under the assumption ℓ_c=0, it further decomposes into blocks of equal orbital angular momentum ℓ_r.

A bound state |ψ_ν f⟩ at energy E_ν f is obtained by solving

    
    
    M = 0, 
     M = tanβ⃗ 11 +  K,


where the elements of β⃗, β_i = π(ν_i - ℓ_r,i), are related to the relative principal quantum numbers ν_i defined with respect to the associated ionization threshold of the i-th channel, i.e.

    E=E_i-R_μ/ν_i^2,  ∀ i,

for total energy E and thresholds E_i.
At a bound state energy E_ν f, the coefficients for each channel A_i are defined via the one-dimensional null space

    
    
    A⃗ = ( (ν⃗^-3/2cosβ⃗)^⊤ .  M ), 
    A⃗^2 = 1,


where the exponentiation of ν⃗, containing elements ν_i, as well as the matrix multiplication (.) are assumed to be element-wise operations.
Then, the total MQDT wave function is given as

    |ψ_ν f⟩ = ∑_i A_i^(ν f) P_ν_iℓ_r,i(r) |ϕ_i⟩,

with the channel function |ϕ_i⟩ representing the ionic core as well as the spin and angular configuration of the Rydberg electron.
The radial representation of the Rydberg electron is given by

    P_ν_iℓ_r,i(r) = ( f_i cosβ_i + g_i sinβ_i ) ν_i^-3/2,

with the regular and irregular Coulomb functions f_i and g_i, respectively.
Eq. <ref> guarantees the Rydberg wave function to go to zero asymptotically.

The total wave functions (Eq. <ref>) containing the radial representation only for the Rydberg electron (Eq. <ref>), but angular representation for both the Rydberg electron and the ionic core, are used for the calculation of all matrix elements, relevant for constructing the Hamiltonians discussed in the previous sections, particularly the electric multipole moments of order κ

    ⟨ψ_ν f|r_κ q|ψ_ν'f'⟩ = ∑_i,j A_i^(ν f) A_j^(ν' f') R_ij^κ⟨ϕ_i|Y_κ q|ϕ_j⟩,

with the spherical harmonics Y_κ q and the radial integrals

    R_ij^κ = ∫dr  r^κ P_ν_iℓ_r,i(r) P_ν'_jℓ'_r,j(r),

as well as the magnetic dipole moment

    ⟨ψ_ν f|μ⃗|ψ_ν'f'⟩ = ∑_i,j A_i^(ν f) A_j^(ν' f') R_ij^0⟨ϕ_i|μ⃗|ϕ_j⟩.

This assumes the approximation that contributions from the dipole operators acting on the core are negligible compared to contributions from the Rydberg electron because the corresponding wave function differ vastly in size.
Here, the representation in the fragmentation channel is beneficial: Rydberg and core quantum numbers are well separated therein.



 §.§ Strontium 87 


For the Rydberg structure of , we use the experimentally reported quantum defects for ^88Sr in the LS frame as listed in <cit.>.
We are interested in the 5snℓ Rydberg series close to the ionization threshold at relatively large principal quantum numbers n⪆30.
In Sr, some doubly excited states of the 5pnp-, 4dnd-, and 4dnp-series lie energetically below the first ionization threshold and are particularly relevant in order to accurately describe the spectrum at low n⪅15 <cit.>.
However, for our purposes, the influence of these series is sufficiently captured by considering the quantum defects of the associated 5snℓ series as slowly varying functions of energy <cit.>.
In the case of ^+ with its fermionic core and neglecting electron excitations, i.e. ℓ_c=0, the finite nuclear spin I=9/2 couples to the electronic angular momentum j_c=s_c=1/2 giving rise to two core hyperfine configurations f_c with energies separated by roughly 5 GHz.
The energies of the ionic ground states with different hyperfine configurations serve as distinct ionization thresholds E_f_c for the Rydberg electron.

To visualize which pair-interaction potentials can possibly be excited via an intermediate level |c>, we use a color code highlighting the potentials according to the dipole coupling of Rydberg-pair states |Ψ⟩ to the pair state |c,t⟩, where |t⟩ is the Rydberg target state to be specified and we have exemplary chosen |c⟩ to be the ^3P_0 f=9/2 clock state.
Thus, the coloring corresponds to the matrix element D=|⟨Ψ|r⃗⊗ 11|c, t⟩|.
Note that the state |c, t⟩ is not symmetrized, therefore the coloring is not sensitive to being in the dark- or bright-state sector of the Hamiltonian.
In pure Born-Oppenheimer approximation, these sectors are well separated, however, considering motional degrees of freedom, there is some finite coupling.
We consider this choice for coloration to be more instructive than coloring according to the overlap with a specified target state, because the latter neglects the impact of other dipole coupled states with for example differing hyperfine configuration or different orbital angular momentum.
For e.g. blockade experiments, it is crucial to find an energy window as well as a region of internuclear distances, which is “clean”, i.e. free of dipole-coupled potentials.
Unless otherwise specified, we choose for all pair-interaction potential figures in this article, D>10^-5 ea_0 for colored points and D>10^-9 ea_0 for gray, high-opacity points to inform about the encountered level densities.



 §.§ Remarks on basis sets 


A major challenge for the exact diagonalization of Rydberg pair Hamiltonians is the choice of the basis set and the corresponding convergence of eigenstates.
For van-der-Waals type interaction, the pair states relevant to converge a given pair potential can be energetically distant, defining a requirement for the considered energy window for diagonalization.
In general external field configurations, when M is not a conserved quantity, the pair state level density becomes large and the number of required basis states is typically the limiting factor.
Adding to this precarity, for multi-channel atoms, the single-atom level density is generally larger than e.g. in alkali atoms due to the relevance of the hyperfine interaction.

To illustrate this, let us consider the number of bound states within a single Rydberg manifold restricting the maximal orbital angular momentum to ℓ_max=3.
In ^87Rb, where I=3/2, there are technically 32 levels with average degeneracy of 4 each.
However, since the hyperfine interaction scales as n^-3, the nuclear spin can be disregarded for sufficiently large principal quantum numbers shrinking the number of levels to 7, or 32 in the case of resolved degeneracy.
Contrary for , hyperfine interaction cannot be disregarded[
In alkali metals, the hyperfine interaction scales with the principal quantum number due to the overlap of the electron wave function with the nuclear core.
While in alkaline-earths the Rydberg electron is similarly disconnected from the nucleus, the second valence electron is always close to the core probing its structure.
The electron-spin coupling of the two valence electrons prevents decoupling of the hyperfine structure and the corresponding scaling.
]
and the number of levels is 64 with average degeneracy 10 each, due to the large nuclear spin I=9/2.
A factor of 20 in the single atom level density has devastating consequences for the diagonalization of pair states, where the number of states scales quadratically and the number of matrix elements to construct the Hamiltonian scales with the power of 4.

In order to achieve convergence on the sub-MHz scale for the figures in this article, the typical basis sizes range from 10k to 25k states, after symmetrization.
For external fields not aligned with the z axis (not discussed here), we were only able to converge potentials to the tens-of-MHz scale using around 45k basis states.
Unless otherwise specified, we take into account 3 to 4 neighboring n manifolds, i.e. 7 to 9 manifolds in total, and orbital angular momenta ℓ from 0 to 3 in the single atom basis.
States further separated from the target state typically do not have the coupling strength to significantly contribute, although even energetically distant states could, in principle, make a degenerate pair.

For calculations with finite electric field, we compare the results including higher ℓ up to 4 or 5, however this does not serve as a proper convergence check in the case of high electric fields, because contributions from degenerate high-ℓ states could, in principle, add up to modify the level structure, hence, the focus on small electric fields, where we expect those contributions to be small.
Combinations of such single atom eigenstates are considered in the pair basis, if their total energy is within a specified energy window of the target state.
We found 20 GHz to be sufficient for our convergence goal for the considered states with n≈60, however, in arbitrary external field configurations, this may lead to intractable basis sizes and the convergence criterion may need to be loosened.



 §.§ Pairinteraction software 


In order to account for the increased requirements associated with calculating pair-interaction potentials of multi-channel atoms discussed in the previous sections, we extended the pairinteraction software.
A corresponding open-source release is currently pending.[
We refer to the software release and the accompanying publication for a more complete description of the improvements and the corresponding capabilities.
The software will be available at <https://pairinteraction.github.io/>.]
Notably, the software supports sophisticated diagonalization routines such as the FEAST algorithm <cit.>.
It is planned to offer cloud access to the pairinteraction software making use of CUDA for distributed computation on GPUs <cit.>.
Furthermore, we propose a database standard for atomic state data.
Such a database contains the bound state energies and appropriate state labeling according to the relevant quantum numbers as well as, importantly, the reduced matrix elements required for constructing Hamiltonians with high-performance <cit.>.

Such a database separates the calculation of pair-interaction potentials from the more fundamental calculation of atomic spectra and, particularly, removes any requirement for the software to handle MQDT wave functions.
Instead, we provide a Julia <cit.> implementation of MQDT using frame transformations following Robicheaux et al. <cit.> and outlined above as a utility tool for pairinteraction that will be available as open-source software.
An MQDT spectrum is generally sensitive on the underlying quantum defect data and the considered channel couplings.
Therefore any experimental advances in the spectroscopy of multi-channel atoms and improved efforts to fit quantum defects will, in principle, lead to new databases that can be used with the pairinteraction software.



§ RESULTS 


In the following we will show that the multi-channel couplings stemming from the hyperfine splitting of the ionic core and resulting in singlet-triplet mixed Rydberg series extends the complexity and versatility of Rydberg pair interactions.
Specifically, we focus on an intrinsic equal-state Förster resonance in the 5sns f=9/2 series of .
We can control the corresponding Rydberg-pair interactions in four ways, namely by the choice of the principal quantum number, by the magnitude of homogeneous magnetic and electric fields, and by the internuclear distance between atoms.
In contrast, the 5sns f=11/2 series of  is effectively described in a single channel approximation and provides the features known from Rydberg states in alkali metals with properties regularly scaling with the principal quantum number.



 §.§ Overview of the Förster resonance 




Fig. <ref> illustrates the level structure and coupling elements of  relevant for the composition of Rydberg pair-interaction potentials as functions of the effective principal quantum number ν.
(For this series n≈ν+3.5, where n counts the electronic orbital.)  
Accounting for the relevant scaling associated with these properties, we show the level separation of the 5sns f=9/2 series to its dipole-coupled states, i.e. states of 5sn'p f'={7/2, 9/2, 11/2} character.
Those states determine both the polarizabilty and the character of the pair-interaction potentials.
Each Rydberg manifold provides in total ten relevant states (not all of which are resolved in the figure), that can be grouped according to their total electronic character S, which can be mixed or pure.
A subset of three states has very similar mixed singlet-triplet character to the 5sns f=9/2 series and their dipole coupling is correspondingly strong.

Importantly, the position of the states of neighboring manifolds within the spectrum relative to the s series slowly changes as a function of ν and around ν≈61 the states of the neighboring manifolds are equidistant, cf. Fig. <ref>(b-c).
In other words, the Förster defects, i.e. the energy differences, of pairs (n-1)p+np and ns+ns have zero crossings giving rise to a broad Förster resonance.
Additionally, the contribution of these transitions to the Stark effect cancel giving rise to near zero polarizabilty.
Both of these effects have been predicted by perturbative considerations <cit.>.
Fig. <ref> reveals the complex interplay of the contributing states.
The more weakly coupled states as seen in Fig. <ref>(a), which have strong triplet character, contribute to the overall structure and illustrate the significant energy dependence of quantum defects, which originates in the hyperfine-split ionization threshold of ^+.

In the following, we analyze the conditions for this Förster resonance, indicating that it is intrinsic to an equal-state pair in the s series strongly coupling to the adjacent p levels at the zero crossing of the associated Förster defect, which is unique to specific Rydberg series in divalent atoms.
In Rydberg series with slowly varying energy dependence of the quantum defects, for example the 5sns f=11/2 series in , the functional behavior of the Förster defect features the regular n^-3 scaling.
This scaling and the consequential absence of a zero crossing of the Förster defect prohibits the existence of a Förster resonance.
Particularly, this scaling is generally present in alkali atoms, where Förster resonances are not typically found in equal-state pairs and rarely without additional tuning of energy levels with external fields <cit.>.
Instead, the existence of a Förster resonance requires the analysis of many spectral properties and the careful tuning of the quantum numbers to different values for the involved atoms in order to achieve degeneracy of a pair <cit.>.
Examples are p series in which the fine structure sufficiently shifts the state to be resonant with adjacent s states or an nsn'd pair or inter-species resonances using different principal quantum numbers for each species <cit.>.

In hyperfine-split ionic-core species, however, the existence of a mixed single-triplet s series featuring a Förster resonance is guaranteed, because the hyperfine interaction leads to an exchange in electronic character as a function of n, when the intrinsic splitting of adjacent Rydberg states is comparable to the ionic hyperfine splitting.
For example, we expect such a resonance for the 6sns f=1/2 series of , however, the uncertainty of quantum defects of the 5snp series makes a quantitative prediction of its location in the spectrum currently dispensable[
On the other hand, the quantum defects for the s and d series in Yb are rather well known.
It is, however, inherently difficult to probe the p series in heavy divalent atoms, in which Rydberg spectroscopy is typically performed using two-photon transitions, because the single photon transition is too deeply ultra violet.
We hope to see new Yb spectroscopy data soon or that this article can serve as a motivation for additional experimental efforts.
].

The unique combination of the features makes the Förster resonant state an interesting candidate for quantum technology applications:
The bare Rydberg line is largely insusceptible to external fields and their fluctuations.
Beyond the small polarizability, the mixed electron-spin character of the state also leads to a suppressed linear Zeeman splitting, because singlet states have a vanishing g factor, and the response to magnetic field is mostly quadratic driven by diamagnetic interaction requiring large fields.
At the same time, the pair state is subject to dipole-dipole interactions that provide a larger blockade radius than typically encountered for pair states subject to induced-dipole (i.e. van-der-Waals) interactions.
And finally, the equal-state pair can be easily addressed in a global beam setup requiring only a single wavelength as opposed to Förster states involving pairs of states differing in n or ℓ or even interspecies resonances as recently demonstrated <cit.>.



 §.§ Field-free pair-interaction potentials 




In Fig. <ref> we show the Rydberg-pair interaction potentials for equal pairs in the 5sns f=9/2 series in the vicinity of the Förster resonance at n=64.
Here, the asymptotic energy of the 5sns + 5sns pair is set to zero and the coloring indicates dipolar coupling to the ^3P_0 f=9/2 clock state.
Over a relatively short range of principal quantum numbers, n={59, 62, 64, 66, 69} shown in subfigures (a) through (e), the potentials drastically change character.
At low n far from resonance, the potential is repulsive and scales as R^-6 with the internuclear distance, indicating induced-dipolar interactions, and the pair interaction is well described using perturbation theory.
Closer to resonance, the scaling changes to R^-3, indicating permanent-dipole interactions due to the mixing of pair states with s and p character.
At n=64, the s and p pairs are asymptotically degenerate and the molecular eigenstates are symmetric superpositions, which leads to symmetrically attractive and repulsive potentials as is characteristic for a Förster resonance.
However, the fact that several pairs of p character contribute, leads to some non-generic level crossings and state mixings at large internuclear distances and at small detunings.
Furthermore, the significant p state character of the potentials gives rise to yet more additional couplings to other pair states of either s or d character at larger energies (e.g. around E/h=150 MHz and E/h=250 MHz) that lead to avoided crossings.
At larger n, the dominant potential remains attractive and the scaling transitions to dominantly R^-6 again.
At short internuclear distances R<4 µm, the potential is modified due to significant dipolar interactions stemming from other pair states at smaller energies.
The larger n, the more likely such modifications become, due to the increasing level density and the increasing dipolar interaction strength.



 §.§ Magnetic field response 




The multi-channel character of the different Rydberg series in  gives rise to diverse external field responses depending on the principal and orbital quantum numbers.
High level densities and the requirement for magnetic field strengths sufficient to lift the degeneracy in the ground (^1S_0) or clock (^3P_0) manifold, which is often the case for quantum technology applications, can lead to non-generic level shifts and avoided crossings in the Rydberg spectra of single atoms and pairs.

Fig. <ref> shows magnetic field maps for the Förster resonant pair and surrounding states (a) asymptotically in the non-interacting limit and (b) at 4 µm internuclear spacing.
Here, the magnetic field is aligned with the internuclear axis such that the total projection of the hyperfine state is still a conserved quantity and only the M=-9 symmetry sector is shown.
For the 5s64s equal-state pair, highlighted in black in Fig. <ref>(a), the linear magnetic field response is weak, due to a strong contribution from the singlet channel in that state, and the quadratic response due to diamagnetism dominates even at relatively small field strength.
For such states that are field-seeking, i.e. they lower their energy with respect to the magnetic field strength due to Zeeman interaction, there exists a local minimum in the magnetic field map, where the pair state is insensitive to magnetic field fluctuations due to a balance between attractive Zeeman and repulsive diamagnetic interaction.
For the state of interest this occurs at approximately B=50 G.

Multiple pair states with p character, which are close in energy at zero field, are shifted to smaller energies due to their strong linear response, i.e. Zeeman effect.
Note that most of the states shown in Fig. <ref> are field seeking because of M being negative.
Only in the case of f<I, such states are shifted to larger energies.
However, some pairs of p states, while initially being shifted out of resonance for small fields, become resonant with the s pair again at B>150 G.
This is striking and highlights the importance of diamagnetism:
If only linear shifts were considered, all p pairs would be shifted out of resonance, which would drastically change the character of the pair interaction potentials.

At short internuclear distances in Fig. <ref>(b), dipole-dipole interactions are additionally relevant, and the diverse magnetic field responses of the pair states leads to non-trivial functional behavior of interaction potentials, with a multitude of avoided crossings, and of their dipole addressability.
In particular, at B=200 G, one of the p pairs initially resonant at zero field, becomes degenerate with a different pair state.
This pair, which is detuned by approximately -200 MHz at zero field (cf. subfigure (a)), is from the same 5sns f=9/2 Rydberg series, however, for different n, namely n'=n±2, and one of the two states has slightly different singlet-triplet configuration due to its multi-channel nature.
In a magnetic field, this pair remains approximately equidistant to the Förster resonant state.
However, at B=200 G, the two s pairs become strongly coupled:
Essentially, two Förster resonances occur simultaneously at approximately 200 MHz energy splitting and the involved p states mediate coupling between the s pairs.



The corresponding pair potentials are shown in Fig. <ref> for B=50 G (a) and B=200 G (b).
Dipolar interactions exceed the energy scale of the asymptotic splitting and the pair interaction potentials feature a broad avoided crossing (cf. Fig. <ref>(b)) giving rise to an essentially flat dipole-coupled potential over the full range of R.
While for blockade experiments, this situation needs to be avoided, such a potential is highly unusual and may lead to facilitation, where Rydberg pairs are addressed resonantly.

This example illustrates that the magnetic field can effectively act as a knob for competing Förster resonances.
The potentials in Fig. <ref> can be qualitatively reproduced by a toy model that only includes the four properly symmetrized pair states that contribute dominantly:

    {|nsns⟩, |(n-1)pnp⟩, |n'sn”s⟩, |(n-1)p'np'⟩},

where only s and p pairs are coupled, respectively, by dipolar interaction (of similar strength) and two different Förster defects control the asymptotic detunings as functions of the magnetic field.



 §.§ Non-perturbative polarizability calculations 




After considering the effects of external magnetic fields in the previous section, we now turn our attention towards electric fields.
Generally, electric fields pose a computational challenge because contributions from high-orbital-angular-momentum states become relevant even for moderate field strengths.
Additionally, the influence of electric fields steeply scales with the principal quantum number (∼ n^5).
Therefore, here we restrict our analysis to electric field strengths of few hundreds of mV/cm.
We find that even these small electric fields can cause responses not expected from perturbation theory.

This is illustrated in Fig. <ref> for R=6 µm internuclear spacing and electric fields aligned with the internuclear axis.
While pairs of the pure triplet state 5s64s f=11/2 show the expected linear response of the derivative of its eigenenergy with respect to the electric field strength dE/dF as a function of the electric field strength F, the behavior of the Förster resonant pairs 5s64s f=9/2 indicates the presence of a hyperpolarizability that becomes relevant for relatively small field strengths F∼ 150 mV/cm.
In fact, this can be exploited to achieve extremely high degrees of line stability, when the electric field is tuned to a local extremum of dE/dF.
Overall, the f=9/2 pairs show a much larger polarizability than the f=11/2 pairs.
This can be attributed to the strong p-state admixture at resonance.
Asymptotically, the pure 5s64s f=9/2 has near zero polarizability, approximately 2 orders of magnitude smaller than the 5s64s f=11/2 state.



 §.§ Single channel Rydberg series 




As mentioned previously, the 5sns f=11/2 Rydberg series in  provides, in a lot of aspects, less complexity than the f=9/2 series.
It is a pure triplet series, well described in a single channel approximation and, hence, shows a lot of the regular scaling behaviors with respect to the principal quantum number expected from alkali metals or other single-channel Rydberg series.
In this series, an equal-state Förster resonance is not expected and the polarizability scales as n^7, while induced-dipolar interaction scales as n^11.

Fig. <ref> shows the pair interaction potentials (a) at zero magnetic field and (b) at B=200 G aligned with the internuclear axis.
Here, the M=-11 symmetry sector is shown and the coupling to the clock state is achieved by σ_+ driving.
Essentially, the magnetic field does not have a strong impact on the potential.
At B=200 G the C6 coefficient is reduced by approximately 5% and the spherically symmetric structure of the potential is maintained (not shown in the figure).
However, to achieve, e.g., Rydberg blockade for pairs within that series, still care must be taken for the choice of the principal quantum number in order to avoid accidental resonances.



§ CONCLUSION AND OUTLOOK 


We have analysed the structure of Rydberg pair-interaction potentials of , a multi-channel, divalent atomic species.
The ionic core's hyperfine structure gives rise to intriguing features within the Rydberg series, particularly states with vanishing polarizability and resonant dipolar interaction.
While these properties make those states very promising candidates for application in quantum technology, the arising complexity requires careful consideration of external field configurations.
We have shown that magnetic fields can undermine Rydberg blockade because of competing Förster resonances that give rise to avoided crossings in the potential energy landscape and to unanticipated potential shapes, such as an almost completely flat potential with respect to the internuclear distance.
On the other hand, configurations can be found that offer insensitivity to electric fields with pair states that have zero polarizability.

The results from pair-interaction calculations are readily available as input for error models and fidelity estimations in the context of entangling gates in quantum computation <cit.>.
Specifically, the results provide quantitative information about leakage into doubly excited states.
Depending on the available channels, this undesired behavior can potentially be mitigated by appropriate pulse shaping.

We have integrated the tools to analyze the spectra of multi-channel divalent atoms based on quantum defect measurements into the pairinteraction software.
Due to the increased level density of fermionic species, the calculation of pair potentials in arbitrary external field configurations remains challenging, particularly for  due to its large nuclear spin.

Recently,  has emerged as an interesting candidate for application in quantum computation.
With a nuclear spin of I=1/2 it is a natural choice for nuclear qubit encoding.
Due to a similar multi-channel structure, we expect the Rydberg spectrum of  to share a lot of features with .
Quantitative predictions have been enabled by related work including spectroscopy of the ^1P_1 and ^3P_J series in   <cit.>, which we became aware of during the compilation of this manuscript.



S.W. and J.M. acknowledge funding by the Federal Ministry of Education and Research (BMBF) under the Grants QRydDemo, MUNIQC-Atoms, and by the Horizon Europe program HORIZON-CL4-2021-DIGITALEMERGING-01-30 via the project 101070144 (EuRyQa).







 Paper Title: Ergotropy and Capacity Optimization in Heisenberg Spin Chain Quantum Batteries.

























    /łϕøϱ̂∂Ω∇Φ
        
        
        asal68826@hbku.edu.qaQatar Centre for Quantum Computing (QC2), College of Science and Engineering, Hamad Bin Khalifa University, Qatar Foundation, Doha, Qatarsmalkuwari@hbku.edu.qaQatar Centre for Quantum Computing (QC2), College of Science and Engineering, Hamad Bin Khalifa University, Qatar Foundation, Doha, QatarQatar Centre for Quantum Computing (QC2), College of Science and Engineering, Hamad Bin Khalifa University, Qatar Foundation, Doha, QatarNew York University Shanghai, NYU-ECNU Institute of Physics at NYU Shanghai, Shanghai Frontiers Science Center of Artificial Intelligence and Deep Learning, 567 West Yangsi Road, Shanghai, 200126, ChinaState Key Laboratory of Precision Spectroscopy, School of Physical and Material Sciences, East China Normal University, Shanghai 200062, ChinaCenter for Quantum and Topological Systems (CQTS), NYUAD Research Institute, New York University Abu Dhabi, UAEDepartment of Physics, New York University, New York, NY 10003, USAQatar Centre for Quantum Computing (QC2), College of Science and Engineering, Hamad Bin Khalifa University, Qatar Foundation, Doha, QatarThe University of Adelaide, SA 5005, AustraliaFaculty of Physics, Semnan University, P.O. Box 35195-363, Semnan, Iranhaddadi@semnan.ac.irFaculty of Physics, Semnan University, P.O. Box 35195-363, Semnan, Iran
        
            /łϕøϱ̂∂Ω∇Φ§ ABSTRACT 
                This study examines the performance of finite spin quantum batteries (QBs) using Heisenberg XX, XY, XXZ, and XYZ spin models with Dzyaloshinsky-Moriya (DM) and Kaplan–Shekhtman–Entin-Wohlman–Aharony (KSEA) interactions. The QBs are modeled as interacting quantum spins in local inhomogeneous magnetic fields, inducing variable Zeeman splitting. We derive analytical expressions for the maximal extractable work, ergotropy and the capacity of QBs, as recently examined by Yang et al. [https://doi.org/10.1103/PhysRevLett.131.030402Phys. Rev. Lett. 131, 030402 (2023)]. These quantities are analytically linked through certain quantum correlations, as posited in the aforementioned study. Different Heisenberg spin chain models exhibit distinct behaviors under varying conditions, emphasizing the importance of model selection for optimizing QB performance. In antiferromagnetic (AFM) systems, maximum ergotropy occurs with a Zeeman splitting field applied to either spin, while ferromagnetic (FM) systems benefit from a uniform Zeeman field. Temperature significantly impacts QB performance, with ergotropy in the AFM case being generally more robust against temperature increases compared to the FM case. Incorporating DM and KSEA couplings can significantly enhance the capacity and ergotropy extraction of QBs. However, there exists a threshold beyond which additional increases in these interactions cause a sharp decline in capacity and ergotropy. This behavior is influenced by temperature and quantum coherence, which signal the occurrence of a sudden phase transition. The resource theory of quantum coherence proposed by Baumgratz et al. [https://doi.org/10.1103/PhysRevLett.113.140401Phys. Rev. Lett. 113, 140401 (2014)] plays a crucial role in enhancing ergotropy and capacity. However, ergotropy is limited by both the system's capacity and the amount of coherence. These findings support the theoretical framework of spin-based QBs and may benefit future research on quantum energy storage devices.
                
                Ergotropy and Capacity Optimization in Heisenberg Spin Chain Quantum Batteries
                
                    Saeed Haddadi0000-0002-1596-0763
                    August 5, 2024
                ===============================================================================
                
                
                
                § INTRODUCTION
                
                Ongoing battery research faces critical challenges, such as low energy density, slow charging speeds, limited shelf life, and environmental concerns. This introduces a pressing demand for innovative solutions to address these issues <cit.>. 
                The operation and architecture of chemical batteries are purely classical, completely disregarding the underlying useful quantum resource with the potential to leverage a new paradigm in the energy sector. 
                
                Recent theoretical proposals on quantum battery (QB), based on the principles of quantum mechanics, present compelling heuristics <cit.>. These proposals attempt to harness quantum traits such as superposition and quantum correlations <cit.> to tackle the challenges faced by conventional classical batteries by, e.g., offering superior energy densities and fast charging rates.
                This could potentially offer crucial insights into quantum thermodynamics, particularly in overcoming challenges related to environmental-induced superselection (einselection) in quantum mechanics <cit.>. Recent findings also emphasize the importance of QBs for truly reversible quantum gates on quantum computers <cit.>. Moreover, studying QBs can illuminate the quantum mechanical origins of thermodynamic quantities such as heat, internal energy, ergotropy and entropy.
                
                Although experimental realization of QBs in the literature is scarce, 
                a recent study <cit.> reported experimental findings that demonstrate a QB employing superconducting qutrits, which have been optimized to maintain stable charging states. Furthermore, the authors highlighted a self-discharge mechanism reminiscent of supercapacitors, suggesting efficient energy storage capabilities in superconducting circuits.  
                Similarly, recent experimental tests confirm a quantum advantage in charging QBs using NMR star-topology spin systems <cit.> and verified the charging behavior in organic QBs <cit.>.
                Moreover, the authors in <cit.> realized Dicke QBs, showcasing superextensive power storage capabilities in their experimental setup. More recently, QB charging utilizing single photons within a linear optics setup has been demonstrated <cit.>.
                
                
                
                QBs are fundamentally realized through two distinct designs utilizing quantum cells (QCs) as their basic units, as depicted in Fig. <ref><cit.>. The first design, the isolated QC-based QB, conceptualizes each QC as an independent, non-interacting entity, all subjected to a uniform global field for charging. These QCs only remain entangled during the charging process via a global entangled charger field, as shown in Fig. <ref>(a). This approach primarily focuses on the intrinsic properties of each QC, such as quantum superposition and a non-degenerate energy spectrum. In contrast, the second design, the interacting QC-based QB, illustrated in Fig. <ref>(b), considers each QC as part of a system where inter-QC couplings are pivotal in determining the QB’s overall performance in terms of energy storage and retrieval, acknowledging that no quantum system operates in isolation. This design evolves from isolated systems to a coupled many-body quantum system. QBs can be modeled using spin-chain models, which represent many-body quantum systems with couplings between constituent spins that function as QCs. Unlike non-interacting QC-based QBs, the interacting QC-based QB model necessitates at least two QCs or spins and emphasizes the importance of inter-spin couplings, as seen in one-dimensional spin chains with realistic two-body couplings <cit.>.
                
                Heisenberg spin chains (HSCs), which serve as foundational models in condensed matter physics and quantum information technologies, exhibit a variety of multipartite couplings and potential non-classical correlations such as quantum entanglement and quantum discord. These HSCs have been used extensively in quantum technologies including quantum transistors <cit.>, cold atoms <cit.>, interferometry <cit.>, optical lattices <cit.>, and quantum communications  <cit.>. Since the introduction of the QB concept, researchers have been actively pursuing theoretical and experimental implementations across diverse quantum platforms, with the spin chain-based QB emerging as particularly promising <cit.>. By leveraging spin chains as QBs, we can examine the interplay between exchange couplings and the charging process, potentially unlocking quantum advantages in terms of power and efficiency. Unlike earlier studies that focused on global entangling operations during charging, spin chain-based QBs facilitate an in-depth analysis of the role of local charging fields and the contribution of intra-system couplings to the charging process.
                
                
                In this paper, we focus on QBs based on the HSC with anti-symmetric exchange coupling, also known as Dzyaloshinsky-Moriya (DM) and symmetric Kaplan–
                Shekhtman–Entin-Wohlman–Aharony (KSEA) coupling <cit.>, which has not been addressed in the literature yet to the best of our knowledge. There are different sub-categories of 1D Heisenberg spin models, such as the Ising, XX, XY, XXX, XXZ, XYZ, and different research groups have addressed these specific cases of HSC models with intrinsic spin-spin couplings without DM and KSEA couplings <cit.>. Therefore, our major emphasis is on exploiting DM and KSEA couplings for extracting ergotropy production since in previous work the spin-spin coupling in different subcategories of HSC such as Ising, XX, XY, XXZ, XYZ has been reported. By exploring the spin-chain QB with added DM and KSEA couplings, we initiate a new avenue for experimentalists to design experimental schemes or working substances for spin chain-based efficient QBs, paving the way for the development of efficient and practical QBs tailored for quantum technologies.
                
                
                The rest of this paper is organized as follows: Section <ref> covers the modeling of QBs in a spin-1/2 chain, their initialization, and the charging process. It also introduces the performance indicators for QBs, including work done, ergotropy, power, and efficiency. Section <ref> presents the results and discussion. Finally, Section <ref> concludes this paper. 
                
                
                
                
                § MODELING QUANTUM BATTERIES WITH HEISENBERG SPIN CHAIN
                
                
                 §.§ Classes of Heisenberg spin models
                
                A spin-based QB is typically a system of quantum mechanical spins' collection with associated couplings to each other. 
                The general 1D Heisenberg spin-1/2 chain with nearest neighbor coupling can be modeled via a Hamiltonian
                
                    ℋ̂ = 1/4∑_j=1^N-1 J[(1+γ)σ̂_j^x ⊗σ̂_j+1^x 
                    + (1-γ)σ̂_j^y ⊗σ̂_j+1^y]
                    + 1/4∑_j=1^N-1Δσ̂_j^z ⊗σ̂_j+1^z,
                
                here, γ is called the anisotropy parameter, and J and Δ are the nearest neighbor coupling parameters in the xy- and z-plane respectively. When both J and Δ are positive, this model exhibits anti-ferromagnetic (AFM) behavior, whereas when both J and Δ are negative, it demonstrates ferromagnetic (FM) characteristics. The standard Pauli spin operators denoted as σ̂_i^k (for k = x, y, z), define the spin vector on the ith site as σ_i = (σ̂_i^x, σ̂_i^y, σ̂_i^z). The different range of values of γ and Δ determines how the model falls in different categories of the one-dimensional Heisenberg model as shown in Table <ref>.
                
                In this paper, we focus on four out of the six potential cases. Specifically, we analyze the XX, XY, XXZ, XYZ models. This selection allows us to thoroughly investigate the key dynamics and characteristics pertinent to our research objectives.
                
                For a more comprehensive understanding of the other cases not considered in this study, we direct the reader to the interesting work presented in <cit.>. We note that, to date, no research has incorporated DM and/or KSEA coupling(s) in the QBs based on various subclasses of HSCs.
                
                
                
                
                 §.§ The QB Hamiltonian
                
                
                
                
                
                
                The Hamiltonian ℋ̂_𝒮 for working substance of our system reads  
                
                    ℋ̂_𝒮 = 1/4∑_j=1^N-1{J[(1+γ)σ̂_j^x ⊗σ̂_j+1^x  + (1-γ)σ̂_j^y ⊗σ̂_j+1^y]  + Δσ̂_j^z ⊗σ̂_j+1^z +𝒟.(σ̂_j×σ̂_j+1)+ σ̂_j.Γ.σ̂_j+1},
                
                where, the DM coupling strength is denoted by 𝒟=(D_x,D_y,D_z) and the KSEA coupling Γ is a symmetric traceless tensor which can be written as 
                
                    Γ = [   0 G_z G_y; G_z   0 G_x; G_y G_x   0;     ].
                
                The DM coupling, stemming from spin-orbit coupling, introduces an antisymmetric exchange coupling between neighboring spins, which is crucial for understanding phenomena like weak ferromagnetism. Simultaneously, the often-overlooked KSEA coupling restores symmetry disturbed by the DM term, providing a more accurate depiction of spin dynamics. Neglecting these couplings would obscure crucial aspects of quantum coherence and correlation, hindering our understanding of spin systems under external magnetic fields. Thus, their inclusion forms a fundamental basis for advancing QB technologies and deepening our understanding of correlated quantum materials-based QBs where DM and KSEA couplings are dominant.
                
                Now, to guarantee the non-degeneracy in the energy state of the working substance to act as QB, we
                apply Zeeman splitting to the spins using the Hamiltonian ℋ̂_ℱ. 
                We consider inhomogeneous magnetic fields along the chain, necessitating a site-dependent Hamiltonian formulation to capture this variation accurately. This approach is crucial for comprehensively understanding the system's dynamics and behaviors. The magnetic Hamiltonian takes the following form
                
                
                    ℋ_ℱ = ∑_j=1^N-1[ ℬ_j  (σ̂_j⊗Î_2) + ℬ_j+1  (Î_2  ⊗σ̂_j+1) ],
                
                here, the ℬ_j=(B_j^x,B_j^y,B_j^z) represents the magnetic field applied in an arbitrary direction to induce Zeeman splitting at any j-th and (j+1)-th site we set magnetic field along x- and y-direction to be zero. 
                
                Now that the Hamiltonian of the working substance is given as in Eq. (<ref>) and the Zemaan splitting Hamiltonian in Eq. (<ref>), we can establish the Hamiltonian for a QB
                
                    ℋ̂_𝒬=ℋ̂_𝒮+ℋ̂_ℱ.
                
                
                Given the intricate nature of the model, characterized by many parameters, maintaining complete generality becomes daunting. We add that not all subcategories of Heisenberg spin models are exactly solvable such that the analytical spectrum of energies and eigenstates is known. Thus, we opt for a tractable approach, focusing on a scenario involving two neighboring spins. Within this framework, we investigate the influence of a Zeeman splitting due to an inhomogeneous sinusoidal magnetic field acting along the conventional z-direction, denoted as ℬ_1=(0,0, B cosθ) and ℬ_2=(0,0, Bsinθ).  Let us define 0≤θ≤π/2 such that when θ=0 (or equivalently θ=π/2 ), the Zeeman splitting is applied to only one spin. When θ=π/4, the magnetic field is applied equally to both spins, and the same amount of Zeeman splitting is applied to both QCs. Furthermore, we consider the DM and KSEA couplings aligned with the z-direction, specified by 𝒟=(0,0, D_z), with the constraint G_x=G_y=0 and G_z≠0. Consequently, Eq. (<ref>) together with Eq. (<ref>) through Eq. (<ref>) can be written to accommodate any pair of consecutive spin sites, as demonstrated
                
                    Ĥ_𝒬   = J/4[(1+γ)σ̂_1^x ⊗σ̂_2^x  + (1-γ)σ̂_1^y ⊗σ̂_2^y] 
                       +Δσ̂_1^z ⊗σ̂_2^z+ B cosθ(σ_1^z⊗Î_2) + B sinθ(Î_2⊗σ_2^z) 
                       + D_z(σ̂_1^x⊗σ̂_2^y -σ̂_1^y⊗σ̂_2^x) + G_z (σ̂_1^x⊗σ̂_2^y + σ̂_1^y⊗σ̂_2^x).
                
                For convenience, we set B=1 henceforth.
                
                
                
                 §.§ The initial state of QB
                
                In the context of QBs, we may consider two initial states: the uncharged state and the charged state. For the uncharged QB's initial condition, it can either be in its ground state |00⟩⟨ 00|, representing absolute zero temperature, or in a thermal state with finite temperature. In the latter case, the system is initially in its ground state, and through the unitary operator, it undergoes a transition to an excited state, effectively fully charging the battery over time for subsequent power or ergotropy extraction. Since in the real world no system can reach absolute zero as the third law of thermodynamics and the Heisenberg uncertainty principle doesn't allow, we choose the second option and find Gibbs thermal state at equilibrium temperature T from the Hamiltonian expressed in Eq. (<ref>), we get
                
                    ϱ̂_th = e^-ℋ̂_𝒬/k_BT/Z= ∑_μ=1^4𝒫_μ|ψ_μ⟩⟨ψ_μ|.
                  
                In the expression, Z = Tr[exp(-ℋ̂_𝒬/k_BT)] is the partition function and 𝒫_μ = exp(-ν_μ/k_BT) denotes the probability of encountering a system in a specific state |ψ_μ⟩, where ν_μ stands for the eigenenergy corresponding to the state |ψ_μ⟩. Here, k_B signifies the Boltzmann constant, set to 1 for convenience, and T denotes the temperature. The state |ψ_μ⟩ represents the energy eigenstates of the system's Hamiltonian, as depicted in Eq. (<ref>)
                    ϱ̂_th = 1/Z(
                    [ ϱ_11    0    0 ϱ_14;    0 ϱ_22 ϱ_23    0;    0 ϱ_32 ϱ_33    0; ϱ_41    0    0 ϱ_44;      ]).
                
                The density matrix elements and the partition function can be expressed as 
                
                 in (<ref>) are the following:
                
                    Z    = 2 {cosh(Δ/T) [cosh(ℰ/T) + cosh(ℛ/T)] 
                          + sinh(Δ/T) [cosh(ℛ/T) - cosh(ℰ/T)]},
                
                    ϱ_11   = e^(ℰ - Δ) / T [ℰ - √(2)sin(θ + π/4)]/2 ℰ
                          + e^-(Δ + ℰ) / T [ℰ + √(2)sin(θ + π/4)]/2 ℰ,
                
                    ϱ_14 = ϱ_41^* = 2i χ e^-Δ / Tsinh(ℰ / T)/ℰ,
                
                    ϱ_22 = e^Δ / T(√(2)sinh(ℛ / T) sin(θ - π/4)/ℛ + cosh(ℛ / T)),
                
                    ϱ_33 = e^Δ / T(√(2)sinh(ℛ / T) cos(θ - π/4)/ℛ + cosh(ℛ / T)),
                
                    ϱ_44   = e^(ℰ - Δ) / T [ℰ + √(2)sin(θ + π/4)]/2 ℰ
                          - e^-(Δ + ℰ) / T [-ℰ + √(2)sin(θ + π/4)]/2 ℰ,
                
                    ϱ_23 = ϱ_32^* = -2i ϕ^* e^Δ / Tsinh(ℛ / T)/ℛ,
                
                where, ℛ= √(4 D_z^2 - sin(2 θ) + 4 J^2 + 1), ϕ = D_z + i J, ℰ = √(1 + 4 G_z^2 + 4 J^2 γ^2 + sin(2 θ)) and χ = G_z + i γ J. 
                
                
                 §.§ Charging the QB
                
                The QB can be charged by applying a local external magnetic field in the x-direction with charging magnetic field strength Ω=Ω(t) which may or may not be dependent on time. For convenience, we will assume that charging magnetic field strength does not depend on time. We consider that the charging Hamiltonian has the following form:
                
                    ℋ̂_𝒞 = Ω(t) (σ̂_1^x⊗𝕀̂_2 + 𝕀̂_2 ⊗σ̂_2^x).
                
                
                Furthermore, it is possible to charge QCs with site-dependent Ω values.
                Typically, the QB is charged by applying the uniform magnetic field to all QCs uniformly. We therefore assumed the Hamiltonian of the form expressed in Eq. (<ref>). Once the QB is charged, i.e. t→τ, its stored energy will reach its peak value. During this time τ, the charging field along the x-axis must be disconnected to prevent the charged state from cyclically reverting to an uncharged state. This is crucial because if the charging field remains connected, it will continuously cycle the QB between charged and uncharged states, thereby losing the objective.
                
                
                This transverse Pauli-X operator-based charging Hamiltonian functions as a NOT gate, driving the interacting spin chain out of its initial state. This Hamiltonian can be implemented as a uniform local magnetic field along the x-axis, perpendicular to the original Zeeman splitting magnetic field applied in z-direction. This effect can be physically generated by applying an additional external magnetic field relative to the existing magnetic field along the z-direction, as shown in Eq. (<ref>).
                
                In a closed system, the battery's charging process can be implemented as a unitary operation
                
                    Û_𝒞 (t) = exp(-i ℋ̂_𝒞t).
                
                The unitary operator expressed in Eq. (<ref>) guides how the battery will be charged in time cyclically. The expression for the unitary operators Û_𝒞 (t) expressed in Eq. (<ref>) for given charging hamiltonian as shown in Eq. (<ref>) can be explicitly written as
                
                    Û_𝒞 (t) = (
                        [   α β_- β_- 1-α; β_+   α α-1 β_-; β_+ α-1   α β_-; 1-α β_+ β_+   α;     ]),
                
                where α=cos ^2(Ω t) and β_±=±1/2sin (2Ω t ).
                
                
                
                
                
                
                 §.§ Performance indicators for quantum batteries
                
                Let ϱ̂ be a quantum state of QB with Hamiltonian ℋ̂_𝒬. The question we are interested in is: how much work can be extracted from this QB through a cyclic unitary process? In this context, cyclicity implies that the Hamiltonian of the QB must be the same at the start and end of the process, i.e., ℋ̂_𝒬 = ℋ̂_𝒬(0) = ℋ̂_𝒬(τ)<cit.>. Since the evolution is unitary, any change in the internal energy ⟨ℋ̂_𝒬⟩ is attributed to the work done on or by the QB.
                
                To analyze the QB's Hamiltonian, we express it in terms of its increasing spectral decomposition:
                
                    ℋ̂_𝒬 := ∑_μ=1^4ν_μ |ψ_μ⟩⟨ψ_μ|,  where ν_μ+1≥ν_μ ∀μ.
                
                
                In this expression, ν_μ denotes the energy eigenvalues, and |ψ_μ⟩ represents the corresponding eigenstates. This formulation enables the evaluation of energy variations and potential for work extraction in a cyclic unitary process, where the QB's Hamiltonian remains unchanged at the start and end of the cycle.
                
                The state ϱ̂, on the other hand, is expressed in its decreasing eigen-decomposition 
                
                
                    ϱ̂ := ∑_κ=1^4 r_κ |r_κ⟩⟨ r_κ|,  with  r_κ+1≤ r_κ ∀κ.
                
                
                The goal is to transform ϱ̂ into a state with lower internal energy, extracting the difference in internal energy in the process.
                
                After maximal cyclic, unitary work extraction, no further work can be extracted, and the system ends up in a so-called passive state π̂<cit.>. A passive state is unique up to degeneracy in the Hamiltonian. Passive states are diagonal in the Hamiltonian's eigenbasis, with decreasing populations for increasing energy levels <cit.>. That is, a state ϱ̂, as defined above, is passive if |r_n⟩ = |ψ_n ⟩ ∀ n. Gibbs thermal states are consequently passive <cit.>; therefore, we set π̂=ϱ̂_th. 
                
                The maximum work that can be extracted from a non-passive state ϱ̂ for a Hamiltonian ℋ̂_𝒬 via a cyclic unitary process ϱ̂→ρ̂_th (ϱ̂_th=Û_𝒞ϱ̂Û_𝒞^†) is called ergotropy which is found to be <cit.>
                    ξ(t) :=∑_m,n r_m ν_n [|⟨ψ_n | r_m ⟩|^2 - δ_mn]. 5
                
                where δ_mn is the Kronecker delta or equivalently <cit.>, we can write
                
                    ξ(t) := Tr[ϱ̂ℋ̂_𝒬 - ϱ̂_thℋ̂_𝒬].
                
                If the final state is another non-passive state with relatively higher energy than π̂, say Π̂, then work done by the QB can be defined as <cit.>
                    𝒲(t)=Tr[ϱ̂Ĥ_𝒬-Π̂Ĥ_𝒬].
                
                
                The average power is ⟨ p(t)⟩=𝒲(t)/t, and the peak average power is ⟨ p_max(t)⟩= max_t𝒲(t)/t<cit.> .
                The efficiency η = 𝒲(t)/ξ(t) quantifies the fraction of stored energy 𝒲(t) that can be extracted as ergotropy ξ(t). Efficiency is typically constrained to η≤ 1, with η = 1 indicating 𝒲(t)=ξ(t) for which the initial state should be a passive state.
                
                
                § RESULTS AND DISCUSSION
                
                The study of temporal ergotropy evolution in various Heisenberg model subclasses, detailed in Section <ref>, provides key insights into the charging and discharging dynamics of spin-based QB systems, facilitating their output optimization. The closed-form expression for ergotropy is given by
                
                
                    ξ(t) =    2 sin ^2(Ω t)/√(ab)(2 h ϵ _a+d^2+1){c [-2 √(ab) h ϵ _a+√(a)cos (2 Ω t) (-2 √(b) d g ϵ _a+√(b)(d^2+1)+2 γ(d^2-1) J)+2 √(ab) d ϵ _b]  
                        + 2 √(b) f_a [h (a+2 J (c-2 J))+2 c d g J cos (2 Ωt)+sin (2 θ ) (h-d g)]+2 √(a) d f_b [b+2 γ  J (c-2 γ  J)]},
                
                where 
                a = ℛ^2,
                b = ℰ^2,
                c = -Δ +γ  J+J,
                d = e^√(b)/T,
                ϵ _a = cosh(√(a)/T), ϵ _b = cosh(√(b)/T),
                f_a = sinh(√(a)/T), f_b = sinh(√(b)/T),
                g = e^2 Δ/T, and
                h = e^2 Δ + √(b)/T.
                
                
                 §.§ Effect of variable Zeeman Spliting
                
                We investigate the temporal evolution of ergotropy in an inhomogeneous magnetic field parameterized by θ in Eq. (<ref>), focusing on QB performance over time Ω t and θ, with DM and KSEA couplings set to zero for various Heisenberg model subclasses. Setting B=1 and 0 ≤θ≤π/2, Zeeman splitting affects one spin at θ=0 or θ=π/2, while both spins experience equal splitting at θ=π/4. We constrain Ω t to 0 ≤Ω t ≤ 2π, allowing the QB to undergo multiple charging and discharging cycles at T = 0.1, illustrating cyclic unitary charging operation. The strategy involves deactivating the x-axis charging field upon reaching maximum ergotropy, as the quantum NOT-Gate based charging Hamiltonian in Eq. (<ref>) would otherwise return the system to a thermal state, causing oscillations between charged and passive states over time.
                
                The density plot in Fig. <ref> shows the variation of ergotropy ξ(t) in the AFM-scenario, with J≥0, Δ≥0, D_z=0, G_z=0, and T=0.1, as a function of time Ω t and Zeeman splitting inhomogeneity parameter θ. Maximum ξ(t→τ) occurs when the magnetic field affects one spin at θ=0 or θ=π/2, while the lowest ξ(t) is observed when both spins are equally affected at θ=π/4. Although trends are similar across HSC cases, the XY model exhibits distinct behavior around θ=π/2, maintaining non-zero ξ(t) at θ=π/4, suggesting slightly better maximum work performance compared to the XX, XXZ, and XYZ models, which reach zero at θ=π/4.
                
                Within the interval 0 ≤Ω t ≤ 2π, the QB experiences two complete charge-discharge cycles. The maximal charge states are denoted by red regions along the Ω t-axis. For the given set of parameters, the optimal charging time is determined to be τ = π/2Ω. The peak value of ξ_max(t) remains invariant in time due to the unitary nature of the charging process.
                
                We have analyzed the effect of inhomogeneous Zeeman splitting on the QB in the AFM case over time. Next, we will investigate the FM case. The density plot in Fig. <ref> shows the temporal evolution of ξ(t) as a function of time Ω t in the FM scenario, where J and Δ are negative, and the inhomogeneity parameter θ, with D_z=0, G_z=0, and T=0.1. The results in the FM case differ significantly from those in the AFM case.
                
                
                Unlike the AFM case illustrated in Fig. <ref>, where the highest values of ξ(t→τ) are attained at τ=π/2Ω for all considered HSC cases and at θ=0 or π/2, the FM scenario presents a distinct behavior. In the XX model (Fig. <ref>(a)) and the XY model (Fig. <ref>(b)), the peak value of ergotropy, or first full-charging, occurs at τ=π/4Ω. This peak is reached four times within the interval 0 ≤Ω t ≤ 2π, indicating that the charging rate in the FM XX and XY models in Fig. <ref>(a)-(b) is twice as fast as in the AFM scenario for all considered HSC cases in Fig. <ref>(a)-(d). For the AFM case, the peak value of ergotropy is achieved for the Zeeman splitting parameter at θ=0 (or equivalently π/2), whereas in the FM scenario, the ergotropy behavior remains consistent for all values of 0 ≤θ≤π/2, with maxima at θ=0 and θ=π/2. Thus, the inhomogenous Zeeman splitting in the FM XX and XY models does not substantially alter the maximum ergotropy. 
                
                Considering XXZ and XYZ for which from Fig <ref>(c) and <ref>(d), we can see that unlike Fig <ref>(a) and <ref>(b) where the impact of Zeeman splitting parameter θ was roughly independent on ergotropy, here we find that the maximum value of ξ(t→τ) occurs at θ=π/4 wheres minimum value but not zero occurs at θ=0 or equivalently π/2. Furthermore, for this case unlike  Fig <ref>(a) and <ref>(b), first peak charging 
                is achieved at τ=π/4Ω here first peak value of ξ(t) occurs at τ=π/2Ω, which means the charging rate in XX and XY case is twice as XXZ and XYZ case.  
                
                A key observation is that the overall peak ergotropy in all HSC categories under the FM scenario is nearly three times greater than the peak ergotropy in the AFM case. Notably, the Heisenberg XY model is particularly robust for QB under AFM and ferromagnetic (FM) cases.
                
                In optimizing a QB using HSC models, the strategies vary significantly between FM and AFM scenarios. For FM cases, where spin polarizations are parallel, the XX and XY models achieve the fastest charging rates, with peak ergotropy occurring at τ = π/4Ω and repeating four times within 0 ≤Ω t ≤ 2π. In contrast, the XXZ and XYZ models reach maximum ergotropy at θ = π/4 and have slower charging, peaking at τ = π/2Ω. This indicates efficient energy transfer in the FM scenarios, but the choice of coupling model affects the timing of optimal charging.
                
                For AFM scenarios, where spins are oppositely aligned, maximum ergotropy for the XX, XXZ, and XYZ models occurs at θ = 0 or π/2. The XY model shows minimal ergotropy at θ = π/4, suggesting that AF settings benefit from specific spin chain couplings to enhance performance. Thus, we will use θ = π/4 for ferromagnetic and θ = 0 or π/2 for AF cases in our analysis.
                
                
                 §.§ Effect of Temperature
                
                Let's study how temperature T affects ξ(t) without DM and KSEA couplings in AFM (θ = π/2) and FM (θ = π/4) cases.
                
                In the AFM scenario illustrated in Fig. <ref>, the maximum ξ(t→τ) is observed at T ≈ 0. Specifically, for XX, XXZ, and XYZ, peak charging can be observed from t=0 in integer multiples of π/2. The maximum value of ergotropy remains constant in the temperature range around 0≤ T ≤ 1.5, except for the XY case. 
                
                
                
                In the XY case, ξ(t) does not peak at T=0 but instead increases from a smaller value as T rises, reaching a peak around T ≈ 0.8. This indicates a higher charging rate compared to XX, XXZ, and XYZ, where temperature has less impact. This is shown in Fig. <ref>(b). Essentially, the charging rate in the XY model is about twice that of the XX, XXZ, and XYZ models. Additionally, the peak ergotropy is highest in the XY scenario, followed by XX, then XXZ and XYZ.
                
                In the FM case, maximum ξ(t→τ) is attained at θ = π/2. Unlike the AFM case, where ξ(t) shows a different trend with temperature, in the FM scenario, ξ(t) generally decreases with rising temperature due to increased thermal fluctuations and reduced quantum coherence. As shown in Fig. <ref>, the peak ergotropy in the FM case at T ≈ 0 is about twice as large as in the AFM case for all HSC models. However, ξ(t) declines significantly with temperature, more so than in the AFM case. 
                
                The charging rate for the XXZ and XYZ models remains consistent, but in the FM XXZ case, it is doubled compared to the AF case. Overall, temperature negatively impacts ergotropy more significantly in the ferromagnetic case across all models (XX, XY, XXZ, XYZ) than in the AFM case, indicating worse performance of QBs in the FM scenario with rising temperature. The strategy for optimal performance should be to operate at low temperatures and leverage model-specific strengths like the high charging rates of the AF XY and FM XXZ models.
                
                
                 §.§ Symmetric-antisymmetric exchange coupling
                
                
                  §.§.§ Effect of KSEA Coupling
                
                
                Given that a high ergotropy in both AFM and FM cases requires low temperatures, it is crucial to apply Zeeman splitting fields effectively. In AFM scenarios, applying the field to one spin is advantageous, while in FM contexts, applying it to both spins is beneficial. Next, we investigate how DM and KSEA couplings can be utilized to enhance ergotropy in HSC-based QBs, using predefined parameters from previous analyses.
                
                Fig. <ref> shows the density plot of ξ(t) in the AFM-case as a function of Ω t and the z-component of KSEA coupling at T=0.1, θ=π/2, D_z=0. It shows that G_z positively influences all HSC cases, including XX, XY, XXZ, and XYZ. The full-charging, which occurs at τ=π/2Ω during the first charge, is unaffected by G_z. Notably, G_z significantly boosts ergotropy in the AFM case, with peak values around 75 near G_z≈ 20 as seen in Fig. <ref>. Beyond G_z≈ 20, ergotropy sharply drops to zero for G_z>20, indicating that G_z has maximized the QB's charge, and further increases in G_z do not enhance ergotropy. A similar trend is observed in the FM case, which is omitted here for brevity.
                
                
                    
                    
                      
                
                
                
                  §.§.§ Dzyaloshinsky-Moriya Coupling
                
                The relationship between ξ(t), Ω t, and D_z for the AFM case with θ = π/2, G_z = 0, and T = 0.1 is shown in Fig. <ref>. The density plots show that variations in DM coupling do not affect the charging rate across the cases depicted in Fig. <ref>(a) through <ref>(d). Notably, increasing the DM coupling enhances ergotropy values up to a threshold near D_z ≈ 35, consistent across all configurations from XX to XYZ. Beyond this threshold, ξ(t) suddenly drops down to zero, similar to the effect of G_z on ξ(t) in Fig. <ref>. This pattern where ξ(t) increases with D_z up to a point, then abruptly drops to zero persists across all AFM cases, including XX, XY, XXZ, and XYZ, indicating a consistent trend.
                
                In the FM case, as shown in Fig. <ref>, the variation of ξ(t) with D_z and Ω t (using the same parameters as the AFM case, i.e., G_z = 0 and T = 0.1, but with θ = π/4) reveals a similar behavior. Like the AFM scenarios in Figs. <ref>(a) and <ref>(c), where ξ(t) drops suddenly after exceeding a certain threshold of D_z, the FM case depicted in Figs. <ref>(a) and <ref>(c) also shows these sudden drops.
                
                
                To further extend our analysis of why this is happening, we will focus on the XYZ case as an example of complete anisotropy. Both the FM and AFM scenarios exhibit similar behaviors in the XYZ configuration, including the abrupt transition of ξ(t) with changes in D_z. Our goal is to determine the maximum ergotropy ξ_max(t) achieved by the QB in the AFM scenario and investigate its variation with D_z at different temperatures. We will maintain the other parameters as in Figs. <ref> and <ref>, and perform the analysis using numerical optimization.
                Various temperature values have been chosen to show the dependence of ξ_max(t) on D_z in Fig. <ref>a-d.
                
                
                The plots are generated under the AFM scenario with parameters G_z=0, J=1, Δ=0.5, and γ=0.5. These figures illustrate how the maximum ergotropy ξ_max(t) varies with D_z at different temperatures. 
                
                At a low temperature such as T=0.01 (Fig. <ref>(a)), ξ_max(t) initially increases with D_z up to a threshold value of D_z ≈ 2.25, then sharply drops to zero. The inset in Fig. <ref>(a) shows a narrow window of oscillations before the value reaches zero, and ξ_max(t) does not recover for any further increase even until D_z→∞. Similar behavior is observed at higher temperatures, with ξ_max(t) showing larger values and a higher threshold of D_z as shown in Figs. <ref>(b) and <ref>(c). The maximum value of ξ_max(t) corresponding to the threshold D_z increases with temperature.
                
                Recently, a new measure, known as the capacity of QB is proposed <cit.>, defined as 
                
                    K = Tr[Ĥ_𝒬ϱ̂_↑] - Tr[Ĥ_𝒬ϱ̂_↓],
                
                where ϱ̂_↓ denotes the lowest energy state of the QB, which in our case is the Gibbs thermal state defined in Eq. (<ref>), and ϱ̂_↑ is the highest energy state, ϱ̂_↑=|11⟩⟨ 11|. The ergotropy, which is the maximum amount of work extractable from the QB, cannot, in general, be equal to K except when T=0 or if we take the vacuum state instead of the thermal state. The evaluation of the capacity of the QB does not involve time; it depends only on the lowest and highest energy states, allowing it to be evaluated for both unitary and non-unitary quantum batteries. This measure, not involving any optimization, can be functionally connected with certain entanglement and quantum coherence measures, as done in <cit.>, and its values cannot be compared to the peak value of ergotropy.
                
                To support the trend observed in Fig. <ref> for the peak value of ergotropy versus D_z, we evaluate and plot the K for the same parameter values used in Fig. <ref>. With    
                S = sin(θ)+cos(θ), 
                the closed-form expression for the capacity of QB turns out to be
                
                
                
                    K = 2 √(a) f_a h+√(b)(d^2-1)+d^2 S+2 h ϵ  (2 Δ +S)+S/d^2+2 h ϵ +1.
                
                
                
                Figure <ref> shows K as a function of D_z at various temperatures: T=0.01 in Fig. <ref>(a), T=0.1 in Fig. <ref>(b), and T=1.0 in Fig. <ref>(c). The plots are generated under the AF scenario with parameters G_z=0, J=1, Δ=0.5, and γ=0.5. 
                
                These plots reveal how K varies with D_z at different temperatures. At T=0.01 (Fig. <ref>(a)), K initially increases with D_z up to a threshold value of D_z ≈ 2.25, then sharply drops to zero. The inset in Fig. <ref>(a) highlights a narrow, sharp oscillation window before K reaches zero, and it does not recover for any higher D_z. Similar behavior is observed at higher temperatures, with K showing larger values and a higher threshold of D_z as shown in Figs. <ref>(b) and <ref>(c). The maximum value of K corresponding to the threshold D_z increases with temperature.
                
                
                We observe a similarity between the capacity of the QB and the peak value of ergotropy with respect to the threshold value of D_z, where both measures reach their maximum and then suddenly drop to zero. Although the peak values of these quantifiers differ, they exhibit the same trend concerning the threshold value of D_z at various fixed temperatures. This suggests that the QB has a finite capacity for supplying maximum work, which explains why infinite ergotropy cannot be achieved by continuously increasing D_z. The QB's capacity also reflects this constraint, showing that while increasing D_z up to a certain threshold can enhance the QB's capacity, it cannot be extended beyond that limit.
                
                To investigate whether the observed signature in ergotropy and QB capacity is linked to an abrupt transition in quantum coherence, which encompasses all types of quantum correlations including separable and inseparable ones (such as non-locality, steering, entanglement, and quantum discord), we evaluate the l_1-norm of coherence 𝒬(ϱ̂) for the AFM XYZ scenario. This evaluation uses the same parameter values as depicted in Fig. <ref>(a). Specifically, we consider the density matrix
                
                    ϱ̂=Û_𝒞^†ϱ̂_thÛ_𝒞,
                
                where ϱ̂_th and Û_𝒞 are defined in equations (<ref>) and (<ref>), respectively.
                
                
                The 𝒬(ϱ̂) is defined by <cit.>
                    Q(ϱ̂) = ∑_i≠ j |⟨ i|ϱ̂|j⟩| = ∑_i,j |ϱ̂_ij| - ∑_i |ϱ̂_ii|,
                
                which is the sum of the absolute off-diagonal elements of a quantum state ϱ̂ in the reference basis.
                
                Figure <ref> illustrates the variation in the maximum l_1-norm of quantum coherence for the AFM Heisenberg XYZ model as a function of D_z and G_z, using the same fixed parameter values as in Figs. <ref> and <ref> for comparison. Similar to the plots of ξ_max(t) and the QB capacity shown in Figs. <ref> and <ref>, Q_max (ϱ̂) displays a comparable trend. A sudden drop in Q_max (ϱ̂) around the same threshold values of DM and KSEA interactions is observed, analogous to the behaviors of ergotropy and capacity. This clearly indicates that the amount of ergotropy is explicitly bounded by the capacity of the QB, and capacity is constrained by the availability of quantum coherence, implying that ergotropy, or the maximum work extraction, is ultimately (implicitly) limited by the QB's coherence.
                
                Furthermore, an increase in temperature requires even larger values of D_z to achieve the same maximum quantum coherence (≈ 3 as shown in Fig. <ref>(a) through Fig. <ref>(d)), as also observed in the QB capacity plot in Fig. <ref>. This suggests that indefinitely extracting work from a spin QB by merely increasing the values of DM or KSEA interactions is not feasible. The extractable work is constrained by the QB's capacity and ultimately limited by the availability of quantum coherence.
                
                The occurrence of unexpected oscillation followed by non-analytic behavior in ξ_max(t) and K in Fig. <ref> and Fig. <ref>, especially within the insets, indicates a kind of sudden phase transition. Initially, the ergotropy ξ(t) increases with D_z, reflecting enhanced work capacity due to quantum coherence and correlations. However, beyond a critical D_z, ξ_max(t) abruptly drops to zero, signifying a transition where the essential quantum coherence for work extraction is lost. This behavior is mirrored in the QB's capacity K and the l_1-norm of coherence 𝒬(ϱ̂), which also sharply decline at the same threshold. This consistent pattern suggests a fundamental reorganization of the system's quantum state, highlighting the critical role of quantum coherence in work extraction and underscoring the influence of quantum correlations on the system's energetic properties.
                
                
                
                
                § CONCLUSION
                
                In this paper, we provide a thorough investigation of finite spins quantum batteries based on four different subcategories of Heisenberg spin chains (XX, XY, XXZ, XYZ), integrating various magnetic interactions and couplings such as variable Zeeman coupling, and  DM and KSEA interactions. By exploring the maximum extractable work from diverse spin chain models in both AFM and FM settings, we have uncovered significant insights into the mechanisms of energy storage and extraction in these quantum systems.
                
                Our results indicate that the performance of these quantum batteries is notably affected by several factors:
                
                
                  * Inhomogenous Zeeman splitting and temperature: In antiferromagnetic systems, optimal energy extraction is achieved when the external magnetic field is applied to one of the two spins. In contrast, ferromagnetic systems benefit from applying the magnetic field uniformly to both spins for certain fixed parameter values. Generally, lower temperatures enhance performance across most models. However, the XY model in antiferromagnetic configurations exhibits considerably improved robustness at elevated temperatures. Conversely, in ferromagnetic systems, energy extraction diminishes with increasing temperatures due to heightened thermal fluctuations.
                    
                    
                  * Symmetric-antisymmetric exchange interactions: The incorporation of specific spin interactions, such as Dzyaloshinsky-Moriya and Kaplan-Shekhtman-Entin-Wohlman-Aharony couplings, can substantially boost the energy storage and extraction capabilities of these quantum batteries. Nevertheless, there exists a threshold beyond which further increases in interaction strength result in a sharp decline in energy storage capacity.
                    
                    
                  * Model-specific behavior: Different Heisenberg spin chain models (XX, XY, XXZ, XYZ) exhibit distinct behaviors under varying conditions, highlighting the importance of model selection for optimizing battery performance in specific applications.
                    
                    
                  * Quantum phenomena: The interplay between various quantum phenomena, including coherence and entanglement, significantly influences the performance of the battery. The maximum amount of extractable energy is limited by both the battery's inherent capacity and the level of quantum coherence present within the system.
                
                
                These findings highlight the interplay between energy extraction, quantum battery capacity, and quantum coherence under varying magnetic conditions. Our findings reinforce the theoretical framework for finite spin chain-based quantum batteries and provide practical insights into material selection and experimental setups.
                
                Recognizing the limitations of quantum resources, this study underscores the need for strategies to optimize their use in advanced quantum systems. By controlling magnetic interactions, symmetric and anti-symmetric exchange interactions, temperature, and quantum effects, we can improve the efficiency of energy storage and extraction in quantum batteries.
                
                In summary, this research offers valuable insights into optimizing quantum batteries derived from spin systems, contributing to the development of efficient energy storage devices in quantum technologies.
                
                
                
                § ACKNOWLEDGEMENTS
                
                M.G. and S.H. were supported by Semnan University under Contract No. 21270. 
                
                
                
                § DISCLOSURES
                
                The authors declare that they have no known competing financial interests.


 Paper Title: Scheduling Quantum Annealing for Active User Detection in a NOMA Network.








Scheduling Quantum Annealing for Active User Detection in a NOMA Network
    Romain Piron
Université de Lyon, INSA Lyon, INRIA 

CITI EA 3720

69621 Villeurbanne, France 

romain.piron@insa-lyon.fr
Claire Goursaud
Université de Lyon, INSA Lyon, INRIA 

CITI EA 3720

69621 Villeurbanne, France 

claire.goursaud@insa-lyon.fr


    August 5, 2024
=============================================================================================================================================================================================================================================================



§ ABSTRACT 
Active user detection in a non-orthogonal multiple access (NOMA) network is a major challenge for 5G/6G applications. However, classical algorithms that can perform this task suffer either from complexity or reduced performances. This work aims at proposing a quantum annealing approach to overcome this trade-off. Firstly, we show that the maximum a posteriori decoder of the activity pattern of the network can be seen as the ground state of an Ising Hamiltonian. For N users in a network with perfect channels, we propose a universal control function to schedule the annealing process. Our approach avoids to continuously compute the optimal control function but still ensures high success probability while demanding a lower annealing time than a linear control function. This advantage holds even in the presence of imperfections in the network.



Quantum annealing, local adiabaticity, control function, non-orthogonal multiple access, active user detection




§ INTRODUCTION


One of the major challenges of 5G wireless networks is to achieve ultra-reliable low latency communications (uRLLC) requirements to serve applications as the Industrial Internet of Things <cit.>. However, the allocation of a dedicated time slot or frequency prior to transmission for each user is no longer adapted for typical uRLLC networks because of their growing size <cit.>. Instead, each time that the users wish to transmit a signal, they must complete a random access procedure <cit.>. 

This procedure begins by an activity notification sent by the active users in the uplink channel. To do so, the code division multiple access (CDMA) approach can be used <cit.>. An identification sequence is attributed to each node of the network and all active users simply transmit their code. This approach permits sharing a common time-frequency resource to all the users without interference.

It recently appeared that leveraging non orthogonal multiple access (NOMA) in CDMA systems is promising <cit.>. In practice, it allows to  attribute non orthogonal identification sequences to the nodes. Thus, one can reduce the size of these sequences for a given size of the users' set.

Detecting the active users - or the activity pattern - accurately from the received signal is called active user detection (AUD) <cit.>. It poses a significant challenge in NOMA systems since the cross-correlations between the sequences do not vanish. Thus, the users cannot be detected separately. While methods like the zero-forcing technique <cit.> or traditional correlation receiver decoders <cit.> offer computational efficiency, their performance is limited. Conversely, the maximum a posteriori (MAP) Bayesian estimator <cit.> is the most reliable but computationally demanding, being NP-hard to compute.

In AUD scenarios in which the channel state is known, one can show that the computation of the MAP estimator is a quadratic unconstrained binary optimization (QUBO) <cit.> problem. The equivalence between QUBO problems and Ising Hamiltonians allows one to formulate the AUD problem as a ground state searching problem. However, an exhaustive search over the space of configurations to find the ground state has an exponential cost with the system size. 

Several methods have been investigated to avoid this exhaustive search.  Quantum annealing (QA) originally introduced by Kadowaki and Nishimori as a quantum analogue of thermal annealing in <cit.> is considered today as a promising quantum computing-based strategy for the ground state searching of several classes of Hamiltonians <cit.>. Like many authors today, we use the term quantum annealing to refer to QA-inspired adiabatic quantum computation <cit.>. Much work remains to be done regarding execution time, but QA already outperformed classical thermal annealing in reaching the ground state of some classes of Hamiltonians <cit.>. 

The control function that introduces transverse quantum fluctuations in QA usually needs to be adapted according to the parameters of the target Hamiltonian <cit.>. Unfortunately, in typical AUD models several variables are not deterministic but rather random variables <cit.>. This would require to compute the control function for each received signal. This is why, in this paper, we propose a strategy to build a generic control function to parameterize a QA process for AUD given the size N of the wireless network.

The rest of the paper is organized as follows. Sec. <ref> defines our model and build the Ising Hamiltonian associated with the MAP estimator for AUD. In Sec. <ref>, we briefly review the adiabatic theorem and show how to derive a control function for QA that ensures adiabaticity locally during the annealing process. Then, in Sec. <ref> we adopt a simplified scenario of our model to propose a universal control function to schedule a QA approach for AUD once the size of the network has been fixed. Finally, Sec. <ref> takes into account the channel state information in our AUD model. 



 §.§ Notations


The set of matrices over a field 𝕂 of size N × M is denoted 𝕂^N × M with the usual identification 𝕂^N × 1≡𝕂^N. We respectively denote a scalar, vector, matrix by x, x, X. 

The diagonal matrix whose diagonal elements correspond to the components of a vector x is denoted diag(x). We also introduce 1_N = [1, …, 1] ∈𝕂^N, finally the identity matrix of size M× M corresponds to I_M = diag(1_M).

The multivariate normal distribution with the mean vector μ and the covariance matrix Σ is denoted 𝒩(μ,Σ). The Dirac distribution picked at x is denoted Dirac(x). The uniform distribution over a set Ω is denoted 𝒰(Ω).




§ QA FORMULATION FOR AUD




 §.§ Context and problem defintion


We consider a network with N users. Each user #i is equipped with a single antenna and assigned with a bipolar identification sequence <cit.> - also called code - of length M denoted c_i. This code constitutes a signal that is transmitted by the i-th user. Its power is given by:

    P_signal,i = ‖c_i ‖^2.

It is often convenient to work with normalized signals such that P_signal,i = 1. Hence, the bipolar codes are chosen in the set:

    c_i ∈{-1/√(M), 1/√(M)}^M.

These vectors must also satisfy a quasi-linearly independence to ensure the decodability, which is discussed in Appendix <ref>.

Let us assume that a random subset 𝒩_A of these users wish to transmit their code to a base station (BS) to notify their activity on the wireless channel. This subset constitutes the activity pattern of the network encoded in the bit-string b^(0)∈{0,1}^N so that one can write:

    𝒩_A = {i ∈{1, …, N}   s.t    b_i^(0) = 1}.


Each propagation path between the i-th user and the BS might suffer from attenuation due to the Rayleigh fading coming from the isotropic scattering assumption <cit.>. It is modelled by N fading coefficients (w_i)_i=1,…,N that obey a different probability law depending on the quality of the channels:

    w∼Dirac(1_N)      for perfect channels
    𝒩(0,I_N)      for imperfect channels

where w = (w_i)_i=1,…,N. At the end, one measures the following signal at the receiver side:

    y = ∑_i=1^N w_i b_i^(0)c_i + n,

where n∼𝒩(0,ξ^2 I_M) is an additive white Gaussian noise that models the thermal noise at the BS.

The AUD problem consists in retrieving the activity pattern b^(0) of the network with the knowledge of the received signal y and the fading coefficients w. As mentioned in the introduction, the MAP estimator is the most reliable decoder for such task. It is defined by:

    b̂^MAP = b∈{0,1}Narg max ℙ_𝖻|𝗒,𝗐(b|y,w)

In the next part, we express the probability mass function ℙ_𝖻|𝗒,𝗐(b|y,w) in terms of the variables b_i.



 §.§ QUBO formulation for the MAP detector


Eq. <ref> implies that f_𝗒|𝖻,𝗐 simply corresponds to the probability density function of a Gaussian law. In addition, the Bayes' formula states that <cit.>:

    ℙ_𝖻|𝗒,𝗐(b|y,w) f_𝗒|𝗐(y|w) = f_𝗒|𝖻,𝗐(y|b,w)ℙ_𝖻(b).

From this point, a simple hypothesis consists in assuming a uniform distribution over the activity patterns. In this scenario, the prior distribution ℙ_b is constant which yields:

    ℙ_𝖻|𝗒,𝗐(b|y,w) ∝exp(- ‖y-C·diag(w) ·b‖^2/2 ξ^2),

where the sequences have been concatenated in the matrix C = [c_1 , …, c_N] ∈ℂ^M × N. Hence the MAP detector reads:

    b̂^MAP = b∈{0,1}^Narg min {‖y-C·diag(w) ·b‖^2}.

The computation of b̂^MAP aims at recovering b^(0) with (y,w) as inputs. Developing this objective function yields to the expression:

    ‖y-C·.   . diag(w) ·b‖^2 = ‖y‖^2 - 2 ∑_i=1^N (y· w_i c_i) b_i 
        + ∑_i,j=1^N (w_i c_i · w_j c_j )b_i b_j.

Once one throws away the irrelevant constant ‖y‖^2, our minimization problem corresponds to the general form

    b̂^MAP = b∈{0,1}^Narg min {∑_i,j Q_ij b_i b_j + ∑_i a_i b_i }.

Thus, the MAP detector can be written as a QUBO problem <cit.> expressed in terms of the binary variables b_i. 



 §.§ Building the Ising Hamiltonian


As underlined by the authors of <cit.>, a QUBO problem can be mapped to an Ising Hamiltonian by the change of variable σ_i = 1 - 2 b_i. We defer the details to appendix <ref> and simply give here the final expression of the problem Hamiltonian:

    H_P = - ∑_i h_i σ_i - ∑_i < j J_ijσ_i σ_j,

expressed with classical Ising spins σ_i ∈{+1 ≡ ↑, -1 ≡ ↓} and the parameters:

    h_i    = - w_i (c_i ·ỹ) 
    
        J_ij   = -1/2 (w_i w_j)  (c_i ·c_j)
        ,

where the signal ỹ is defined by:

    ỹ = y - 1/2∑_j=1^N w_j c_j.

In terms of Ising variables, an inactive (resp. active) user corresponds to σ_i =  ↑ (resp. σ_i =  ↓). The MAP estimator is the ground state of this Hamiltonian. It is the spin configuration σ^(0)∈{↑,↓}^N associated to b^(0) that minimizes the energy of the system. The decodability constraint previously mentioned about the codes c_i ensures that the ground state is unique in the idealized situation w = 1_N and ξ = 0. 

Let us quickly mention a simple sanity check that confirms that this Hamiltonian is well suited to our problem. We adopt the scenario w = 1_N, ξ = 0 and relax the NOMA constraint by considering orthogonal identifications sequences, ie. c_i ·c_j = 0 for i  j. Then, the activity pattern is trivially recovered by:

    b^(0)_i = |y·c_i|.

On the other hand, Eq. <ref> in this ideal situation reduces to:

    h_i    = ||c_i||^2 σ_i^(0)
    
        J_ij   = 0
        ,

where σ^(0) is the spin configuration associated to the initial activity pattern b^(0). The ground state of H_P is trivially σ^(0), as one might reasonably expect. 

The ground state of an Ising Hamiltonian can be found with perfect accuracy through an exhaustive search in the space of configurations {↑, ↓}^N. However, such task would require 𝒪(2^N) iterations which is why it is known to be NP-hard. In order to perform ground state searching with QA, we promote the Ising spins σ_i to quantum spins σ̂_̂î^̂ẑ [Recall that σ̂_̂î^̂α̂ denotes the Pauli matrices acting on the i-th qubit, where α = x,y,z]. The problem Hamiltonian is thus encoded in the operator:

    Ĥ_P = - ∑_i h_i σ̂_̂î^̂ẑ - ∑_i < j J_ijσ̂_̂î^̂ẑσ̂_̂ĵ^̂ẑ.




§ STATE OF THE ART FOR QA SCHEDULING






 §.§ Basic principle of QA


Given a quantum Hamiltonian Ĥ_P whose ground state is hard to find, one uses a control Hamiltonian Ĥ_C with a simple ground state to initialize the system in this known configuration. In the initial formulation of QA by Kadowaki and Nishimori <cit.>, Ĥ_C is chosen as the uniform transverse field Hamiltonian:

    Ĥ_C = - ∑_i=1^N σ̂_̂î^̂x̂,

whose ground state is the uniform superposition |+⟩^⊗ N. The QA process slowly evolves the system from an initial dynamic dictated by Ĥ_C towards a final dynamic encoded by Ĥ_P through the global Hamiltonian:

    Ĥ(t) = (1-u(t)) Ĥ_P + u(t) Ĥ_C.


The function u(t) is a control function obeying the initial condition u(0) = 1 so that the system starts in the ground state |+⟩^⊗ N. If the control function decreases slowly towards 0 during a period T, the final state of the system is expected to be the ground state of Ĥ_P with high probability. This period T is called the annealing time and its duration can be estimated thanks to the adiabatic theorem.



 §.§ Adiabatic approximation


The so-called adiabatic theorem is a mathematical statement on the time evolution of the global Hamiltonian Ĥ(t) to ensure that the instantaneous state |ψ(t)⟩ of the system remains close to the instantaneous ground state |ϕ_0(t)⟩. Several formulations exist with different precision levels reviewed in <cit.>. If one denotes {ε_n(t), |ϕ_n(t)⟩} the instantaneous eigensystem of Ĥ(t) and Δ(t) = ε_1(t) - ε_0(t) is the spectral gap between the ground state and the first excited state, the most commonly used adiabatic approximation reads <cit.>:

    |⟨ϕ_1(t) | ∂_t Ĥ(t) | ϕ_0(t) ⟩ |/Δ(t)^2   ≪ 1    ∀ t ∈ [0, T] ⇒ P_QA(T) ≈ 1,

where we introduced the notation used by <cit.>:

    P_QA(t) = |⟨ϕ_0(T) | ψ(t) ⟩ |^2.

The success probability of QA naturally corresponds to P_QA(T). In our case, the time dependence of Ĥ(t) is fully contained in the control function u. Using Eq. <ref>, the adiabatic approximation reads:

    | du/dt|  |⟨Ĥ_C - Ĥ_P ⟩_1,0(u) |/Δ(u)^2≪ 1   ∀ t ∈ [0,T]   ⇒ P_QA(T) ≈ 1.

which constraints the control function u. It is quite easy to show that the numerator |⟨Ĥ_C - Ĥ_P ⟩_1,0(u) | scales at most as 𝒪(Poly(N)) (see <cit.> for instance). Thus, a qualitative discussion of the adiabatic approximation as a function of the system size is often focused on the spectral gap Δ^2. At the end, one usually uses the following constraint on the control function u:

    | du/dt| Δ(u)^-2≪ 1   ∀ t ∈ [0,T].




 §.§ Linear scheduling


Several theoretical analysis of QA use a linear control function <cit.> to parameterize the annealing process:

    u_lin(t) = 1 - t/T_lin.

In this case, T_lin naturally corresponds to the annealing time. However, the first derivative of u is constant which heavily constraints T_lin. Indeed, Eq. <ref> immediately imposes:

    T_lin≫Δ^-2_min,

where Δ^2_min = 0 ≤ u ≤ 1min Δ^2(u). However, if the system encounters a first order quantum phase transition during the annealing process, it is known that the minimum of the gap closes exponentially with the system size <cit.>. Concretely, it means that T_lin∼ e^α N with α > 1 so QA would offer no advantage over an exhaustive search to compute b̂^MAP.

The work <cit.> underlines that in this case one can enhance the annealing time by looking for a control function with a non constant derivative.



 §.§ Optimized control function


Recall that u is a decreasing function from u(0) = 1 to u(T) = 0, which means that du/dt < 0. As it is done in <cit.>, we can fix an arbitrary small constant ϵ > 0 and impose:

    du/dt   = - ϵΔ^2(u(t)) 
    
            u(0)    = 1
        ,

which directly comes from Eq. <ref>. We refer to the solution of ordinary differential equation (ODE) as the optimized control function. Qualitatively, this equation tells that the larger the gap, the faster the control function can decrease. Integrating both sides of this ODE from t=0 to t=T yields the same expression than <cit.> for the annealing time:

    T = 1/ϵ∫_0^1 du/Δ^2(u).

These two equations <ref>, <ref> underline that once the gap is known, one can fully determine the annealing time and the control function associated to a problem encoded in Ĥ_P.



 §.§ An example


Going back to AUD, one can build the quantum Ising Hamiltonian <ref> once the received signal y and the channel coefficients w are known. Thus, we call a pair (y, w) a problem instance that allows to compute the Ising parameters of Eq. <ref>. Since the time dependence of the full Hamiltonian Ĥ is entirely contained in the control function, one can take u ∈ [0,1] as an affine parameter to evaluate the spectral gap as a function of u before even knowing the time dependence u(t). Thus, we denote Δ^2_(y, w)(u) the spectral gap associated to the problem instance (y, w). From Eq. <ref> and <ref>, one defines the associated control function:

    du_(y, w)/dt   = - ϵΔ^2_(y, w)(u_(y, w)(t)) 
    
            u_(y, w)(0)    = 1
            ,

which is the optimal one for (y, w). The associated annealing time reads:

    T_(y,w) = 1/ϵ∫_0^1 du/Δ^2_(y, w)(u).


As an example, we considered a network of N=8 users and generated a problem instance (y,w) in the following scenario:

    Activity pattern:    b^(0) = (1,0,0,0,0,0,0,0) 
    Perfect channels:    w∼Dirac(1_N) 
    No additive noise:    ξ = 0
        .

We show on Fig. <ref> the two first energy levels of the global Hamiltonian Ĥ(u). Fig. <ref> gives the corresponding spectral gap Δ^2_(y,w). These results have been obtained by exact diagonalization of the Hamiltonian using the QuTiP[<https://qutip.org>] library. Then we numerically evaluate the annealing time T_(y,w) and solve Eq. <ref> on the interval [0,T_(y,w)] to obtain the optimized control function (Fig. <ref>). To do so, we fixed our precision parameter to ϵ = 0.1. The evolution of the overlap during the annealing process scheduled with this control function is finally shown on Fig. <ref>. One can see that the success probability P_QA(T_(y,w)) reached at the end of the evolution is very close to 1, as expected.
However, it would be very long and inconvenient to require to compute T_(y,w) and u_(y,w) for each instance (y,w) of our problem. The goal of the following sections is to emphasize the possibility to build a generic approach to parameterize a QA process for AUD.



§ PROPOSITION: GENERIC CONTROL FUNCTION WITH PERFECT CHANNELS


We would like to check if, despite the randomness of the parameters (y, w), we can estimate a generic control function for a given size of the network N. We first consider a simple scenario with perfect channels.



 §.§ Mean gap in the absence of additive noise




Let us begin with a scenario where the size of the network N is fixed and where the additive Gaussian noise is neglected, which amounts to fix ξ = 0. We randomly generated 30 samples of (y, w) to check how the associated gaps Δ^2_(y,w) behave. The signals y are generated from uniform samples of activity patterns b^(0) and the channel coefficients are given by w = 1_N. Interestingly, Fig. <ref> shows that a pattern emerges for the shape of the gap against u despite the dependence of the local magnetic fields h_i in y. 

These results motivate the definition of the mean gap:

    Δ^2_(mean),N(u) = 𝔼_(y,w)(Δ^2).

In our specific case, this quantity can be exactly computed. Indeed, the channel coefficients are fixed to  w = 1_N and the received signal y is fully determined from the initial activity pattern b^(0). Using that b^(0) is uniformly distributed, one has:

    Δ^2_(mean),N = 1/2^N∑_b^(0)∈{0,1}^NΔ^2_(y, w=1_N)(u).

Then, we can use this mean gap to compute the associated annealing time T_(mean),N with Eq. <ref>:

    T_(mean),N = 1/ϵ∫_0^1 du/Δ^2_(mean),N(u),

and the resulting control function u_(mean),N(t) from Eq. <ref>:

    du_(mean),N/dt   = - ϵΔ^2_(mean),N(u_(mean),N(t)) 
    
            u_(mean),N(0)    = 1
        .


Once for all, we adopt the precision level ϵ = 0.1. We reported the shape obtained for u_(mean),N(s) for N = 6,…,9 on Fig. <ref> with respect to the normalized time s = t/T_(mean),N. Then, we must test for each value of N whether these functions are good candidates to schedule a QA process on the period t ∈ [0, T_(mean),N] to estimate b̂^MAP.









 §.§ Definition of the metrics


We will compare for several (y,w) the success probability of a QA process scheduled with our approach with the one that would been obtained with an optimal scheduling for each problem instance. To do so, one need to define appropriate metrics. We first generate a pair (y, w) and simulate a QA process parameterized with u_(mean),N by numerically solving the Schrödinger's equation:

    i ∂_t |ψ(t)⟩ = Ĥ(u_(mean),N(t)) |ψ(t)⟩   ,    t ∈ [0,T_(mean),N],

with the initial condition |ψ(0)⟩ = |+⟩^⊗ N (we work with natural units where ħ =1) using again the QuTiP library. The obtained success probability is denoted with the short-hand notation:

    P_QA(T_(mean),N ; (y, w)) ≡ P_QA^(mean)(y, w).

On the other hand, the reference success probability is the one corresponding to a QA process scheduled with the well-suited optimized control function u_(y,w) (Eq. <ref>) and the corresponding annealing time T_(y,w) (Eq. <ref>). We adopt also a short hand notation:

    P_QA(T_(y,w) ; (y, w)) ≡ P_QA^(opti)(y, w).


Since the instances (y,w) are randomly distributed, P_QA^(mean) P_QA^(opti) are random variables. Thus, we propose to use their expectation values to quantify the reliability of each scheduling strategy. We turn on the additive Gaussian noise to realize the performance analysis, hence the expectation values are given by:

    𝔼_(y,w)(P_QA^(α)) = ∑_b^(0)∈{0,1}^N∫ dn f(b^(0), n) P_QA^(α)(y,w),

for α∈{mean, opti}. The function f(b^(0), n) is the probability density function associated to the instances (y, w). The channel coefficients are still fixed to w = 1_N but the signal y is now generated from the sample of an activity pattern b^(0) and a Gaussian vector n. Those two parameters are independently distributed, respectively following the laws 𝒰({0,1}^N) and 𝒩(0,ξ^2). Thus, the weights of Eq. <ref> are given by:

    f(b^(0), n) = 1/2^N×1/(2πξ)^M/2exp(-‖n‖^2/2 ξ^2).


We also need a metric to quantify the intensity of the Gaussian noise. A well adapted one is the signal-to-noise ratio (SNR) expressed in dB. Since the power of the signal sent individually by each user is normalized to P_signal,i = 1, the SNR roughly reads:

    SNR  = -20 log_10(ξ).




 §.§ Reliability of our strategy


The instances (y, w) have no longer a finite support. Thus, we generated N_samples = 500 pairs (y, w = 1_N) according to the weights of Eq.<ref> for N = 6, …,9 to estimate the distributions of P_QA^(mean) and P_QA^(opti). The expectation values for α∈{mean, opti} are estimated by:

    𝔼_(y,w)(P_QA^(α)) = 1/N_samples∑_(y,w) ∈samples P_QA^(α)(y, w).


We reported on Fig. <ref> the obtained histograms for a noise level corresponding to SNR = 20dB. We can observe that the values of P_QA^(mean) are more spread out along the axis than the values of P_QA^(opti) which is not surprising. Indeed, the control function u_mean,N cannot be well suited for all possible outcomes (y,w). Nevertheless, the expectation value 𝔼_(y,w)(P_QA^(mean)) is always above 0.8 which is reasonably below 𝔼_(y,w)(P_QA^(opti)).

We also reported the obtained distributions for a level of noise corresponding to SNR = 15dB. The gap between 𝔼_(y,w)(P_QA^(mean)) and 𝔼_(y,w)(P_QA^(opti)) is higher but our approach can still offer a mean success probability above 0.7.

Thus, our generic control function appears as a good compromise to schedule a QA approach for any set of parameters (y, w) given a size of the network N








 §.§ Advantage over a linear control function


Once the annealing time T_(mean),N is known thanks to Eq. <ref>, one might question the necessity to use the control function u_(mean),N on the annealing period [0, T_(mean),N]. Indeed, the differential equation <ref> to obtain the optimized control function associated to a given gap results from several approximations. Thus, one might be tempted to simply use a linear control function with an annealing period T_lin,N = T_(mean),N and expect the resulting success probability to be not too low even if the condition of Eq. <ref> is not satisfied. To answer this question, we define the success probability of a QA process planned according to this strategy:

    P_QA(T_(lin),N = T_(mean),N ; (y,w)) ≡ P_QA^(lin)(y,w).


Following the same approach than above, we reported on Fig. <ref> the distributions of P_QA^(lin) and P_QA^(mean) for a level of noise corresponding to SNR = 20dB. The expectation value 𝔼_(y,w)(P_QA^(mean)) is higher than 𝔼_(y,w)(P_QA^(lin)) which confirms the advantage of using u_mean,N.

As the noise increases, Fig. <ref> shows that the gap between the two expectation values 𝔼_(y,w)(P_QA^(mean)) and 𝔼_(y,w)(P_QA^(lin)) is reduced. Nevertheless at SNR = 15dB, our control function is still expected to yield a success probability higher than the one that would been obtained with a linear control function



§ CONTROL FUNCTION IN THE GENERAL CASE


In the previous section we proposed a generic method to evaluate an appropriate control function and demonstrated its validity on random problem instances with thermal noise at the BS. However in more realistic scenarios, the received signal can also be impacted by the channel coefficients. In this section, we take into account the channels between the users and the BS. From now on, the channel coefficients are normal variables to account for Rayleigh fading:

    w∼𝒩(0,I_N).




 §.§ Behavior of the gap









First of all, let us go back to our scenario of N=8 users with an initial activity pattern b^(0) = (1,0,0,0,0,0,0,0). We generate one random problem instance (y,w) with no additional Gaussian noise by independently sampling the N channel coefficients according to the standard normal distribution.

As done before, we reported on Fig. <ref> the eigenvalues ε_0,1 of the global Hamiltonian Ĥ and the corresponding spectral gap. This time, the minimum of the gap is reached at the end of the annealing process when the control function vanishes.

This new behavior can be explained by the modifications induced in the spectrum of the Ising Hamiltonian Ĥ_P. In the case where a user i is strongly attenuated by a coefficient w_i close from 0, the spin flip σ_i → -σ_i will not significantly modify the associated eigenenergy. Thus, taking into account the attenuation on the propagation paths between the users and the access point strongly reduces the gaps between the eigenvalues of the associated Ising Hamiltonian Ĥ_P. It suggests that the mean gap should be estimated again in this new scenario.



 §.§ Mean control function with fading coefficients


As done previously, we first neglect the additive Gaussian noise in order to evaluate the mean gap from several problem instances. From this quantity, we derive the associated mean control function u_(mean),N defined on [0,T_(mean),N]. Since the channel coefficients w are now also randomly distributed, Eq. <ref> is generalized to:

    Δ^2_(mean),N(u) = ∑_b^(0)∈{0,1}^N∫ dw f(b^(0),w) Δ^2_(y,w)(u).

Since b^(0) and w are independently distributed, the above probability density function is simply given by:

    f(b^(0),w) = 1/2^N×1/(2π)^N/2exp(-‖w‖^2/2).

Contrarily to the scenario with perfect channels, the mean gap cannot be evaluated exactly through Eq. <ref>. Thus, we sample the probability distribution f(b^(0),w) with N_samples, gap = 2000 samples and evaluate the mean gap with the estimator:

    Δ^2_(mean),N(u) = 1/N_samples, gap∑_(y,w) ∈samplesΔ^2_(y,w)(u).

As done previously, we reported on Fig. <ref> the shape of the control functions against the normalized time s = t/T_(mean),N for N = 6, …, 9. These shapes indicate that the first derivative is monotonically decreasing over the annealing period, which is expected regarding the previous comments about the behavior of the gaps Δ^2_(y,w). Let us now check whether this approach still offers a good compromise in terms of success probability.







 §.§ Mean success probability


As we did with the perfect channel scenario, we would like to check that the mean control function still offers a good success probability when the noise is taken into account. We use again the metrics P_QA^(mean) and P_QA^(opti). The expression of their expectation values is modified according to the distribution of the instances (y,w):

    𝔼_(y,w)(P_QA^(α)) = ∑_b^(0)∈{0,1}^N∫ dn dw f(b^(0), n,w) P_QA^(α)(y,w).

for α∈{mean,opti}. Using again the independence of the parameters used to generate a pair (y,w), the above probability distribution is given by:

    f(b^(0),n,w)     = 1/2^N×1/(2πξ)^M/2exp(-‖n‖^2/2 ξ^2) 
    ×   1/(2π)^N/2exp(-‖w‖^2/2).


An estimation of the distributions of P_QA^(mean) and P_QA^(opti) at SNR = 15dB is shown on the histograms of Fig. <ref>. These results have been obtained by performing again N_samples = 500 samples of problem instances to evaluate P_QA^(mean)(y,w) and P_QA^(opti)(y,w). Unfortunately, the mean success probability 𝔼_(y,w)(P_QA^(mean)) is around 0.65 while a well-suited control function for each problem instance still ensures a mean success probability 𝔼_(y,w)(P_QA^(opti)) above 0.9. We must then investigate a way to improve the mean control function in this more complex AUD scenario.



 §.§ Time dilatation


The reduction of the mean success probability 𝔼_(y,w)(P_QA^(mean)) is likely due to the fact that u_(mean),N decreases too fast at the end of the annealing process. Indeed, the example previously shown on Fig. <ref> suggests that when the control function approaches 0, the gap closes which means that u must decrease slowly enough to offer a good reliability. In order to force u_(mean),N to decrease slower, we propose to apply a time dilatation factor λ to its associated annealing period [0, T_(mean),N]. In practice, it amounts to use the time-dilated mean control function:

    u_(mean),λ,N(t) = u_(mean),N(t/λ).

Consequently, the annealing period is extended to [0, λ T_(mean),N]. Of course, we aim at keeping λ reasonably low to preserve the order of magnitude of the annealing time. 

Nevertheless, we show on Fig. <ref> that time dilatation factors of λ = 2,3,4 already allow to boost the mean success probability 𝔼_(y,w)(P_QA^(mean)). These results correspond to a network with N=8 users and a level of additive noise fixed to SNR = 15dB. For instance, picking λ = 2 allows to reach a mean success probability above 0.75 which is significantly better than the results obtained in the previous paragraph.

We also compared again our proposal to a linear control function parameterized over the same annealing period [0,λ T_(mean),N]. The results on Fig. <ref> confirm that u_(mean),λ,N offers significantly a higher mean success probability 𝔼_(y,w)(P_QA^(mean)) than a linear control function.

Thus, our strategy is also promising in this more complex AUD scenario. A time dilatation can be applied to the mean control function to improve the performances.



§ CONCLUSION




 §.§ Contributions

In this work, we proposed a generic strategy to parameterize a QA process with the usual transverse Hamiltonian to compute the MAP estimator for AUD in a NOMA network. Despite the randomness of several parameters involved in the couplings and the local fields of the corresponding targeted Ising Hamiltonian, the evolution of the spectral gap of the global Hamiltonian against u(t) follows a recurrent shape. We have shown that our mean control function ensures a reasonable expectation value for the success probability of QA after (if wanted) applying a time dilatation operation. This function has the advantage to be universal once the size of the network N is fixed and permits to reach better performances than the simple linear control function.



 §.§ Scalability with N

The work conducted in this paper is limited to small network sizes. For larger values of N, the computation of the quantum gaps like those involved in Eq. <ref> would be too long with the exact diagonalization method of the full Hamiltonian used in this work. A further improvement would be to use Quantum Monte Carlo methods as done by <cit.> to extract quantum gaps involved in the mean gap estimator in order to compute it for N ≥ 9.



 §.§ Knowledge of the channel coefficients

Regarding the AUD model, another improvement would be to relax some assumptions we made. In fact, the knowledge of the channel coefficients w at the access point is a strong assumption that does not hold true in practice. Some modern formulations of AUD assume that the statistics of the channel vector w (which is a Gaussian law in this work) is known by the access point but not the realizations. A further perspective to this work could be to adapt the signal processing model to such case.




§ ACKNOWLEDGMENT


The authors would like to thank Lélio Chetot and Bruno Fedrici for helpful discussions.

This work was supported by the ANR under the France 2030 program, grant "NF-PERSEUS : ANR-22-PEFT-0004".






§ CHOICE OF THE SEQUENCES



The identification sequences of the users must be chosen such that each activity pattern b^(0) yields a different signal y in the absence of imperfections. It means that in the case ξ = 0 and w = 1_N, one requires:

    {y(b^(0),n = 0, w=1_N) .   = . y(b^(1),n = 0, w=1_N)}
    ⇒   {b^(0) = b^(1)}.

One can show that this condition is ensured if and only if:

    {∃λ∈{-1,0,1}^N   s.t  ∑_i=1^N λ_i c_i = 0}⇒{λ = 0}.

Throughout this work, we used coding schemes constructed by a random selection of N codes of size M followed by an exhaustive verification of the condition <ref>. We give below the matrices C = [c_1 , …, c_N] ∈ℂ^M × N corresponding to the codes we used for N = 6,…,9.

    C_6    = 1/√(5)[ -1 -1  1  1 -1 -1;  1  1 -1  1  1 -1; -1 -1  1  1  1  1;  1 -1  1  1 -1 -1; -1  1  1  1 -1  1 ],


    C_7    = 1/√(6)[  1  1  1  1 -1 -1 -1;  1  1  1  1  1 -1  1;  1 -1 -1  1 -1  1  1; -1  1  1  1  1 -1 -1;  1  1  1 -1 -1  1 -1; -1 -1  1 -1 -1 -1  1 ], 
    C_8    = 1/√(6)[  1  1 -1  1  1 -1  1 -1;  1 -1 -1 -1  1 -1  1  1; -1  1  1 -1  1 -1 -1  1;  1  1 -1 -1  1  1 -1 -1;  1  1 -1  1  1  1 -1  1; -1  1 -1  1 -1 -1 -1 -1 ], 
    C_9    = 1/√(7)[ -1 -1 -1 -1 -1 -1 -1 -1 -1; -1 -1  1 -1 -1 -1  1 -1  1; -1 -1 -1  1 -1  1 -1 -1  1; -1  1  1 -1  1 -1 -1  1 -1; -1 -1  1  1  1 -1  1 -1  1; -1  1  1  1  1  1  1  1  1; -1 -1  1 -1 -1  1  1  1  1 ].




§ QUBO - ISING MAPPING


One uses the change of variable σ_i = 1-2 b_i in the expression of the objective function <ref> without taking into account the irrelevant constant ||y||^2:

    ‖y-C·.   . diag(w) ·b‖^2 ∼ -2 ∑_i=1^N (y· w_i c_i) (1/2(1-σ_i)) 
        + ∑_i,j=1^N (w_i c_i · w_j c_j )(1/2(1-σ_i)) (1/2(1-σ_j)).

We throw away the constant terms (ie. not depending on the σ's) to obtain:

    ‖y-C·.   . diag(w) ·b‖^2 ∼∑_i=1^N σ_i ( w_i y·c_i - 1/2 w_i c_i ·∑_j w_j c_j) 
        + 1/4∑_i,j=1^N (w_i w_j c_i ·c_j) σ_i σ_j.

Using that σ_i^2 = 1, the diagonal contribution of the double sum is also an irrelevant constant. Since the sum is symmetric under i ↔ j, we can use ∑_i  j≡ 2 ∑_i < j. Hence the expression of the couplings in the main text:

    J_ij = - 1/2 (w_i w_j) (c_i ·c_j).

As for the local fields, one can directly identify:

    h_i = -w_i c_i ·(y - 1/2∑_j w_j c_j).

It matches the expression of the main text:

    h_i = -w_i c_i ·ỹ,

with the signal ỹ expressed as in Eq. <ref>
*





 Paper Title: Provably Efficient Adiabatic Learning for Quantum-Classical Dynamics.






Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Simons Institute and Department of Mathematics, University of California, Berkeley, CA 94704, USA

Department of Physics, University of Virginia, Charlottesville, VA 22904, USA

correspondence: diluo@mit.edu
Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
The NSF AI Institute for Artificial Intelligence and Fundamental Interactions
Department of Physics, Harvard University, Cambridge, MA 02138, USA



§ ABSTRACT 
Quantum-classical hybrid dynamics is crucial for accurately simulating complex systems where both quantum and classical behaviors need to be considered. However, coupling between classical and quantum degrees of freedom and the exponential growth of the Hilbert space present significant challenges. Current machine learning approaches for predicting such dynamics, while promising, remain unknown in their error bounds, sample complexity, and generalizability. In this work, we establish a generic theoretical framework for analyzing quantum-classical adiabatic dynamics with learning algorithms. Based on quantum information theory, we develop a provably efficient adiabatic learning  (PEAL) algorithm with logarithmic system size sampling complexity and favorable time scaling properties. We benchmark PEAL on the Holstein model, and demonstrate its accuracy in predicting single-path dynamics and ensemble dynamics observables as well as transfer learning over a family of Hamiltonians. Our framework and algorithm open up new avenues for reliable and efficient learning of quantum-classical dynamics.



Provably Efficient Adiabatic Learning for Quantum-Classical Dynamics
    Di Luo
    August 5, 2024
====================================================================










Introduction.— Efficient simulation of quantum-classical hybrid dynamics is crucial to multi-scale modelings of a wide range of physical systems, opening new avenues for advancements in material science, chemistry, and drug discovery by providing a more comprehensive understanding of molecular interactions <cit.>. A common hybrid dynamics approach relies on the adiabatic approximation, where two well-separated timescales of a system allow one to treat the slow dynamics classically while quantum calculations are used to solve the fast, often electronic, degrees of freedom that adiabatically follow the classical dynamics. A well-known example is the Born-Oppenheimer approximation which is widely used in ab initio molecular dynamics <cit.>.
The significance of quantum-classical hybrid dynamics lies in its potential to revolutionize how we model and predict the behavior of complex systems, ultimately pushing the frontiers of both fundamental research and practical applications.


The simulations of quantum-classical dynamics, however, is computationally challenging due to not only an exponentially large Hilbert space of quantum sub-systems and repeated time-consuming quantum calculations at every time step, but also the nonlinear differential equation coupled both the quantum and the classical variables. In the past two decades, machine learning (ML) has emerged as a powerful tool in developing force fields and inter-atomic potentials for ab initio molecular dynamics <cit.>. ML force-field approaches have recently been generalized to enable large-scale dynamical simulations of condensed-matter lattice systems <cit.>. 
This approach leverages massive datasets of quantum mechanical results to train models that can predict the potential energy surfaces with high precision.  ML-enhanced force fields facilitate the simulation of large intricate systems by accurately capturing the essential quantum mechanical effects while maintaining computational efficiency. Despite intensive studies and wide applications of ML force field models over past decades, critical questions on the error bound, sample complexity and generalizability of the ML methods have remained unresolved. 

In this work, we establish a generic theoretical framework for analyzing quantum-classical adiabatic dynamics with learning algorithms. We start with the analysis of the approximately constant linear model, derive the error bounded condition for the non-linear model, and introduce the relaxation method to check the error bounded property for a generic model, which lays down a solid foundation for the reliability of learning algorithms in quantum-classical adiabatic dynamics. Inspired by the recent development of quantum information theoretic learning theory for quantum many-body systems <cit.>, we develop a provably efficient adiabatic learning (PEAL) algorithm for quantum-classical dynamics, which offers a sample complexity scaling logarithmically with the system size and favorable scaling of evolution time. We benchmark our algorithm on the Holstein model and demonstrate accurate prediction of the single path dynamics and ensemble dynamics observables, as well as transfer learning across different couplings between quantum and classical degrees of freedom. 



 


Adiabatic Quantum-Classical Dynamics Learning.— We consider a general quantum-classical Hamiltonian:


    Ĥ = Ĥ_q + ∑_α,i g_αÔ_α,i G_α,i(P,Q) + H_cl(P,Q),

where Ĥ_q is the Hamiltonian operator for the quantum degrees of freedom, g_α is the quantum-classical coupling coefficient, α is the index of different types of couplings, i∈{1,…,L} is the label of local regions (e.g. lattice sites), Ô_α,i is the quantum operator that enters the coupling, G_α,i(P,Q) is a general function of the classical degree of freedom Q=(Q_1,…,Q_L) and its canonical momentum P=(P_1,…,P_L), and H_cl(P,Q) is the classical Hamiltonian.


We study the adiabatic evolution dynamics driven by the quantum-classical Hamiltonian in Eq. (<ref>). In this adiabatic limit, similar to the Born-Oppenheimer approximation in quantum molecular dynamics, the quantum subsystem is assumed to quickly relax to the ground state of the total Hamiltonian Ĥ, while the classical degrees of freedom follow the Hamilton's equations of motion (EOM):

    d/dt Q_j    = ∑_α,i g_α⟨Ô_α,i⟩∂/∂ P_j G_α,i + ∂/∂ P_j H_cl, 
        
    d/dt P_j    = -∑_α,i g_α⟨Ô_α,i⟩∂/∂ Q_j G_α,i - ∂/∂ Q_j H_cl - γ P_j,

where j∈{1,…,L}, ⟨·⟩ is the ground state expectation, and γ > 0 is the damping coefficient due to dissipation.

The conventional way to solve the above equations is through an iterative scheme, where one first updates {Q_j(t), P_j(t)} based on the classical EOM, and then solve the ground state from Ĥ. The updated ground state is used to compute expectation values ⟨Ô_α,i⟩ which determine the driving terms of the EOM for the next step. However, repeated ground-state calculations of Ĥ at every time step could be computationally costly using quantum state solvers (QSS), such as exact diagonalization (ED) <cit.>, density functional theory <cit.>, DMRG <cit.>, neural network <cit.>, and quantum computers <cit.>  . 


We consider a general quantum-classical ML model for learning such dynamics, which we call Adiabatic Dynamics Model Learning (ADML). ADML consists of two components, which uses machine learning to predict the quantum observables and evolves the classical observables using classical numerical schemes. The ML force-field models widely used in quantum molecular dynamics can be viewed as special classes of ADML <cit.>. Our goal here is to predict the adiabatic dynamics of Eq. (<ref>) with a learning-based approach. Given access to a dataset ⋃_s=1^N_s{(g_α, P(t), Q(t), ⟨Ô⟩(t))_s : t∈𝒯_s}, where 𝒯_s is the set of sampled time steps, N_s is the number of initial conditions {(g_α, P(t_init), Q(t_init))_s}_s=1^N_s sampled from a distribution 𝒟_init, and Ô stands for Ô_α,i in Eq. (<ref>) or other operators of insterest but not in the Hamiltonian, the task is to design ADML for predicting the dynamics starting with other { (g_α, P(t_init), Q(t_init))} from 𝒟_init. 
In the following, we analyze and derive the error bounded conditions for ADML.



(i) Approximately constant linear model. To serve as a starting point, we consider a simple example of Eq. (<ref>),
Ĥ = Ĥ_q + ∑_i g Ô_i Q_i + ∑_i( 1/2M P_i^2 + 1/2 k Q_i^2 ),
where we only consider one type of quantum-classical coupling with G_α,i(P,Q)=Q_i, and the classical Hamiltonian H_cl(P,Q) is for simple harmonic oscillators with mass M and spring constant k. Further assuming that during the dynamical process we are interested in, the response ∂⟨Ô_i⟩/∂ Q_i is approximately a constant, and the off-diagonal response ∂⟨Ô_i⟩/∂ Q_j (i≠ j) is approximately zero, the EOM of the system is then reduced to that of independent simple harmonic oscillators. 


Focusing on the classical degree of freedom Q_i, we can view the quantum-classical coupling g Ô_i Q_i as a driving force on the oscillator. Since the EOM in Eq. (<ref>) and (<ref>) are approximately linear in this case, the accumulated momentum and position errors between ML and the exact simulation using QSS, p(t) = P_i,ML(t)-P_i,Exact(t), q(t) = Q_i,ML(t)-Q_i,Exact(t) (i index suppressed), also satisfy a similar EOM:

    d/dt q(t)    = 1/M p(t),
        
    d/dt p(t)    = F(t) - K q(t) - γ p(t) + o(q(t))
        ,

where we define the error force F(t) = - g δ⟨Ô_i⟩(t) and the error stiffness K = g ∂⟨Ô_i⟩/∂ Q_i + k. δ⟨Ô_i⟩(t) = ⟨Ô_i⟩_ML(Q_ML(t)) - ⟨Ô_i⟩_Exact(Q_ML(t)) is the ML prediction error at the t-th time step. o(q(t)) means higher order terms in q(t) (See Supplemental Material for the derivation). Given a bounded F(t), even if it is tuned to drive the oscillator optimally, as long as K,γ > 0, the oscillator cannot be driven to infinite amplitude, i.e. the accumulated momentum and position errors are bounded. For F(t) to be bounded, it suffices to have δ⟨Ô_i⟩(t) bounded, which motivates us to define an Error Bounded Property such that when δ⟨Ô_i⟩(t) is bounded, p(t) and q(t) are also bounded. More precisely, we define the following


    A model satisfies the Error Bounded Property with respect to δ⟨Ô_i⟩(t) for t∈[t_init,t_end], if and only if the following claim is true: ∃  C_q, C_p>0 such that ∀ϵ>0, if ∀ t∈[t_init,t_end], |δ⟨Ô_i⟩(t)|^2≤ϵ, then there are |q(t)|≤ C_q√(ϵ) and |p(t)|≤ C_p√(ϵ), ∀ t∈[t_init,t_end].



    The approximately constant linear model satisfies the Error Bounded Property if K>0.

The proof of Prop. <ref> is in Supplemental Material.



(ii) Non-linear model. We can generalize the approximately constant linear model to allow non-linearity. We drop the assumption that the response ∂⟨Ô_i⟩/∂ Q_i is approximately a constant, allow G_α,i(P,Q) to be a non-linear function of Q_i, and allow the potential 1/2k Q_i^2 to include non-quadratic component in Q_i, as long as we can Taylor expand the potential at its minimum. These generalizations can be absorbed by a redefined F(t) and a time-dependent error stiffness K(t). Unlike the approximately constant linear model, the oscillator could have infinite amplitude even if K(t)>0 for all the time. We present a condition in Supplemental Material which guarantees a bounded bounded in the worst case scenario. Summarized as an informal theorem, we have:



 



    If K(t)>M(γ/2)^2+C with a positive constant C for all t, and the error stiffness K(t)'s fluctuation, as well as G_α,i(P,Q) and its first derivatives are bounded, then the non-linear model satisfies the Error Bounded Property in Def. <ref>.




The proof for Thm. <ref> is provided in Supplemental Material. We can further apply it to the Hamiltonian in normal mode with the quadratic momentum under Fourier transform, which could appear in a more general setup. 




(iii) General Relaxation Method. We now come back to the most general ADML. We allow arbitrary G_α,i(P,Q) and H_cl(P,Q) in Eq. (<ref>), and we make no assumption on the response ∂⟨Ô_i⟩/∂ Q_j. Because in the most general case the classical degrees of freedom are no longer decoupled, we restore the i index of the accumulated momentum and position errors, p_i(t) = P_i,ML(t)-P_i,Exact(t), q_i(t) = Q_i,ML(t)-Q_i,Exact(t). 


For a general ADML, the EOM for the errors q_i(t), p_i(t) can in general be derived from Eq. (<ref>) and (<ref>):

    d/dt q_i    = ∑_j ( 𝒦_q_j,q_i q_j + 𝒦_p_j,q_i p_j ) + ℱ_q_i + o(q,p),
        
    d/dt p_i    = ∑_j ( 𝒦_q_j,p_i q_j + 𝒦_p_j,p_i p_j ) + ℱ_p_i - γ p_i + o(q,p),

where the error stiffness matrix 𝒦_q(p)_j,q(p)_i and the error force vector ℱ_q(p)_i depend on (P,Q), and ℱ_q(p)_i is linear in δ⟨Ô_α, i⟩(t) (see Supplemental Material for details). 

While it is difficult to write down an error bounded condition for a general ADML, we propose a relaxation method to provide a sufficient (but not necessary) condition to check the Error Bounded Property in Def. <ref>.
The idea is to consider the worst case scenario. If in the worst case scenario the error still converges, then it is safe to use ADML to accelerate our dynamical simulation. The relaxation method is as follows. First, we require 𝒦 elements, as well as G_α,i(P,Q) and its first derivatives, are bounded. Second, assuming |δ⟨Ô_α,i⟩(t)|^2≤ϵ, we identify possible upper and lower bounds for the elements in 𝒦 and ℱ. 
The bounds just only need to be effective during the time range of the simulation. Third, to achieve the worst case scenario, we want to maximize d q_i/dt when q_i is positive, and minimize when negative (the same for p_i). Therefore, we insert the upper bound of 𝒦_q(p)_j,q(p)_i when q(p)_i and q(p)_j have the same sign, and the lower bound if the opposite sign. ℱ_q(p)_i are adjusted to their upper or lower bounds accordingly. Fourth, we perform a classical simulation of the EOM in Eq. (<ref>) and (<ref>), with the worst case scenario stated above. Finally, if the simulation shows there exist constants C_q_i and C_p_i such that for any ϵ>0, there are |q_i(t)|≤ C_q_i√(ϵ) and |p_i(t)|≤ C_p_i√(ϵ) during the time range of interest, then the Error Bounded Property is verified with the relaxation method. 





Provably Efficient Adiabatic Learning.— Next, we present a provably efficient learning algorithm for the above ADML models based on quantum information theory, which we call Provable Efficient Adiabatic Learning (PEAL). The PEAL algorithm, equipped with a learning model ℳ and a classical ordinary differential equation (ODE) solver, consists of the following steps. 

(i) Data collection for training. We sample a set of N_s initial conditions {(g_α, P(t_init), Q(t_init))_s}_s=1^N_s from a distribution 𝒟_init. 
We evolve the system with QSS and ODE solver to get (P(t), Q(t), ⟨Ô⟩(t))_s. 
For each s, we sample a set of time steps 𝒯_s uniformly from [t_init, t_end]. 
(ii) Model training. We use the dataset ⋃_s=1^N_s{(g_α, P(t), Q(t), ⟨Ô⟩(t))_s : t∈𝒯_s} to train a model ℳ:(g_α, P, Q)↦⟨Ô⟩ with the learning algorithm developed in <cit.>. 
(iii) Prediction. For any unseen new initial condition (g_α, P(t_init), Q(t_init))_new∼𝒟_init, PEAL outputs the dynamical trajectory (P(t), Q(t), ⟨Ô⟩(t))_new for t∈[t_init, t_end], by alternatively updating ⟨Ô⟩(t) with ℳ and P(t), Q(t) with ODE solver, integrated with our symmetry-preserving techniques shown later. The sample complexity and error bounds are summarized in the following informal theorem.




    
   When the Error Bounded Property in Def. <ref> is satisfied, for T time steps quantum-classical adiabatic dynamics of an n-qubit gapped system, with sample complexity O((n)), 
   PEAL gives rise to controllable accumulated errors of classical variables and all k-local, bounded quantum observables scaling as (i) O(√(T)) for generic model ℳ (ii) O(√(log T)) for sub-Gaussian ℳ's prediction error (iii) independent on T for bounded ℳ's prediction error. 




We leave the proofs for the above theorem in Supplemental Material. For Theorem <ref>, it can also be applied to an n-qubit system with ground states of exponential-decay correlation functions using recent results <cit.>. We highlight that PEAL works for unseen couplings g_α,new∉{g_α,s}_s=1^N_s, demonstrating transfer learning over a family of Hamiltonian. Furthermore, PEAL can predict any k-local observable Ô even if it does not appear in the Hamiltonian. 





Numerical Experiments.— We consider the Holstein model <cit.> which describes the electron phonon interaction as follows:

    H = - ∑_i,j c_i^† c_j - g ∑_i (c_i^†c_i - 1/2) Q_i + ∑_i(P_i/2M + k Q_i^2/2),

where c_i is related to the fermionic degree of freedom, while Q_i and P_i are the position and momentum of the phonon as classical degrees of freedom. 

In the numerical experiment, we study such model on a 1D periodic chain with L=50 sites. We set M=k=1 and the damping coefficient γ=0.1. The electronic degree of freedom at time t will be the ground state of H_e({Q_i(t)}) due to the fast relaxation, showing a 1D strong Anderson localization <cit.>. Therefore, PEAL's requirement of a ground state with exponential-decay correlation functions is satisfied. Analytically, the 1D adiabatic Holstein model always cools down to a charge density wave (CDW) in zero temperature. In Supplemental Material, we derive the relation between phonon amplitude and CDW response function, showing the error stiffness K is always positive when it's close to a checkerboard configuration. We also numerically measure the error stiffness K(t) during the time range of interest, confirming the Error Bounded Property (see Supplemental Material). Therefore, we can apply PEAL to the 1D adiabatic Holstein model with a bounded error guarantee during the dynamics. 

 



Since Eq. (<ref>) has the U(1) and translation symmetry, we develop a symmetry-preserving PEAL. The U(1) global symmetry is respected by conserving the total electron density, and the translation symmetry by applying the same model on all sites. The U(1) symmetry preserving is a new feature in our PEAL that does not exist in previous literature and we have shown that the symmetry-preserving PEAL maintains provably efficient error bound in Supplemental Material.

 



We begin with the single-path prediction task by training a model using some initial conditions and certain g values. The goal is to predict observable dynamics from different initial conditions under both the training g values (standard learning) and unseen g values (transfer learning). Here, we choose g∈ G_SL = {1.3, 1.32, 1.34, 1.36, 1.38, 1.4} for training. In Figure <ref>, we demonstrate the single-path prediction by PEAL. The solid curves are the exact simulation using exact diagonalization (ED) for QSS and RK4 for classical ODE solver <cit.>, and the dashed curves are with PEAL. Blue curves are for g=1.4 (standard learning), and orange curves for g=1.39 (transfer learning). We present the time evolution of six different observables during the dynamics, which are the total charge density wave (CDW) ∑_i(-1)^i n_i, the electron density at the first site n, the phonon amplitude at the first site Q, the phonon momentum at the first site P, the hopping term ⟨ c_0^† c_1⟩, and the next-nearest-neighbor correlation ⟨ c_0^† c_2⟩. We note that CDW is a sum of local observables, ⟨ c_0^† c_1⟩ does not enter classical EOM, and ⟨ c_0^† c_2⟩ does not even appear in the Hamiltonian. Nevertheless, all observables in the PEAL curves agree well with their corresponding exact simulations, demonstrating that PEAL guarantees a controllable error for k-local observable and well performs in transfer learning. 

In Figure <ref>, we present the sample complexity of PEAL. We use the same training data in the single-path prediction, build the test set with g∈ G_SL but with different initial conditions, and demonstrate transfer learning with g∈ G_TL = {1.31, 1.33, 1.35, 1.37, 1.39}. For illustration, we consider the root mean square of |n_PEAL-n_Exact| as the test error. The test errors for both standard learning and transfer learning show a logarithmic scaling law with the number of samples. The inner panels also demonstrate that n_PEAL in standard learning and transfer learning match well with n_Exact.



Generically, the ensemble dynamics (i.e. the dynamics average over an ensemble of paths from different initial conditions) can provide more information of the underlying physics. In Figure <ref>, we compare the ensemble averages of the correlation function Q_i(t)Q_j(t) at time t=100 for g=1.4 and g=1.7. Here, g=1.4 prediction is based on PEAL training in g ∈ G_SL, and g=1.7 prediction is based on another model from PEAL training in g∈{1.6,1.62,1.64,1.66,1.68,1.7} where long-lasting domain walls exist. For g=1.4, the system has cooled down to a pure checkerboard configuration, and shows an oscillating correlation function. For g=1.7, since the system has domain walls, it shows a decaying correlation function and a large variance. PEAL provides a good agreement with the exact simulations for both the mean and the variance over different g values. 






Conclusion.— In this work, we establish a comprehensive theoretical framework for analyzing quantum-classical adiabatic dynamics using learning algorithms. We provide a systematic analysis for the error bounded properties of the approximately constant linear model, the non-linear model, and generic models, ensuring the reliability of ADML. We develop a provably efficient adiabatic learning algorithm PEAL, demonstrating logarithmic scaling of sample complexity with system size and favorable evolution time scaling. Benchmarking PEAL on the Holstein model, we achieve accurate predictions of single-path dynamics and ensemble dynamics observables, with effective transfer learning across various quantum-classical coupling strengths. Our framework and algorithm opens up new directions for efficient learning in quantum-classical dynamics, including applying PEAL to quantum experiments, integrating advanced machine learning techniques, and extending to generic non-equilbrium processes.





Acknowledgements—  The authors acknowledge helpful discussion with Max Metlitski and Zhi Ren. DL acknowledges support from the NSF AI Institute for Artificial Intelligence and Fundamental Interactions (IAIFI). JPL acknowledges support by the National Science Foundation (PHY-1818914, CCF-1729369), the NSF Quantum Leap Challenge Institute (QLCI) program (OMA-2016245, OMA-2120757), and a Simons Foundation award (No. 825053). GWC acknowledges the support of the US Department of Energy Basic Energy Sciences under Contract No. DE-SC0020330.
The authors acknowledge the MIT SuperCloud for providing HPC resources that have contributed to the research results reported within this paper.



  
  

 

  






    § SUPPLEMENTAL MATERIAL FOR PROVABLY EFFICIENT MACHINE LEARNING FOR ADIABATIC QUANTUM-CLASSICAL DYNAMICS
    
    
    § I. DIMENSIONLESS MODEL
    
    
    Consider the standard spinless Holstein model:
    
        H = -t_nn∑_⟨ ij ⟩ c_i^† c_j - g ∑_i ( c_i^† c_i - 1/2) Q_i + ∑_i ( P_i^2/2M + kQ_i^2/2).
    
    
    And the equation of motion for the phonons is
    
        dQ_i/dt = P_i/M,   dP_i/dt = g n_i - k Q_i,
    
    where n_i = ⟨ c_i^† c_i ⟩ is the on-site fermion number. The mass m and elastic constant k are related by the familiar formula,
    
        ω = √(k/M),
    
    The inverse ω^-1 gives a characteristic time scale for the dynamical problem. Next, one can introduce a “length scale”Q_0 for the displacement of the simple harmonic oscillator. The energy related to Q at a given site is
    
        E(Q) = -g n_i Q_i + k Q^2/2.
    
    
    Assuming electron number n ∼ 1, minimization with respect to Q gives ∂ E/∂ Q |_Q_0 = 0:
    
        Q_0 ∼g/k.
    
    
    From this one can then introduce a scale for the momentum via the relation dQ/dt = P/M
        ω Q_0 ∼P_0/M P_0 = M ω Q_0 = M ω g/k.
    
    
    We can now define the dimensionless time, displacement and momentum as
    
        t̃ = ω t,   Q̃_i = Q_i/Q_0,   P̃_i = P_i/P_0.
    
    
    In terms of dimensionless quantities, the equation of motion is then simplified to
    
        dQ̃_i/dt̃ = P̃_i,   dP̃_i/dt̃ = n_i - Q̃_i.
    
    
    Next, we consider the tight-binding Hamiltonian for the fermions for a given {Q_i} configuration. We factor out the nearest-neighbor hopping constant t_nn and use it as the unit for energy. Also, we use the dimensionless Q̃_i:
    
        H_TB = t_nn[ -∑_⟨ ij ⟩ c_i^† c_j + g Q_0/t_nn∑_i Q̃_i c_i^† c_i ].
    
    
    The coefficient of the second term above gives an important dimensionless parameter for Holstein model. Instead of t_nn, we can introduce the bandwidth of the tight-binding model: W = 4t_nn for the 1D model. We then define a dimensionless electron-phonon coupling
    
        λ = g Q_0/W = g^2/kW.
    
    
    The dimensionless tight-binding Hamiltonian then becomes
    
        H̃_TB = -∑_⟨ ij ⟩ c_i^† c_j + 4λ∑_i Q̃_i c_i^† c_i.
    
    
    One can see that, using these dimensionless quantities, the only adjustable parameter of the adiabatic dynamics of the Holstein model is this dimensionless λ. In general, for real materials λ≲ 1. For example, we can set it to λ = 0.5 or 0.25 in the simulations.
    
    
    
    
    § II. HOLSTEIN MODEL CHARGE DENSITY WAVE RESPONSE ANALYSIS
    
    
    Consider a tight binding model on a 1D lattice:
    
        H    = -t ∑_i (c_i^† c_i+1 + h.c.) - g ∑_i (-1)^i Q c_i^†c_i,
    
    where i∈{0,1,…,L-1}, L is the lattice size and is even. The lattice has periodic boundary condition. The effective potential on the lattice is staggered: [-gQ, +gQ, -gQ, +gQ, …]. The unit cell consists of 2 lattice sites. 
    
    We can solve the single-particle wavefunctions using the ansatz
    
        |ψ_k⟩ = (a, b e^ik, a e^2ik, b e^3ik, …)^T,    k∈{0, 2π/L, …, 2π/L(L/2-1)}.
    
    Note that the range of k is halved because the unit cell is doubled.
    
    The eigenvalue equation H|ψ_k⟩ = E|ψ_k⟩ becomes
    
        - g Q a - 2(cos k) b    = E a, 
        
            - (2cos k) a + g Q b    = E b,
    
    which requires E=±√((gQ)^2+(2cos k)^2) to have nontrivial solutions. 
    
    The solutions form two bands. At half filling, all the states in the lower band, which has negative E, are filled with a particle. The charge density wave amplitude for |ψ_k⟩ is n_k=(a^2-b^2)/2. From the eigenvalue equation and the normalization (L/2)(a^2+b^2)=1, we can solve the CDW amplitude for |ψ_k⟩:
    
        n_k = sgn(gQ)/L√((2cos k/g Q)^2 + 1),
    
    where sgn(gQ) is the sign of gQ.
    
    The total charge density wave at half filling is 
    
        n = ∑_k n_k = ∑_k sgn(gQ)/L√((2cos k/g Q)^2 + 1).
    
    The range of k has been described above.
    
    In the infinite lattice limit, L→∞, the sum over k turns into an integral 
    
        n →∫_0^πdk/2πsgn(gQ)/√((2cos k/g Q)^2 + 1) = sgn(gQ) EllipticK[-(2/gQ)^2]/π,
    
    where EllipticK[m]≡π/2_2 F_1(1/2, 1/2;1;m) is the complete elliptic integral of the first kind, _2 F_1(a, b;c;x) is the hypergeometric function <cit.>.
    
    When gQ→ 0, the result says n→ 0. However, the derivative ∂ n/∂ (gQ) diverges at zero as log (1/gQ). More precisely, n∼ -gQ log (gQ) for small gQ.
    
    The property of n as a function of gQ is important, especially for gQ close to zero. This is because when the system stabilizes, the equation of motion tells us k_springQ = g n, i.e. the forces are balanced. This means a straight line n=(k_spring/g^2)gQ that crosses the origin. The number of crossing points between this straight line and the n vs. gQ function determines whether the system have a stable CDW in the long time. Figure <ref> shows the function curves for different lattice size.
    
    
     
    
    For example, in the infinite lattice limit, the derivative diverges at zero. Therefore, any straight line crossing the origin with a finite slope will cross the function at some other points. This means that there always exists a stable CDW for the infinite lattice. However, we can estimate how large the CDW amplitude should be for a small g. Combine the straight line n=(k_spring/g^2)gQ and the asymptotic behavior n∼ -gQ log (gQ), we can solve a non-trivial crossing point at the phonon amplitude Q∼ e^-k_spring/g^2/g. We see that the amplitude decays exponentially when g→0.
    
    However, for a finite system size L, the situation is different. For L=4N, the function n has a finite limit ±1/L when gQ→0, and has a discontinuity at 0. Therefore, a straight line crossing the origin always has non-trivial intersections with the function curve. i.e. There exists CDW with amplitude at least 1/L for all values of g. For L=4N+2, the function n goes to zero when gQ→0, and has a finite slope at 0. When g is small, (k_spring/g^2) exceeds this slope, the straight line only has intersection with the function curve at zero. When g is large, (k_spring/g^2) smaller than this slope, there are non-trivial intersections. Therefore, there is a phase transition due to the finite system size: When g<g_crit, no CDW; when g>g_crit, there is CDW. The system size we study, L=50, is in this situation.
    
    We can analytically solve g_crit. We can take the derivative of Eq. <ref> at the limit to zero:
    
        .∂ n/∂(gQ)|_gQ→ 0^+ = ∑_k 1/2L|cos k|.
    
    By identifying (k_spring/g_crit^2) with this slope, we get
    
        g_crit = √(2k_springL/∑_k 1/|cos k|).
    
    Some numerical values for k_spring=1 are listed in Table. <ref>
    
    For our choice L=50, g_crit=1.1524. The domain wall formation g value for L=50 is around 1.6. Therefore, we pick g∈[1.3,1.4] in our numerical experiment, to avoid the finite size effect and the domain walls.
    
    In the main text, we see the bounded error requires k_spring - g∂ n/∂ Q > 0. We can prove this is true when the system is close to the stable CDW configuration. We apply the self-consistency equation k_spring=gn/Q. From the concavity of the n vs. gQ function when gQ>0, and the convexity when gQ<0, we see n/gQ>∂ n/∂ (gQ) as long as gQ≠ 0. Therefore, k_spring - g∂ n/∂ Q = g^2 (n/gQ - ∂ n/∂ (gQ)) > 0 almost surely, because the gQ=0 case has measure 0. This completes our proof.
    
    
    
    § III. FROM THE EOM OF Q, P TO THE EOM OF Q, P
    
    
     §.§ Approximately constant linear model
    
    
    In the main text, we introduced the “approximately constant linear model”, which has the Hamiltonian: 
    
        Ĥ = Ĥ_q + ∑_i g Ô_i Q_i + ∑_i( 1/2M P_i^2 + 1/2 k Q_i^2 ).
    
    It has the EOM for Q_i and P_i:
    
        d/dt Q_i(t)    = 1/M P_i(t), 
        d/dt P_i(t)    = - g ⟨Ô_i⟩ -k Q_i(t) - γ P_i(t),
    
    where the quantum observable ⟨Ô_i⟩ is computed based on the configuration Q(t).
    
    The EOM above is equally valid for both the exact dynamical path and the ML dynamical path:
    
        d/dt Q_i,Exact(t)    = 1/M P_i,Exact(t), 
        d/dt P_i,Exact(t)    = - g ⟨Ô_i⟩_Exact -k Q_i,Exact(t) - γ P_i,Exact(t),
    
    and 
    
        d/dt Q_i,ML(t)    = 1/M P_i,ML(t), 
        d/dt P_i,ML(t)    = - g ⟨Ô_i⟩_ML -k Q_i,ML(t) - γ P_i,ML(t).
    
    
    Take the difference between the ML EOM and the exact EOM, define the accumulated position and momentum errors q(t) = Q_i,ML(t)-Q_i,Exact(t), p(t) = P_i,ML(t)-P_i,Exact(t), we get 
    
        d/dt q(t)    = 1/M p(t), 
        d/dt p(t)    = - g (⟨Ô_i⟩_ML - ⟨Ô_i⟩_Exact) -k q(t) - γ p(t).
    
    
    However, here the observables depend on different configurations, ⟨Ô_i⟩_ML = ⟨Ô_i⟩_ML(Q_ML(t)), and ⟨Ô_i⟩_Exact = ⟨Ô_i⟩_Exact(Q_Exact(t)). When we take their difference, we not only need to take care about the difference between the prediction methods, we also need to take care about the difference between the configurations Q_ML and Q_Exact. 
    
    We do a Taylor expansion and apply the assumption that the off-diagonal response ∂⟨Ô_i⟩/∂ Q_j (i≠ j) is approximately zero. We can make the difference between the observables into two terms:
    
        ⟨Ô_i⟩_ML(Q_ML(t)) - ⟨Ô_i⟩_Exact(Q_Exact(t))    = ⟨Ô_i⟩_ML(Q_ML(t)) - ⟨Ô_i⟩_Exact(Q_ML(t)) + ∂⟨Ô_i⟩/∂ Q_i q(t) + o(q(t)) 
            
           = δ⟨Ô_i⟩(t) + ∂⟨Ô_i⟩/∂ Q_i q(t) + o(q(t)),
    
    where δ⟨Ô_i⟩(t) = ⟨Ô_i⟩_ML(Q_ML(t)) - ⟨Ô_i⟩_Exact(Q_ML(t)) is the single-step prediction error.
    
    Insert the two terms back to the EOM, we get the same result as in the main text:
    
        d/dt q(t)    = 1/M p(t),
            
        d/dt p(t)    = - g δ⟨Ô_i⟩(t) - g ∂⟨Ô_i⟩/∂ Q_i q(t) - k q(t) - γ p(t) + o(q(t)).
    
    
     §.§ Non-linear model
    
    
    
    In the main text, we then generalized a bit to the non-linear model, which has the Hamiltonian: 
    
        Ĥ = Ĥ_q + ∑_i g Ô_i G_i(Q_i) + ∑_i( 1/2M P_i^2 + V_i(Q_i) ).
    
    It has the EOM for Q_i and P_i:
    
        d/dt Q_i(t)    = 1/M P_i(t), 
        d/dt P_i(t)    = - g ⟨Ô_i⟩ G_i^'(Q_i) - V_i^'(Q_i) - γ P_i(t),
    
    where the quantum observable ⟨Ô_i⟩ is still computed based on the configuration Q(t).
    
    The EOM above is equally valid for both the exact dynamical path and the ML dynamical path:
    
        d/dt Q_i,Exact(t)    = 1/M P_i,Exact(t), 
        d/dt P_i,Exact(t)    = - g ⟨Ô_i⟩_Exact G_i^'(Q_i,Exact) -V_i^'(Q_i,Exact) - γ P_i,Exact(t),
    
    and 
    
        d/dt Q_i,ML(t)    = 1/M P_i,ML(t), 
        d/dt P_i,ML(t)    = - g ⟨Ô_i⟩_ML G_i^'(Q_i,ML) -V_i^'(Q_i,ML) - γ P_i,ML(t).
    
    
    Take the difference between the ML EOM and the exact EOM, define the accumulated position and momentum errors q(t) = Q_i,ML(t)-Q_i,Exact(t), p(t) = P_i,ML(t)-P_i,Exact(t), expand to first order of q(t), we get:
    
        d/dt q(t)    = 1/M p(t), 
        d/dt p(t)    = - g (⟨Ô_i⟩_ML - ⟨Ô_i⟩_Exact) G_i^'(Q_i,ML) - g ⟨Ô_i⟩_Exact G_i^”(Q_i,Exact) q(t) - V_i^”(Q_i,Exact) q(t) - γ p(t) + o(q(t)).
    
    
    We apply the same technique as in the approximately constant linear model. Insert
    
        ⟨Ô_i⟩_ML - ⟨Ô_i⟩_Exact = δ⟨Ô_i⟩(t) + ∂⟨Ô_i⟩/∂ Q_i q(t) + o(q(t)),
    
    we get
    
        d/dt q(t)    = 1/M p(t), 
        d/dt p(t)    = - g δ⟨Ô_i⟩(t) G_i^'(Q_i,ML) - g ∂⟨Ô_i⟩/∂ Q_i G_i^'(Q_i,ML) q(t) - g ⟨Ô_i⟩_Exact G_i^”(Q_i,Exact) q(t) - V_i^”(Q_i,Exact) q(t) - γ p(t) + o(q(t)).
            
           = F(t) - K(t) q(t) - γ p(t) + o(q(t)),
    
    where we define the effective driving force and the effective spring constant as
    
        F(t)    = -g δ⟨Ô_i⟩(t) G_i^'(Q_i,ML(t)), 
        
            K(t)    = g ∂⟨Ô_i⟩/∂ Q_i G_i^'(Q_i,ML(t)) + g ⟨Ô_i⟩_Exact G_i^”(Q_i,Exact(t)) + V_i^”(Q_i,Exact(t)).
    
    
    This EOM is equivalent to a damped spring with time dependent driving force and spring constant. We can prove the amplitude of the spring will not diverge if F(t) is bounded and K(t) does not fluctuate largely.
    
    
    
     §.§ Generic model
    
    
    The generic model has the Hamiltonian 
    
        Ĥ = Ĥ_q + ∑_α,i g_αÔ_α,i G_α,i(P,Q) + H_cl(P,Q),
    
    and the EOM
    
        d/dt Q_j    = ∑_α,i g_α⟨Ô_α,i⟩∂/∂ P_j G_α,i + ∂/∂ P_j H_cl, 
            
        d/dt P_j    = -∑_α,i g_α⟨Ô_α,i⟩∂/∂ Q_j G_α,i - ∂/∂ Q_j H_cl - γ P_j.
    
    
    We can repeat the techniques above: writing down the EOM for ML and the exact simulation, making a difference of the EOM, defining q_i and p_i (here the index i cannot be suppressed), doing Taylor expansion, and combining terms up to the first order. After similar derivations, we can get the result
    
        d/dt q_i    = ∑_j [ q_j 𝒦_q_j,q_i(P,Q) + p_j 𝒦_p_j,q_i(P,Q) ] + ℱ_q_i + o(q,p),
            
        d/dt p_i    = ∑_j [ q_j 𝒦_q_j,p_i(P,Q) + p_j 𝒦_p_j,p_i(P,Q) ] + ℱ_p_i - γ p_i + o(q,p),
    
    where
    
        𝒦_q_j,q_i   = ∑_α,k g_α( ∂⟨Ô_α,k⟩/∂ Q_j∂ G_α,k/∂ P_i + ⟨Ô_α,k⟩∂^2 G_α,k/∂ Q_j ∂ P_i) + ∂^2 H_cl/∂ Q_j ∂ P_i,
            
        𝒦_p_j,q_i   = ∑_α,k g_α( ∂⟨Ô_α,k⟩/∂ P_j∂ G_α,k/∂ P_i + ⟨Ô_α,k⟩∂^2 G_α,k/∂ P_j ∂ P_i) + ∂^2 H_cl/∂ P_j ∂ P_i,
            
        𝒦_q_j,p_i   = - ∑_α,k g_α( ∂⟨Ô_α,k⟩/∂ Q_j∂ G_α,k/∂ Q_i + ⟨Ô_α,k⟩∂^2 G_α,k/∂ Q_j ∂ Q_i) - ∂^2 H_cl/∂ Q_j ∂ Q_i,
            
        𝒦_p_j,p_i   = - ∑_α,k g_α( ∂⟨Ô_α,k⟩/∂ P_j∂ G_α,k/∂ Q_i + ⟨Ô_α,k⟩∂^2 G_α,k/∂ P_j ∂ Q_i) - ∂^2 H_cl/∂ P_j ∂ Q_i,
            
        ℱ_q_i   = ∑_α,j g_αδ⟨Ô_α,j⟩(t)∂/∂ P_iG_α,j(P,Q),
            
        ℱ_p_i   = -∑_α,j g_αδ⟨Ô_α,j⟩(t)∂/∂ Q_iG_α,j(P,Q).
    
    
    § IV. PROOF OF PROPOSITION <REF>
    
    
    Here we prove that the approximately constant linear model satisfies the Error Bounded Property in Def. <ref> if K>0. We use the analogy to a damped harmonic oscillating spring, phrasing the error force F(t) in the main text as the “driving force” and the error stiffness K in the main text as the “spring constant”. For a damped spring, given a bounded driving force, even if the force is tuned to drive the spring optimally, as long as K, γ>0, the spring cannot be driven to infinite amplitude. This can be seen by the following worst-case analysis. In a worst case scenario, the driving force is set to be the maximal value in the direction of the spring movement. We know a constant driving force means a shift of the reference point of the spring. Therefore, by shifting the reference point back and forth, the driving force at most linearly increases the amplitude with the motion cycles of the spring. However, a finite damping decreases the amplitude of the spring by a constant factor in each cycle. Therefore, for a large enough initial amplitude, the decrement of the amplitude due to the damping must exceed the increment of the amplitude due to the driving force in the cycle, and thus the amplitude in the next cycle must be smaller than the initial one, which tells us that the damped spring cannot be driven to infinite amplitude.
    
    Now we analyze the relation between the maximal amplitude q, the maximal momentum p, and the maximal driving force F, using the dimensional analysis. At t_init, there are no accumulated errors, i.e. q(t_init)=p(t_init)=0. Therefore, the only physical quantity that carries the dimension of length is the maximal driving force [F]=[M][L][T]^-2, where [M], [L], and [T] are the dimension of mass, length, and time, respectively. Note that the maximal amplitude [q]=[L] and the maximal momentum p=[L][T]^-1 both carries power one of the length dimension [L]. Therefore, by dimensional analysis, they are both proportional to the maximal driving force F. i.e. ∃ C_q, C_p such that q = C_q F, p = C_p F.
    
    For any ϵ>0, if |δ⟨Ô_i⟩(t)|^2≤ϵ, by the definition F(t)=-gδ⟨Ô_i⟩(t), we have |F(t)|≤ |g|√(ϵ). By the analogy above, it means that the maximal force F=|g|√(ϵ). Therefore, by the meaning of the maximal amplitude and the maximal momentum, there are |q(t)|≤q=C_q F=C_q|g|√(ϵ) and |p(t)|≤p=C_p F=C_p|g|√(ϵ). Define two new constants C_q = |g|C_q and C_p = |g|C_p, we get |q(t)|≤ C_q√(ϵ) and |p(t)|≤ C_p√(ϵ), which is what we want to show. This completes the proof of Prop. <ref>. 
    
    § IV. PROOF OF THEOREM <REF> (ERROR BOUNDED CONDITION FOR NON-LINEAR MODEL)
    
    
    
    Here we present the formal version of Thm. <ref> in the main text. We use the analogy to a damped harmonic oscillating spring, phrasing the error force F(t) in the main text as the “driving force” and the error stiffness K(t) in the main text as the “spring constant”.
    
    
    
        Consider a damped harmonic oscillator with the following EOM:
        
        d/dt q(t)    = 1/M p(t), 
        d/dt p(t)    = F(t) - K(t) q(t) - γ p(t) + o(q(t)),
    
        where q is the classical coordinate, p is the classical momentum, t is time, M is the mass of the oscillator, F(t) is a time-dependent driving force, K(t) is a time-dependent spring constant, γ is the damping coefficient.The initial condition is q(t_init)=p(t_init)=0.
    
        Given K_max>K_min>M(γ/2)^2 satisfying the following inequality:
        
        K_max/K_min < exp[2(arctan ω_min/ω_min + π - arctan ω_max/ω_max)],
    
        where ω_max=√(K_max/M(γ/2)^2-1), ω_min=√(K_min/M(γ/2)^2-1).
    
        If K(t)∈[K_min,K_max] for all t, and G_α,i(P,Q) and its first derivatives are bounded, then the Error Bounded Property in Def. <ref> is satisfied, i.e. ∃ C_q, C_p>0 such that ∀ϵ>0, if |δ⟨Ô_i⟩(t)|^2≤ϵ for all t, then there are 
        
        |q(t)| ≤ C_q√(ϵ),    |p(t)| ≤ C_p√(ϵ),   for all  t.
    Proof. Consider a damped harmonic oscillator with the following EOM:
    
        d/dt q(t)    = 1/M p(t), 
        d/dt p(t)    = F(t) - K(t) q(t) - γ p(t) + o(q(t)),
    
    where q is the classical coordinate, p is the classical momentum, t is time, M is the mass of the oscillator, F(t) is a time-dependent driving force, K(t) is a time-dependent spring constant, γ > 0 is the damping coefficient. We assume F(t)∈[-F,F], K(t)∈[K_min, K_max], where K_max > K_min > M(γ/2)^2. Note that with Eq. <ref>, F(t) = -g δ⟨Ô_i⟩(t) G_i^'(Q_i,ML(t)), the assumption |δ⟨Ô_i⟩(t)|^2≤ϵ implies F≤ O(√(ϵ)) given bounded G_i^'.
    
    We consider the worst case scenario. The mass on the spring starts from one side with a large amplitude. As the mass moving towards the other side, the driving force always keeps the maximal value towards the other side. Before the mass crosses the origin, the spring constant is set as K_max to maximize the drag towards the other side. After the mass crosses the origin, the spring constant is set as K_min to minimize the burden for its moving as far as possible. In this worst case scenario, the mass will stop at its largest amplitude on the other side. If this amplitude on the other side is smaller than the one it started with, than the spring cannot have diverging amplitude, and therefore the error converges.
    
    We now analyze this worst case scenario. Assume the mass starts at q(-t_0)=-q_0, q(-t_0)=0. At time t=0 it crosses the origin, q(0)=0, with a velocity q (0) = v. At time t=t_1 it stops on the other side at q(t_1)=q_1, q(t_1)=0.
    
    Before the mass crosses the origin, the spring constant is K_max. The EOM is
    
        q(t) = - γq(t) -K_max/M q(t) + F/M,    t∈[-t_0, 0].
    
    The EOM has a general solution
    
        q(t) = A e^-γ/2 tsinΩ_max t + B e^-γ/2 tcosΩ_max t + F/K_max,
    
    where Ω_max = √(K_max/M - (γ/2)^2).
    
    Inserting the boundary condition at t=0, we get equations
    
        q(0)    = 0 = B + F/K_max, 
        q(0)    = v = Ω_max A - γ/2 B,
    
    from which we solve
    
        A    = 1/Ω_max(v-γ/2F/K_max), 
        
            B    = -F/K_max.
    
    
    Inserting the boundary condition at t=-t_0, we get equations
    
        q(-t_0)    = -q_0 = - A e^γ/2 t_0sinΩ_max t_0 + B e^γ/2 t_0cosΩ_max t_0 + F/K_max, 
        q(-t_0)    = 0 = A (γ/2 e^γ/2 t_0sinΩ_max t_0 + Ω_max e^γ/2 t_0cosΩ_max t_0 ) + B( - γ/2 e^γ/2 t_0cosΩ_max t_0 + Ω_max e^γ/2 t_0sinΩ_max t_0 ).
    
    We can solve t_0 from the second equation:
    
        t_0    = 1/Ω_maxarccot(γ/2)A+Ω_maxB/-Ω_maxA+(γ/2)B
           = 1/Ω_maxarccot(-(γ/2)v - F/M/Ω_maxv),
    
    where we have inserted the solution of A and B. We can then insert everything into the first equation to solve q_0, but we do not do it now.
    
    Let's also take a look at the time after the mass crosses the origin. The spring constant changes to K_min. The EOM is 
    
        q(t) = - γq(t) -K_min/M q(t) + F/M,    t∈[0, t_1].
    
    The EOM has a general solution
    
        q(t) = C e^-γ/2 tsinΩ_min t + D e^-γ/2 tcosΩ_min t + F/K_min,
    
    where Ω_min = √(K_min/M - (γ/2)^2).
    
    Inserting the boundary condition at t=0, we get equations
    
        q(0)    = 0 = D + F/K_min, 
        q(0)    = v = Ω_min C - γ/2 D,
    
    from which we solve
    
        C    = 1/Ω_min(v-γ/2F/K_min), 
        
            D    = -F/K_min.
    
    
    Inserting the boundary condition at t=t_1, we get equations
    
        q(t_1)    = q_1 = C e^-γ/2 t_1sinΩ_min t_1 + D e^-γ/2 t_1cosΩ_min t_1 + F/K_min, 
        q(t_1)    = 0 = C ( - γ/2 e^-γ/2 t_1sinΩ_min t_1 + Ω_min e^-γ/2 t_1cosΩ_min t_1 ) + D( - γ/2 e^-γ/2 t_1cosΩ_min t_1 - Ω_min e^-γ/2 t_1sinΩ_min t_1 ).
    
    We can solve t_1 from the second equation:
    
        t_1    = 1/Ω_minarccot(γ/2)C+Ω_minD/Ω_minC-(γ/2)D
           = 1/Ω_minarccot(γ/2)v - F/M/Ω_minv,
    
    where we have inserted the solution of C and D. Note that comparing to the case before, there is no minus sign in the arccot function. 
    
    To have a bounded error, we want that for large enough q_0, there is q_1/q_0<1, where q_0 is the starting amplitude and q_1 is the stopping amplitude. i.e. The amplitude never diverges. Iteratively we consider q_1 as the next starting amplitude, and we can see the sequence of amplitudes converges in the long time. Note that the amplitude has the dimension of length, and among all given parameters, only the maximal force F carries the dimension of length. Using dimension analysis, we see that the maximal amplitude the system could reach will be proportional to the maximal force F. The maximal momentum the system could reach will also be proportional to F from dimension analysis. 
    i.e. q(t) ≤ O(F), p(t) ≤ O(F). Combining with the assumption F≤ O(√(ϵ)), it leads to q(t) ≤ O(√(ϵ)), p(t) ≤ O(√(ϵ)), which is what we want to show.
    
    
    Now the final step is to simplify the condition q_1/q_0<1. Note that for arbitrarily large q_0, the velocity v can also be arbitrarily large. Therefore, we can consider the limit F/v→ 0. Under this limit,
    
        A/v    →1/Ω_max, 
        
            B/v    → 0, 
        
            C/v    →1/Ω_min, 
        
            D/v    → 0, 
        
            t_0    →1/Ω_maxarccot(-(γ/2)/Ω_max) = π - arctan(2Ω_max/γ)/Ω_max, 
        
            t_1    →1/Ω_minarccot(γ/2)/Ω_min = arctan(2Ω_min/γ)/Ω_min.
    
    
    Finally,
    
        q_1/q_0   = C e^-γ/2 t_1sinΩ_min t_1 + D e^-γ/2 t_1cosΩ_min t_1 + F/K_min/A e^γ/2 t_0sinΩ_max t_0 - B e^γ/2 t_0cosΩ_max t_0 - F/K_max
           →(1/Ω_min) e^-γ/2 t_1^* (Ω_min/√(K_min/M)) /(1/Ω_max) e^γ/2 t_0^* (Ω_max/√(K_max/M)) 
           = √(K_max/K_min) e^-γ/2 (t_0^* + t_1^*),
    
    where we have used the equality: sin[arccot(-(γ/2)/Ω_max)] = Ω_max/√(K_max/M), sin[arccot(γ/2)/Ω_min] = Ω_min/√(K_min/M). Here t_0^* and t_1^* are the limiting values in Eq. <ref> and <ref>, respectively.
    
    We define ω_max = Ω_max/(γ/2) = √(K_max/M(γ/2)^2 - 1) and ω_min = Ω_min/(γ/2) = √(K_min/M(γ/2)^2 - 1). After rearranging the terms, we simplifies the condition q_1/q_0<1 into
    
        K_max/K_min < exp[2(arctan ω_min/ω_min + π - arctan ω_max/ω_max)],
    
    which is the inequality we see in Eq. <ref> in Thm. <ref>. This completes the proof of Thm. <ref>. 
    
    
    Note that the inequality in Eq. <ref> in Thm. <ref> is satisfied in the limit K_max-K_min/K_min→ 0. We can see this by take a log on both sides of the inequality and get
    
        log(1+K_max-K_min/K_min) < 2(arctan ω_min/ω_min + π - arctan ω_max/ω_max),
    
    whose left hand side is O(K_max-K_min/K_min) and right hand side is O(1) because ω_min≈ω_max. Therefore, in the limit K_max-K_min/K_min→ 0, the inequality always holds. This agrees with the Error Bounded Property of the approximately constant linear model stated in Prop. <ref>. (Actually the approximately constant linear model satisfies the Error Bounded Property under a more general condition: instead of K>M(γ/2)^2 we only need K>0 for the approximately constant linear model.)
    
    
    
    
    If we transform the coordinates of the system to their normal modes, because the normal modes are also described by damped harmonic oscillators, all the analysis works the same for normal modes. This addresses the comment under Thm. <ref> in the main text.
    
    In Figure <ref>, we show an example of the fluctuation of the error stiffness K(t)=k-g∂ n/∂ Q during the Holstein model dynamics. We see that the error stiffness K(t)>M(γ/2)^2=0.0025, where M=1 and γ=0.1. The highest and lowest points in the plot are 0.7104 and 0.5011. We insert these numbers into Eq. <ref>. The left hand side of the condition results in 1.418, and the right hand side 1.501. Therefore, we confirm that the inequality in Eq. <ref> in Thm. <ref> is satisfied during the time range we are interested in, and thus the Error Bounded Property is satisfied in the Holstein model.
    
    
    
    § V. PROVABLY EFFICIENT ADIABATIC LEARNING
    
    
    Here we provide more details about how the PEAL algorithm is performed.
    
    Step I: Collecting data. To be able to train a learning model, we first need to collect data samples from the distribution we are interested in. For example, one can sample a few different initial conditions, and use either classical solver or quantum computer to obtain a few dynamical evolution paths. The “configuration”-“observable” pairs sampled from those paths can serve as the data set we need. If one wants to predict other k-local observable that is not involved in the dynamics, one can compute that observable along the paths and collect the data for the next training step as well. 
    
    Step II: ML training. A provable efficient ML algorithm for predicting ground state properties was presented recently <cit.>. Here we apply this algorithm to adiabatic dynamics. On the data set collected in Step I, a nonlinear feature map with geometrically local region information is performed. A model is trained with an l_1-regularized regression (LASSO) on the features. Hyperparameters in the model are properly chosen. One can train a single model for one type of observable if the system has translation symmetry. The same training process can be applied to the data set of observable that is not involved in the dynamics. Moreover, the learning model can be transferred to unseen physical parameter g_α in the Hamiltonian. An example is shown in our numerical experiment, also illustrated in Fig. <ref> in the main text.
    
    Step III: ML prediction. For dynamical evolution paths with unseen initial conditions, we can use the PEAL prediction to speed up the simulation, instead of repeatedly using costly classical solver or quantum computer. The dynamical simulation is realized by iteratively updating the classical degree of freedom and the quantum degree of freedom. The classical updating step is done by classical ODE solver. The quantum updating step is done by ML prediction. In each iteration, we can add an extra correction step to achieve symmetry-preserving PEAL (see details in the following section). The provably efficiency and controllable error of the PEAL algorithm will be presented later in Thm. <ref>.
    
    Before we formally present Thm. <ref>, we would like to mention two other theorems introduced in other works. In Ref. <cit.>, the authors prove two theorems:
    
    
    Consider any family of n-qubit geometrically-local Hamiltonians  
    {H(x): x ∈ [-1,1]^m } in a finite spatial dimension, such that each local term in H(x) depends smoothly on x, and the smallest eigenvalue and the next smallest eigenvalues have a constant gap ≥Ω(1) between them. Then the ground state properties can be efficiently predicted. 
    
        
    Given n, δ > 0, 1/e > ϵ > 0 and a training data set {x_t, y_t}_t=1^N of size
    
        N = log(n/δ)2^polylog(1/ϵ),
    
    where x_t is sampled from an unknown distribution D and |y_t - Tr(O ρ(x_t))| ≤ϵ for any observable O with eigenvalues between -1 and 1 that can be written as a sum of geometrically local observables. With a proper choice of the efficiently computable hyperparameters δ_1, δ_2, and B, the learned function h^*(x) = 𝐰^* ·ϕ(z) satisfies
    
        𝔼_x ∼ D| h^*(z) - Tr(O ρ(z)) |^2 ≤ϵ
    
    with probability at least 1 - δ. The training and prediction time of the classical ML model are bounded by 𝒪(nN) = n log(n/δ) 2^polylog(1/ϵ).
    
    
    
    The request in the first one, that the Hamiltonian has a constant gap, can be soften into that the correlation length has a finite upper bound. 
    
    
    
    
    Now we present the formal version of Thm. <ref> in the main text.
    
    
        Given n,δ,η>0, 1/e>ϵ>0 and a training data set {(g_α, P,Q)_l,⟨Ô_α,i⟩_l}_l=1^N of size
        
        N=log(n/δ)2^polylog(1/ϵ),
    
        where l∈{1,2,…,N} is the index of data points, (g_α)_l is the coupling constant used when collecting the data, (P,Q)_l is classical variables sampled from exact simulation data, Ô_α,i is an observable with eigenvalues in [-1,1] that can be written as a sum of geometrically local observables, ⟨Ô_α,i⟩_l is the ground state expectation value with an n-qubit gapped geometrically local Hamiltonian Ĥ((P,Q)_l). Apply an ML predicted model ℳ learned with a proper choice of the efficiently computable hyperparameters. When the Error Bounded Property in Def. <ref> is satisfied, a T-step PEAL prediction Q_i,PEAL(T) comparing to the exact dynamical process Q_i,Exact(T), i.e. the accumulated error, has an error bound
        
        |Q_i,PEAL(T) - Q_i,Exact(T)| ≤ O(√(Tϵ/η))
    
        with probability ℙ≥ 1-η-δ.
    
        If further assume the learning error of model ℳ is sub-Gaussian distributed, the error bound can be improved into 
        
        |Q_i,PEAL(T) - Q_i,Exact(T)| ≤ O(√(2ϵlog(2T/η)))
    
        with probability ℙ≥ 1-η-δ.
    
        If further more assume the learning error of model ℳ is bounded by √(ϵ) almost surely, the error bound can be further improved into
        
        |Q_i,PEAL(T) - Q_i,Exact(T)| ≤ O(√(ϵ))
    
        with probability ℙ≥ 1-δ, i.e. the error is bounded by a constant.
    
        The accumulated error of classical variables P_i and all k-local, bounded quantum observables also have the same scaling as the accumulated error of Q_i.
    Proof. To be able to apply the Error Bounded Property in Def. <ref>, we request that the prediction errors |δ⟨Ô⟩| = |⟨Ô⟩_PEAL(P_PEAL,Q_PEAL) - ⟨Ô⟩_Exact(P_PEAL,Q_PEAL)| are upper bounded for all T number of time steps. From the Thm. <ref> above, we learn
    
        𝔼 |δ⟨Ô⟩|^2 ≤ϵ,
    
    from which we can estimate the probability of a single-shot prediction error going beyond a threshold:
    
        ℙ[|δ⟨Ô⟩| ≥ A √(ϵ)]    = ℙ[|δ⟨Ô⟩|^2 ≥ A^2 ϵ] 
            
           ≤𝔼 |δ⟨Ô⟩|^2/A^2 ϵ
           ≤1/A^2,
    
    where A ≥ 0 is a constant factor we choose to describe the threshold. In the second line, we apply the Markov's inequality because |δ⟨Ô⟩|^2 is non-negative.
    
    For a dynamical simulation process with T number of prediction stpes, the probability of any prediction error going beyond the threshold is controlled by the union bound:
    
        ℙ[⋃_i=1^T{|δ⟨Ô⟩(t_i)| ≥ A √(ϵ)}] ≤∑_i=1^Tℙ[|δ⟨Ô⟩(t_i)| ≥ A √(ϵ)] ≤T/A^2.
    
    
    Taking into account that the trained model could have at most δ probability of failure, we get the probability of not being able to apply our PEAL method is at most (T/A^2 + δ) by another union bound. 
    
    When we are able to control all the prediction errors under the threshold A√(ϵ), we have 
        |q(t)|=|Q_i,PEAL(T) - Q_i,Exact(T)|≤ O(A√(ϵ))
    
    by the Error Bounded Property in Def. <ref>.  From the analysis above, we see this is applicable with a probability at least (1 - T/A^2 - δ).
    
    For given η>0, we choose the threshold factor A=√(T/η), such that η = T/A^2. Therefore, we get our most general bound
    
        |Q_i,PEAL(T) - Q_i,Exact(T)|≤ O(√(Tϵ/η))
    
    with probability ℙ≥ 1 - η - δ.
    
    This (1 - η - δ) behavior in the probability is expected, because we have an adiabatic hybrid algorithm with both quantum and classical updates. The δ failure comes from approximating the quantum algorithm, and the η failure comes from the adiabatic quantum-classical dynamics.
    
    This most general bound is showing a √(T) behavior, which happens to be the same as a diffusion model. 
    
    We can improve this bound if we can acquire more knowledge on the distribution of the prediction error δ⟨Ô⟩. For example, if δ⟨Ô⟩ has a sub-Gaussian distribution with a moment-generating function (MGF) 
    
        M_δ⟨Ô⟩(s) = 𝔼[ e^sδ⟨Ô⟩] ≤expϵ s^2/2,
    
    then we can apply the Chernoff bound and get
    
        ℙ[|δ⟨Ô⟩| ≥ B √(ϵ)] ≤ 2exp(-(B√(ϵ))^2/2ϵ) = 2 exp(-B^2/2),
    
    where B is also a threshold factor.
    
    Repeat the union bound argument, we can get |Q_i,PEAL(T) - Q_i,Exact(T)|≤ O(B√(ϵ)) with a probability at least (1 - 2 T exp(-B^2/2) - δ). For given η>0, we choose B = √(2log(2T/η)), such that η = 2 T exp(-B^2/2). Therefore, we get our bound with assuming sub-Gaussian prediction error
    
        |Q_i,PEAL(T) - Q_i,Exact(T)|≤ O(√(2ϵlog(2T/η)))
    
    with probability ℙ≥ 1 - η - δ.
    
    This improves the general √(T) behavior to a better √(log(T)) behavior.
    
    We can improve this bound even further if we add even stronger assumption on the distribution of the prediction error δ⟨Ô⟩. If we assume there exists a constant factor C such that |δ⟨Ô⟩| ≤ C√(ϵ) almost surely, then we can directly apply our Thm. <ref> and get
    
        |Q_i,PEAL(T) - Q_i,Exact(T)|≤ O(√(ϵ))
    
    with probability ℙ≥ 1 - δ.
    
    This is a constant bound which is independent on T. i.e. The accumulated position error of the dynamical simulation is bounded by a constant.
    
    Because we also have a controlled |p(t)| in Thm. <ref>, all the arguments above are also true if replacing Q by P. 
    
    With the proof of the following lemma, we complete the proof of Thm. <ref>.  
        All k-local, bounded observables have the same provably efficient bound under PEAL, no matter whether they are involved in the dynamics or not. For such an observable Ω̂, given a training data set {(P,Q)_l,⟨Ω̂⟩_l}_l=1^N of size N same as in Thm. <ref>, under various assumptions in Thm. <ref>, the accumulated error |⟨Ω̂⟩_PEAL(P_PEAL(T),Q_PEAL(T)) - ⟨Ω̂⟩_Exact(P_Exact(T),Q_Exact(T))| holds similar bounds as in Eq. <ref>, <ref>, and <ref>.
    Proof. The accumulated error for an observable is defined as 
    
        |⟨Ω̂⟩_PEAL(P_PEAL(T),Q_PEAL(T)) - ⟨Ω̂⟩_Exact(P_Exact(T),Q_Exact(T))|.
    
    
    There are two source of error in this operator learning: one from the PEAL path (P_PEAL(T),Q_PEAL(T)) deviate from the exact simulation (P_Exact(T),Q_Exact(T)), the other from the single-step ML prediction error. In terms of formula, we have
    
        |⟨Ω̂⟩_PEAL(P_PEAL(T),Q_PEAL(T)) - ⟨Ω̂⟩_Exact(P_Exact(T),Q_Exact(T))| 
        ≤    |⟨Ω̂⟩_PEAL(P_PEAL(T),Q_PEAL(T)) - ⟨Ω̂⟩_Exact(P_PEAL(T),Q_PEAL(T))|
            
            + |⟨Ω̂⟩_Exact(P_PEAL(T),Q_PEAL(T)) - ⟨Ω̂⟩_Exact(P_Exact(T),Q_Exact(T))|.
    
    
    We first analyze the second term by the perturbation theory. We have
    
        ⟨Ω̂⟩_Exact(P_PEAL(T),Q_PEAL(T))    = ⟨GS_PEAL|Ω̂|GS_PEAL⟩, 
            
        ⟨Ω̂⟩_Exact(P_Exact(T),Q_Exact(T))    = ⟨GS_Exact|Ω̂|GS_Exact⟩,
    
    where |GS_PEAL/Exact⟩ = |GS(P_PEAL/Exact(T),Q_PEAL/Exact(T))⟩ is the ground state wavefunction for the Hamiltonian with classical parameters (P_PEAL/Exact(T),Q_PEAL/Exact(T)), respectively.
    
    Note that (P,Q) are served as parameters of the Hamiltonian. For slightly different (P_PEAL(T),Q_PEAL(T)) and (P_Exact(T),Q_Exact(T)), the corresponding Hamiltonians are slightly different, and thus the difference between the ground states can be analyzed by first order perturbation
    
        |GS_PEAL⟩ - |GS_Exact⟩ = ∑_n≥1⟨ n_Exact|V̂|GS_Exact⟩/E_GS,Exact - E_n,Exact |n_Exact⟩ + o(p, q),
    
    where |n_Exact⟩ is the n-th excited state for the Hamiltonian with classical parameters (P_Exact(T),Q_Exact(T)). E_GS,Exact and E_n,Exact are energies of |GS_PEAL⟩ and |n_Exact⟩, respectively. We define the accumulated errors of classical parameters p=P_PEAL(T)-P_Exact(T), q=Q_PEAL(T)-Q_Exact(T). V̂ is the first order Taylor expansion of the Hamiltonian
    
        V̂ = ∑_i (∂Ĥ/∂ Q_i q_i + ∂Ĥ/∂ P_i p_i).
    
    
    Note the the dependence on (p, q) in Eq. <ref> only appears in V̂, which is linear in (p, q). Moreover, Eq. <ref> is bounded by looking at its left hand side. Therefore, we see that overall |GS_PEAL⟩ - |GS_Exact⟩ is of order O(p, q), and is controlled by the various bounds in Thm. <ref>.
    
    Therefore, the second term can be controlled by
    
        |⟨Ω̂⟩_Exact(P_PEAL(T),Q_PEAL(T)) - ⟨Ω̂⟩_Exact(P_Exact(T),Q_Exact(T))| 
            
        
            =     |⟨GS_PEAL|Ω̂|GS_PEAL⟩ - ⟨GS_Exact|Ω̂|GS_Exact⟩|
            
        ≤    |⟨GS_PEAL|Ω̂|GS_PEAL⟩ - ⟨GS_PEAL|Ω̂|GS_Exact⟩| + |⟨GS_PEAL|Ω̂|GS_Exact⟩ - ⟨GS_Exact|Ω̂|GS_Exact⟩|
            
        
            =     |⟨GS_PEAL|Ω̂ (|GS_PEAL⟩ - |GS_Exact⟩ )| + | (⟨GS_PEAL| - ⟨GS_Exact| ) Ω̂|GS_Exact⟩|,
    
    where (|GS_PEAL⟩ - |GS_Exact⟩ ) and (⟨GS_PEAL| - ⟨GS_Exact| ) is of order O(p, q) and everything else is O(1).
    Therefore the last line in Eq. <ref> is of order O(p, q) because both terms are of order O(p, q). Hence, the second error term satisfies the same bounds as Q does in Thm. <ref>, which in the proof we analyze the probability of all T steps' predictions having errors under a certain threshold.
    
    The first term is the single-step prediction error δ⟨Ω̂⟩ at T-th step, which is analyzed in the proof of Thm. <ref>. Combining the first term and the second term is equivalent to requiring (T+1) steps' prediction having errors under a certain threshold, which can be analyzed with the same technique we used in the proof of Thm. <ref>. More precisely, we have (T+1) terms in the union bound in Eq. <ref>. This is equivalent to having results in Thm. <ref> by changing T into (T+1). However, such change does not affect the scaling behaviors of the bounds with T.
    
    Therefore, the first and second terms combined also satisfy the same bounds as Q does in Thm. <ref>. This completes the proof of Lemma <ref>, that the accumulated error of any k-local observable satisfy the same bounds in Thm. <ref>. With this lemma proved, we complete the entire proof of Thm. <ref>. 
    
    § VI. PEAL IMPLEMENTATION
    
    
    Here we provide more details about how we implement the PEAL algorithm in our numerical experiment on the Holstein model.
    
    PEAL Step I: Collecting data. For each of the six training values g∈{1.3, 1.32, 1.34, 1.36, 1.38, 1.4}, we randomly sampled 18 independent initial conditions Q(t=0)iid∼𝒩(0,Q_v)^⊗ L with the standard deviation Q_v=0.2. L=50 is the system size. We set P(t=0)=0. For each initial condition, We iteratively perform quantum steps and classical steps, for 10,000 epochs, to generate a dynamical evolution path. We set the simulation time step dt=0.01 and the total time is 100. In the quantum steps, we use exact diagonalization (ED) to obtain the quantum observables. In the classical steps, we use fourth order Runge-Kutta method. For each path, we randomly sampled 500 pairs of (Q_shift-i(t), n_i(t)) data, where t is a random time step and i is a random site. Q_shift-i is the Q vector with elements rolled by a shift of i, so that the i-th element is put on the first place. Overall, we have 6×18×500=54,000 data pairs, which effectively serve as 1,080 samples on the size L=50 system.
    
    The purpose to use this kind of site-shifting data pairs is to respect the translation symmetry of the system. With periodic boundary condition, the system has translation symmetry. Therefore, we can use the same model to predict n_i on any location in the system, as long as we also translate the Q vector accordingly.
    
    PEAL Step II: ML training. We use the provably efficient ML algorithm introduced in <cit.>. For each length-50 vector Q, we generate 50 local regions, each local region having radius 1 and consisting of 3 elements. We perform a random Fourier feature map on the local regions, generate R number of cosine features and R number of sine features for each local region. R is a hyperparameter chosen from {5, 10, 20, 40, 80, 160}. The frequency factor γ_ω for the random Fourier feature map is another hyperparameter chosen from {0.3, 0.6, 1, 2, 3, 6, 10, 20}. We trained an l_1-regularized regression (LASSO) on the non-linear features. The strength of regularization α is determined by LassoLars method with 4-fold cross validation <cit.>. For 1,024 samples, the grid-search best hyperparameters are R=20, γ_ω=6, and α=4.98×10^-6. The best model has 316 non-zero linear coefficients after the LASSO feature selection. 
    
    PEAL Step III: ML prediction. To speed up the dynamical simulation, we replace the ED steps by the prediction from the learning model. We take care of the U(1) global symmetry of the system. After each ML boosted quantum step, we subtract the electron density n_i by the mean of their excess over 1/2. Therefore, we can make the total electron number conserved at half filling, and respect the U(1) global symmetry of the system. See more details in the following section.
    
    
    
    § VII. DETAILS OF SYMMETRY-PRESERVING PEAL
    
    
    In the application of PEAL to specific systems with symmetries, we would like to preserve the symmetries during our numerical simulation. Here we present symmetry-preserving PEAL for two types of symmetries: The U(1) global symmetry and the translation symmetry.
    
    To preserve the U(1) global symmetry, we would like to have the total charge conservation at each time step during the dynamics. When we sum up the model's prediction of the electron charge density at each site of the system, often case the result is not the same as the total charge in the previous time step. We can make a correction on the electron charge density at each site during each time step of the dynamics, in order to conserve the total charge.
    
    There are two possible ways to do the correction. The first one is to uniformly add or subtract a constant on the electron charge density at each site. The second is to uniformly multiply a factor to the electron charge density at each site. However, the second way does not treat an electron (charge density equals to 1) and a hole (charge density equals to 0) equivalently. Therefore, we use the first way to do the correction.
    
    Suppose the learning model ℳ predicts the electron charge density n_i=⟨ c_i^† c_i⟩ on each site i. In order to preserve the U(1) global symmetry, we would like to have the corrected electron charge density n_i such that ∑_i n_i = L/2, assuming at half filling for a system with L sites. It is straightforward to see that we can achieve this charge conservation by doing the correction n_i→n_i = n_i + Δ n with Δ n = 1/L∑_i(1/2-n_i).
    
    Now we prove the error bounds for our PEAL algorithm are still valid with this correction. Suppose each n_i has a prediction error δ n_i=n_i-n_i^*, where n_i^* is the electron charge density computed with QSS, which satisfies ∑_i n_i^* = L/2. We can see Δ n=-1/L∑_i δ n_i, and the error of the corrected electron charge density is δn_i = n_i - n_i^* = δ n_i + Δ n = δ n_i - 1/L∑_i δ n_i.
    
    In the proof of PEAL error bounds, we analyzed all situations by turning the problem into the calculation of the probability of |δ⟨Ô⟩| ≤ D √(ϵ), where the constant D stands for A, B, or C in the proof of Thm. <ref>. Note that when all |δ n_i|≤ D√(ϵ), we have 
    
        |δn_i| = |δ n_i - 1/L∑_i δ n_i| ≤ |δ n_i| + 1/L∑_i |δ n_i| ≤ 2  D √(ϵ),
    
    which means that we are safe to apply every error bounds after we raise the corresponding constant D by a factor of 2. This completes the proof.
    
    When the system has a translation symmetry, such as the one for the periodic chain we studied in the main text, every site i is on equal footing. Therefore, we will get the identical error distribution when we apply a model ℳ_i trained for n_i to another site j to predict n_j. This means that we can save our cost by applying one model repeatedly on every site i instead of training an independent model for every n_i, and we have the error bounds unaffected. Moreover, by doing so, we preserve the translation symmetry in the PEAL prediction: If instead of the original initial condition Q(0)=(Q(0)_0, Q(0)_1, …, Q(0)_L-2, Q(0)_L-1) we input a shifted initial condition (Q(0)_1, Q(0)_2, …, Q(0)_L-1, Q(0)_0), then all outputs from the symmetry-preserving PEAL algorithm will be also shifted correspondingly comparing to the original outputs.
    
    
    
    § VIII. DETAILS OF NUMERICAL EXPERIMENTS
    
    
    We are interested in the quantum dynamics of the Holstein model in Eq. <ref> with a random initial distribution of Q_i(0), and we set initial momentum P_i(0)=0. Under the adiabatic approximation, the quantum dynamics can be further described by three equations:  
    
        Q^'_i(t)    = 1/M P_i 
        
            P^'_i(t)    = -k Q_i + g (n_i - 1/2) - γ P_i 
        
            H_e({Q_i(t)})    = -t ∑_i,j c_i^† c_j - g ∑_i (c_i^†c_i - 1/2) Q_i
    
    where n_i = ⟨ c_i^†c_i ⟩ and γ is the damping coefficient.
    
    In the experiment, we observed three regimes of g values. For a small g value, the final stable configuration of the system has no CDW. This is due to the finite size effect. For the system size L=50 we used, the critical value to generate stable CDW is g_crit=1.152. This critical value can be analytically solved and we present the calculation already in earlier section. For a large g value, domain walls can be generated in the system. The characteristic length of the domain walls decreases when g increases. Therefore, when the system size is much larger than the domain wall characteristic length, domain walls can be relatively far from each other, which makes their interaction exponentially small and the equilibrium time exponentially large. For the system size L=50 we used, the value to generate long-lasting domain walls is g_DW∼ 1.6. For the value g_crit < g < g_DW, we can see a clear CDW developing during the dynamics. Therefore, we collect a data set for g∈{1.3, 1.32, 1.34, 1.36, 1.38, 1.4}, mimicking a uniform distribution in [1.3,1.4].
    
    For each g value, we generate 18 ED dynamical evolution paths with random initial conditions, serving as the training data. We also generate 10 more ED dynamical evolution paths with random unseen initial conditions for each g∈{1.3, 1.32, 1.34, 1.36, 1.38, 1.4} and for each g∈{1.31, 1.33, 1.35, 1.37, 1.39}, serving as the standard learning test set and the transfer learning test set, respectively. In Figure <ref> in the main text, we show how the test error scales with the number of samples used in model training. We trained models with number of samples from 1, 2, 4, ..., to 1,024. (Each sample contains 50 data pairs based on the system size we used.) We see an error scaling law agrees with the predicted logarithmic scaling. The transfer learning test error is only slightly higher than the standard learning test error. The scatter plots of the target values show the model has learned nicely for both the normal case and the transfer learning case.


 Paper Title: Parallel ergotropy: Maximum work extraction via parallel local unitary operations.



APS/123-QED






riccardo.castellano@sns.it

Scuola Normale Superiore, I-56126 Pisa, Italy

ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain

Dipartimento di Fisica dell’Università di Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy




ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain




Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain






donato.farina@unina.it
Physics Department E. Pancini- Università degli Studi di Napoli Federico II, Complesso Universitario Monte S. Angelo- Via Cintia- I-80126 Napoli, Italy

ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain











             



§ ABSTRACT 

Maximum quantum work extraction is generally defined in terms of the ergotropy functional, no matter how experimentally complicated is the implementation of the optimal unitary allowing for it, especially in the case of multipartite systems.

In this framework, 
we consider a quantum battery made up of many interacting sub-systems and study the maximum extractable work  via concurrent local unitary operations on each subsystem.

We call the resulting functional parallel ergotropy.



Focusing on the bipartite case, 
we first observe that
parallel ergotropy outperforms
work extraction via egoistic strategies,
in which
the first agent A extracts locally on its part the maximum available work and the second agent B, subsequently, extracts what is left on the other part.

For the agents, this showcases the need of cooperating for an overall benefit.


Secondly, from the informational point of view, we observe that the parallel capacity of a state can detect entanglement and compare it with the statistical entanglement witness that exploits fluctuations of stochastic work extraction.

Additionally, we face the technical problem of computing parallel ergotropy. 

We derive analytical upper bounds for specific classes of states and Hamiltonians and provide  receipts to obtain numerical upper bounds
via semi-definite programming in the generic case.



Finally, extending the concept of parallel ergotropy, we demonstrate that 
system's free-time evolution and application of local unitaries
allow one to saturate the gap with the ergotropy of the whole system.





                              
Parallel ergotropy: Maximum work extraction via parallel local unitary operations
    Donato Farina
    August 5, 2024
=================================================================================







§ INTRODUCTION


In the realm of classical thermodynamics, storage and transformation of energy embody its very essence. Rapid development of quantum technologies, however, requires revisiting foundational thermodynamic principles in order to address them at a microscopic level. In particular, counter-intuitive phenomena such as quantum superposition and entanglement offer many-body quantum systems as energy storage devices that overcome limitations of classical batteries <cit.>. Indeed, such quantum batteries are characterized by increasing charging power and energy storage capacity. The usefulness of a quantum battery can be captured by two quantities: 1) ergotropy, the maximal amount of work that can be extracted from its state via cyclic unitary dynamics generated by the Hamiltonian of the quantum battery <cit.>, and 2) quantum battery capacity, the maximal amount of work that can be transferred via such unitary cycles <cit.>.

Experimental implementation of a quantum battery composed of several elements can be demanding since the resulting unitary operations may be highly non-local. This limitation can be faced by focusing on a subsystem instead of the entire compound, hence, addressing ergotropy with respect to unitary operations on this subsystem <cit.>. However, it raises the following question: How much work can be extracted from the whole compound via concurrent local work extracting unitaries with respect to each subsystem? In this paper, we explore it by introducing a more experiment-friendly notion of parallel ergotropy that captures concurrent work extraction from each subsystem composing a quantum battery. In particular, we provide its general lower and upper bounds and calculate it explicitly for relevant specific classes of quantum states. In a similar manner, we introduce and study parallel capacity, i.e., quantum battery capacity under concurrent local unitaries. We show that  parallel capacity can serve as an entanglement witness. Finally, we prove that combining free time evolutions and local unitaries on the subsystems, we can always close the gap with the global ergotropy of the system.

The paper is organized as follows. In Sec. <ref>, we set the necessary framework and introduce the ergotropy, local ergotropy, and quantum battery capacity functionals. In <ref>, we introduce the figure of merit of the paper, the parallel ergotropy and parallel capacity functionals. 
In Sec. <ref> we establish general lower and upper bounds for this quantity. In Sec. <ref>, we calculate parallel ergotropy for locally maximally mixed states, 

or for arbitrary states but with null local Hamiltonian terms,

providing an analytic upper bound for quantum systems of arbitrary dimension and the exact value for two-qubit systems. 
In Sec. <ref> we introduce a general numerical technique for upper bounding parallel ergotropy and we perform a comparison among different estimates for two-qubit and two-qutrit systems. In Sec. <ref>, we demonstrate that the parallel capacity functional can be used as an entanglement witness. A summary of the results together with an outlook to possible future developments are presented in Sec. <ref>.





§ PRELIMINARIES



We start by recalling the definition of ergotropy. Let us consider a d-dimensional quantum system S prepared in a state ρ∈ D(ℋ), where D(ℋ) denotes the space of density operators on the Hilbert space ℋ of S, and equipped with a Hamiltonian H. Under cyclic control of the Hamiltonian the state ρ evolves due to a certain unitary operation U inducing a change 

    [ρ H] →[U ρ U^† H]

in the mean energy of S. The latter can be associated with extraction of work from ρ, and the maximal amount of work that can be extracted via unitary operations is defined by the ergotropy functional






    ℰ(ρ, H)    :=   [ρ H] - min_U ∈𝚄(d)( [ U ρ U^† H ] )
       =   [ (ρ - UρU^†) H] ,


where 𝚄(d) is the unitary group of degree d, and U̅ is the optimal unitary operation, whose closed form can be provided in terms of the eigenstates of ρ and H <cit.>. Therefore, (<ref>) establishes an upper bound on energy that can be removed from S in the state ρ. Substituting minimization with maximization in (<ref>) provides a lower bound on change (<ref>) in the mean energy of S characterizing how much it can be charged in the state ρ. The difference between these bounds characterizes, hence, the maximal amount of work that S can transfer as a quantum battery. It can be formalized as quantum battery capacity functional

    𝒞(ρ, H)    :=   max_U ∈𝚄(d) (  [U ρ U^† H ] ) - min_U ∈𝚄(d) (  [U ρ U^† H ] ) 
       =   [ (UρU^† - UρU^† ) H ]  ,

where U and U are the optimal unitaries achieving upper and lower bounds on change (<ref>) in the mean energy of S, respectively, with U being defined in (<ref>).


Experimental implementation of work extracting unitary operation U (as well as work injecting U) can be challenging, for example, if S is multipartite, so that U can result to be highly non-local. Such limitations for global work extraction have been recently explored in Ref. <cit.> and motivate to consider local work extraction, where the unitary cycle is performed on a subsystem of S. In particular, let us consider a bipartite system S shared between two agents A and B. It is equipped with a Hamiltonian H and initially prepared in a state ρ∈ D(ℋ_a ⊗ℋ_b), where ℋ_a and ℋ_b are the Hilbert spaces of subsystems of S associated to A and B, respectively. Then the maximal amount of work that A can extract from S is given by the local ergotropy functional on A
<cit.>








    ℰ_ La(ρ, H)    :=   max_U_a∈𝚄(d_a)( [ ( ρ - (U_a⊗1_b)ρ (U^†_a⊗1_b)) H ] ) 
       =   [ ( ρ - (U_a⊗1_b)ρ (U^†_a⊗1_b)) H ]   ,

where U_a is the optimal unitary operation performed by A, d_a is dimension of the subsystem of S associated to A, and 1_b is an identity operator on ℋ_b. Therefore, for a multipartite system S composed of n subsystems shared among {A_i}_i=1^n agents, we formalize the maximal amount of work that can be extracted from S by A_i as local ergotropy functional on A_i

    ℰ_ La_i(ρ, H)    :=   max_U_a_i∈𝚄(d_a_i)( [ ( ρ - U_ La_iρ U_ La_i^† ) H ] ) 
       =   [ ( ρ - U_ La_iρU_ La_i ) H ]   ,

where we denote U_ La_i := 1_a_1⊗…⊗ U_a_i⊗…⊗1_a_n and U_ La_i := 1_a_1⊗…⊗U_a_i⊗…⊗1_a_n, with U_a_i being the corresponding optimal unitary operation on A_i.
































§ PARALLEL ERGOTROPY






Let us consider a multipartite quantum battery consisting of many interacting subsystems S_1, …, S_N shared between N agents. As anticipated, performing generic global unitaries on the whole system might be experimentally challenging, and it is reasonable to question work extraction under restriction to operations performed locally by each agent. 
While local ergotropy (<ref>) quantifies work that can be extracted by a single agent acting on its subsystem, it cannot catch cooperative work extracting strategies of the agents. In order to fill this gap, we introduce the parallel ergotropy (PE) as the maximum energy extractable via concurrent local unitary operations,

    ℰ_ P(ρ,H) := max_U_P∈𝒰_P^N[(ρ - U_Pρ U_P^†) H],

where 𝒰_ P^N denotes the set of N-parallel unitaries, i.e., unitaries of form U_P = ⊗_i=1^N U_i, with U_i ∈𝚄(d_i) acting on i-th subsystem.
A schematics of parallel ergotropy for a bipartite system, compared to (global) ergotropy <cit.> and local ergotropy from Ref. <cit.>, 
is reported in Fig. <ref>.





A comment on the definition (<ref>) is necessary. As for the local ergotropy (<ref>), the realization of such local unitary evolutions is intended on time scales that are much smaller than any other time scale in the problem.
Put differently,
functional (<ref>) does not allow for slow driving of the local Hamiltonian and thus is time-dependent on the free dynamics of the system. In formulas, defining U_0(t):=exp (-i H t),  


    ℰ_ P(U_0(t) ρ  U_0(t)^†,H) ≠const .
 

Therefore, PE can be regarded as the figure of merit for work extraction only if operations are carried out instantaneously with respect to the dynamics of the system.


We point out two main reasons for which one may stick to this restricted class of operations instead of the bigger set 𝒰_Pex introduced in Section <ref>, which exploits both parallel (hence, local on each subsystem) Hamiltonian control and internal evolution induced H. First, one may need to charge and discharge the battery on a faster time-scale than the one characterizing the dynamics of the system.

Second, the precise time-keeping necessary to perform free evolution operations has its own energy and thermodynamic cost <cit.>, which might be higher than the advantage of reaching generic global unitaries over the local ones. Notice that both arguments are also valid for the local ergotropy introduced in Ref. <cit.>.



In the rest of the paper, we mainly consider a bipartite quantum battery (i.e., N=2). Denoting both parties A and B with associated Hilbert spaces ℋ_a and ℋ_b, respectively. It is described hence by a bipartite state ρ_ab∈ D(ℋ_a ⊗ℋ_b) and equipped with a Hamiltonian

    H = H_a + H_b + V_ab := H_ab ,

where V_ab is the interaction term acting on the joint Hilbert space ℋ_a⊗ℋ_b, and H_a and H_b are the local terms on A and B, respectively. Without loss of generality, we will choose the tree quantities such that _ a[V_ab] = _ b[V_ab] = 0. In what follows, we study the properties of the considered quantum battery with respect to action of 2-parallel unitaries U_a ⊗ U_b, where U_a ∈𝚄(d_a) and U_b ∈𝚄(d_b). In particular, the PE functional (<ref>) reduces to 

    ℰ_ P(ρ_ab, H_ab) = max_U_a, U_b  [ (ρ_ab - (U_a ⊗ U_b) ρ_ab (U_a^†⊗ U_b^†))H_ab] .


Our choice is due to two main reasons. On the one hand, it allows for a more straightforward comparison with the local ergotropy functional (<ref>). On the other hand, despite its higher experimental accessibility compared with the global ergotropy (<ref>), both the analytical and numerical treatments of PE appear to be already rather involved for N=2. 




§ GENERAL PARALLEL BOUNDS AND EGOISTIC WORK EXTRACTING STRATEGIES



In order to simplify expressions involving several unitary operations, given a state ρ, we sometimes represent action of a unitary operator U as a superoperator 𝖴[ρ] := Uρ U^†. By construction, the following hierarchy of the ergotropic quantities introduced above can be established,



    ℰ(ρ, H) ≥ℰ_ P(ρ, H) ≥ℰ_ La(ρ, H) + ℰ_ Lb(𝖴_a[ρ], H) ≥ℰ_ La(ρ, H) ,



where 𝖴_a corresponds to the optimal unitary U_a on A resulting from maximization in (<ref>). Notice that the hierarchy (<ref>) is constructed with respect to the party A. Due to the symmetry, it remains naturally valid with respect to B as well, i.e., when a and b are exchanged in the ergotropic quantities of (<ref>). 

The new quantity ℰ_ La(ρ, H)+ℰ_ Lb(𝖴_a[ρ], H) in (<ref>) represents the maximum work that the agents A and B can extract from the system under an egoistic strategy. This kind of strategy implies a fixed order in which agents act on the bipartite system, so that A applies the optimal unitary U_a maximizing local work extraction with respect to its subsystem, and then B extracts work from the resulting state. This contrasts with the PE implying a cooperative strategy, under which A and B agree on applied local unitaries, optimizing thereby total extracted work. This leads to a natural question: can A and B follow egoistic strategies in order to extract work guaranteed by PE? Generally speaking, the answer is no despite a misleading intuition suggested by commutativity of local unitaries of A and B. 





Let us assume that the competing agents A and B share a 2-qubit battery equipped with a Hamiltonian (<ref>) and can either cooperate (achieving the value of PE) or not on choosing the local work extracting unitary operations. We proceed to show that, for most H_ab, there exist states, for which a cooperative strategy is strictly necessary to achieve the maximal work ℰ_ P(ρ_ab, H_ab), and the following strict inequalities hold,






    ℰ_ P(ρ_ab, H_ab)    >   ℰ_ Lb(ρ_ab, H_ab)+ℰ_ La(𝖴_b[ρ_ab], H_ab) ,  
    ℰ_ P(ρ_ab, H_ab)    >   ℰ_ La(ρ_ab, H_ab)+ℰ_ Lb(𝖴_a[ρ_ab], H_ab)  .



An interesting feature is that, sometimes, the first agent acting on the system has to inject work in order to enable the other agent to complete the optimal cooperative gate. To show this, let us we pick here a specific pair of states and Hamiltonians (see Appendix <ref> for analysis of egoistic strategies for a larger class of states and Hamiltonians). 

Let us assume that the quantum battery is equipped with the Hamiltonian H_ab characterized by the following eigenvectors,




    |E_0⟩:=|00⟩  , |E_1⟩ :=|11⟩, 
          |E_2⟩:=|10⟩, |E_3⟩:=|01⟩ ,


where E_j denote the corresponding energies fulfilling E_0 < E_1 < E_2≤ E_3, and {|0/1⟩} is the computational basis of each qubit. In turn, the state of the battery is given by ρ_ ab = |E_1⟩⟨E_1|. In this case, the egoistic strategies are highly inefficient: it is straightforward to verify that any single local unitary increases the energy of the battery. Therefore, the optimal egoistic strategy in this setup is not implementing any work extracting unitary by each agent at all, and 

    ℰ_ La(ρ_ab, H_ab) = ℰ_ Lb(ρ_ab, H_ab) = 0.



On the other hand, if A and B are allowed to cooperate, a bit flip on each subsystem X ⊗ X can be performed, reaching thereby the ground state and extracting PE (which is equal to global ergotropy (<ref>) as well)

    ℰ_ P(ρ_ab, H_ab) = ℰ(ρ_ab, H_ab) = E_1-E_0>0.


Apart from providing an example of a setup that opens a gap between egoistic and cooperative strategies via (<ref>)–(<ref>), this result demonstrates several interesting features. First of all, we notice that it does not involve coherence, being hence a classical result. 

On the other hand, it suggests that, depending on the order in which the gates are applied, an optimal strategy to extracted the maximal total work requires that one agent has to invest work, while another receives the entire work gain. Indeed, let us assume that A acts first on its subsystem. Then it can apply a bit flip X, reaching thus the state |E_3⟩ and paying an energy cost E_3-E_1. After that, B applies a bit flip X on its subsystem, reaching the ground state and receiving the gain E_3 - E_0. In turn, the total gain is E_1-E_0 as anticipated. Generally speaking, this results form the fact that




    E(ρ_ab)-E(𝖴_a[ρ_ab]) ≠  E(𝖴_b[ρ_ab]) - E((𝖴_a⊗𝖴_b)[ρ_ab]) ,


where E(ρ):= tr [H ρ].

At first, this might seem in contrast with the non-signaling principle. Let us assume that A and B find themselves in laboratories separated by a distance L. If A performs U_a just after Bob has performed U_b, amount of extracted work by A could indicate the choice of U_b. However, it is necessary to take into account that the non-local Hamiltonian H_ab is a valid approximation of the real interaction only on time scales longer than L/c. Therefore, the effective non-relativistic H_ab does not provide a valid description of the dynamics of the system, so that the energy of the quantum battery is not given by tr[H_abρ] if A and B perform the local work extracting unitaries in space-separated events. On time scales longer than L/c the previous example is instead valid, and energy can be transferred among the agents. Indeed, LOCC protocols for energy transmission are known as quantum energy teleportation and extensively studied, e.g., in <cit.>.
 
 











§ PARALLEL BOUNDS IN SPECIFIC CASES
 


Despite its operational simplicity, generally speaking, PE is a complicated functional to compute. 

In what follows, we provide analytic formulae and bounds for PE for a relevant class of states and Hamiltonians which are computable. 



 §.§ Locally maximally mixed states and/or Hamiltonians with null local terms


First, we focus on the class of quantum batteries, which can be described (i) by locally maximally mixed (LMM) states, that play an important role in quantum information, and/or (ii) in the limit of asymptotically strong Hamiltonian interaction with respect to the Hamiltonian local terms. 


In the case (i), we consider a bipartite LMM state ρ_ab∈ D(ℋ_a ⊗ℋ_b), where dimℋ_a = dimℋ_b = d.

This means that the reduced density operators correspond to maximally mixed states <cit.>, 



    _a[ρ_ab] = 1_b/d ,   _b[ρ_ab] = 1_a/d .




The set of LMM states contains locally maximally entangled states <cit.>, which are pure states ρ_ab satisfying (<ref>), and form a convex set including pure maximally entangled states as well as mixed states such as Werner states. 

Qubit Bell states are known to be the only extremal points of the set of LMM states for d=2. 
For higher dimensions, the set of LMM states is strictly larger: e.g., for d = 3 its extremal points are not necessarily locally equivalent <cit.>.


As detailed in Appendix <ref>, the analysis carried below holds in the case (ii) as well, corresponding to the condition 

    H_a=H_b=0

on the Hamiltonian. 
In our notation (<ref>), it can be seen as the limit of asymptotically strong interaction, in which local contributions are neglected with respect to the interaction term.

A paradigmatic example of a Hamiltonian that satisfies these conditions is provided by the antiferromagnetic Hamiltonian 

    H_Ant := ω (σ_a·σ_b),

where σ_a/b are vectors, whose components σ_a/b^j are generalized Pauli operators <cit.> on ℋ_a/b, which reduce to the usual Pauli operators for d=2.





In order to analyze PE for a quantum battery satisfying (i) and/or (ii), we do the following observation. Exploiting the generalized Pauli operator expansion <cit.>, the action of work extracting unitaries U_ P in (<ref>) can be associated to rotations in real space (see Appendix <ref>). 

The latter form a subgroup 𝒮𝒪(d^2-1) of the corresponding rotation group 𝚂𝙾(d^2 - 1), i.e. 𝒮𝒪(d^2-1) ⊆𝚂𝙾(d^2 - 1). 

Therefore, the optimization over parallel unitaries in (<ref>) can be substituted by optimization over 𝒮𝒪(d^2-1), and PE in the case of systems satisfying (i) and/or (ii) is given by 


    ℰ_ P(ρ_ab, H_ab) = tr[𝐕𝐓] + max_𝐎_a, 𝐎_btr[(-𝐕)𝐎_a𝐓𝐎_b^T],


where 𝐓 and 𝐕 are two-body components of the state ρ_ab and the Hamiltonian H_ab, respectively, i.e.,


    𝐓_i,j   :=  [ρ_ab(σ_a^i ⊗σ_b^j)]. 
    𝐕_i,j   :=  1/4[V_ab(σ_a^j ⊗σ_b^i)],


and 𝐎_a,b∈𝒮𝒪(d^2-1). 







The optimization in (<ref>) can be carried out analytically (details are reported in Appendix <ref>) over the entire rotation group 𝚂𝙾(d^2 - 1), establishing the following upper bounds on PE,



    ℰ_ P(ρ_ab, H_ab) ≤[ρ_ab V_ab] + (λ⃗^|𝐕|·λ⃗^|𝐓|)    if (-𝐕𝐓) ≥ 0, 
    [ρ_ab V_ab ] + (λ⃗^|𝐕|·λ⃗^|𝐓|-2λ_0^|𝐕|λ_0^|𝐓|)    if (-𝐕𝐓) < 0,



where λ⃗^|𝐌| = {λ_0^|𝐌|≤…≤λ_d^2-2^|𝐌|} denotes a non-decreasingly ordered set of singular values of a generic matrix 𝐌, i.e., eigenvalues of |𝐌| = tr[√(𝐌^†𝐌)]. Since there is a one-to-one correspondence (up to a global phase) between qubit unitary transformations and 3D rotations due to the isomorphism 𝚄(2)/𝚄(1) ≅𝚂𝙾(3), the upper bound (<ref>) is saturated in the case of a two-qubit system.



 §.§ Werner states




Werner states W are bipartite LMM states that are invariant under parallel unitary operations of form U ⊗ U,

    (𝖴⊗𝖴)[W] = W,

for any U acting on the d-dimensional Hilbert space. Werner states provide an important example of states whose PE and local ergotropies coincide, i.e., for a bipartite system AB,

    ℰ_ P(W, H_ab) = ℰ_ La(W, H_ab) = ℰ_ Lb(W, H_ab).


In order to see this, let us assume U⊗V to be an optimal parallel unitary that achieves the value of PE ℰ_ P(W). Taking into account the invariance condition (<ref>) with U = V and U = U, we obtain

    (𝖴⊗𝖵)[W]    =    (𝖴𝖵^†⊗1)[(𝖵⊗𝖵)[W]] 
       =    (𝖴𝖵^†⊗1) [W],

and

    (𝖴⊗𝖵)[W]    =    (1⊗𝖵𝖴^†)[(𝖴⊗𝖴)[W]] 
       =    (1⊗𝖵𝖴^†)[W],

so that the same amount of work can be extracted via a local unitary U^†V on the single subsystem A as well as V^†U on the subsystem B, hence, proving the property (<ref>). 






As an example, let us consider a Werner state of two qubits, which can be written explicitly as


    W_p := (1-p)1_ab/4+p|ψ_max⟩⟨ψ_max| ,


with p ∈ [0,1] and |ψ_max⟩ being any of the four Bell states |Φ^±⟩ = 1/2(|00⟩± |11⟩) and |Ψ^±⟩ = 1/2(|01⟩± |10⟩). 

The state in (<ref>) is characterized by 𝐓 = - pO, where O ∈𝚂𝙾(3) <cit.>.



Therefore, applying the upper bound (<ref>), which is saturated since a 2-qubit state is considered, we obtain PE (as well as local ergotropy <cit.>) of the Werner state (<ref>),






    ℰ_ P(W_p, H_ab) = 
    
    p  ( ⟨ψ_max|V_ab|ψ_max⟩ + tr[|𝐕|]  )    if (𝐕) ≥ 0, 
    
    p ( ⟨ψ_max|V_ab|ψ_max⟩ + (tr[|𝐕|] - 2λ_0^|𝐕|)  )    if (𝐕) < 0.
    


In particular, for the antiferromagnetic Hamiltonian (<ref>), which is characterized by 𝐕 = ω1, it reads:

    ℰ_ P(W_p, H_Ant) = 
    
    4pω   if |ψ_max⟩∈{|Φ^±⟩, |Ψ^+⟩}, 
    
    0    if |ψ_max⟩ =|Ψ^-⟩.





§ NUMERICAL APPROXIMATION VIA SEMIDEFINITE PROGRAMMING



Beyond the assumptions required for the bound on PE provided by Eq. (<ref>), a more generally applicable approximation method can be obtained with semidefinite programming (SDP). SDP is a subclass of convex optimization problems that allow for constraints in the form of linear matrix inequalities and for which efficient algorithms exist for its numerical solution <cit.>, with polynomial complexity in the number of constraints and on the size of the matrix. 

Directly performing the optimization required to compute the PE can be a difficult task due to its nonlinear nature and might depend on an adequate parametrization of the local unitaries according to the system under consideration. Certain relaxations of the original problem might come at no cost, such as optimizing over the convex hull of the set of products of unitaries, since the functional (<ref>) is itself convex and, therefore, will have as extreme points pure products of unitaries. Approximating the original optimization problem by an SDP has the advantage of producing a reproducible upper bound, since local optima are also global for convex optimization problems, but also the constraints allowed retain some nonlinearity, since linear matrix inequalities correspond to polynomial inequalities in the entries of the matrices. Furthermore, it is known that certain complex problems can be exactly approximated by SDPs of increasing sizes <cit.>. As such, required approximations to formulate the optimization of Eq. (<ref>) as an SDP aim to replace constraints such as U U^† = 1 and U = U_a ⊗ U_b, which are nonlinear in nature, with relaxed forms that include the ideal target unitary channels. Since the optimization is performed on a larger family of channels than the ones ideal for PE, the result is also an upper bound on its actual value.

To optimize over channels and impose the relevant constraints, the Choi-Jamiołkowski isomorphism is used to describe the superoperator Λ as a state-like operator J_Λ <cit.>, such that if Λ is a channel acting on ℒ(⊗_i=1^n ℋ^𝚒𝚗_i), mapping operators that act on input Hilbert spaces to operators in ℒ(⊗_i=1^n ℋ^𝚘𝚞𝚝_i), then J_Λ is an operator in the combined space ℒ[(⊗_i=1^n ℋ^𝚒𝚗_i) ⊗(⊗_i=1^n ℋ^𝚘𝚞𝚝_i)]. Assuming finite dimensions and choosing a particular basis for the input space, the equivalence is established by the relations

    J_Λ = ∑_μ,ν |μ⟩⟨ν| ⊗Λ(|μ⟩⟨ν|) , 
    Λ(ρ) = Tr_𝚒𝚗[ J_Λ (ρ^T ⊗1) ],

where the transposition is performed in the chosen basis and the trace is over the collection of input spaces, denoted here as 𝚒𝚗. Similarly, the collection of output spaces will be denoted as 𝚘𝚞𝚝, and individual spaces by 𝚒𝚗^(i) and 𝚘𝚞𝚝^(i).




We notice that the optimization of novel local unitaries can be replaced with products of convex combinations of local unitaries, in fact, the functional is convex, ensuring that the maximum is always reached on extremal points. For sub-systems of dimension 2, the latter set coincides with unital channelsb<cit.>, thus, to simplify further the constraints, we could optimize over the (slightly larger for local dimension d>2) set of local unital channels.
This constitutes our first source of error when approximating PE.
These constraints can be mapped on the Choi-state of the channel: 
 ...
(Explicit forumalas)
...
Once again, since the functional is convex we can replace the tensor product of Choi-states of unital channels with convex combinations of them, without increasing our upper bound.
However, in order to write efficiently the problem as an SDP is convenient to replace the strict separability constraint on the choice states with the converging hierarchy of simpler tests introduced in <cit.>...


This constitutes our second source of error...






Inspired by the technique of Ref. <cit.>, we replace the unitary property of the channel with unitality, adapted here for the action on multiple input systems. Although unital channels are, at first glance, a rough approximation to unitary maps, they satisfy some properties that reveal they are good candidates for numerical approximation: not only can they be described in terms of a set of simple linear matrix equations (as described below), but they form a superset to the ideal channels–more precisely, to convex combinations of unitary channels. Not only convexifying the set of unitary channels is a costless approximation for computing PE, since the functional of Eq. (<ref>) is convex in the unitaries applied, but furthermore, for channels mapping qubits to qubits, the sets of unital channels and convex combinations of unitaries coincide <cit.>. In the last case, an extremal point of the unital set should be reproducible by a unitary map, and the approximation can be exact. For higher dimensions, the sets do not coincide anymore and this approximation would constitute a source of deviation to the exact value of the PE. Given that the target channel is composed by local unitaries, here we require the channel to be locally unital, meaning it should be independently unital on each subsystem it acts upon, a property that can be described in the following formula:

    Λ(ρ⊗1_i) = ρ' ⊗1_i,

where ρ is a density matrix in ℬ(⊗_j ≠ iℋ^j_𝚒𝚗) and ρ' the resulting state also acting on all but the i-th output Hilbert space. 

Another approximation required by the structure of the SDP takes into account the fact that, ideally, Λ should converge to a product of local unitaries. The ideal J_Λ should also decompose in a tensor product of local Choi operators acting each on each subsystem ℋ^𝚒𝚗_i ⊗ℋ^𝚘𝚞𝚝_i. Since ensuring such a structure is also usually a difficult task computationally <cit.>, a double relaxation is employed on J_Λ that approximates the set of ideal channels from outside: first, we admit that J_Λ can be a convex combination of product Choi operators, a relaxation similar from moving from pure product states to general separable states in entanglement theory; secondly, we approximate the resulting convex set with the converging hierarchy of simpler tests introduced in <cit.>, based on symmetric extensions of the state with positive partial transpose (PPT). In particular, although higher levels of the hierarchy are too resource demanding to implement computationally, already the “level zero" of just requiring PPT for J_Λ results usually in a good approximation to the target set in practical cases. And since the hierarchy of approximations converges to the actual set of separable states, the higher the level of the hierarchy implemented, the closer should be the map to an actual combination of product of unital maps.



With all approximations combined, the optimization problem can be structured as the following SDP:

    Minimize     Tr[J_Λ (ρ^T ⊗ H)] 
    s.t.      J_Λ≥ 0,
         Tr_𝚘𝚞𝚝[J_Λ] = 1_𝚒𝚗, 
    ∀ i ∈ (1,…,n)      Tr_𝚒𝚗^(i)[J_Λ] = Tr_𝚒𝚗^(i), 𝚘𝚞𝚝^(i)[J_Λ] ⊗1_𝚘𝚞𝚝^(i)/d_i, 
          J_Λ ∈ 𝖣𝖯𝖲_k,

where the first two constraints ensure J_Λ represents a proper CPTP quantum channel, the third line is a family of constraints ensuring local unitality, and the last line corresponds to the constraints imposed by the k-th level of the Doherty-Parrilo-Spedalieri (DPS) hierarchy for separable states. The level k=0 corresponds to J^T_i_Λ≥ 0, meaning partial transposition on the space of each subsystem (𝚒𝚗^(i), 𝚘𝚞𝚝^(i)) should preserve positive-semidefiniteness. The last two families of constraints are explained in more detail in Appendix <ref>. The result of the above optimization should lower bound the application of the more restricted class of channels that are pure product of unitaries, since the optimization occurs on a superset containing the latter. As such, the difference Tr[ρ H] - Tr[J^*_Λ (ρ^T ⊗ H)], with J^*_Λ being the result of the above optimization, should provide an upper bound to the actually attainable PE with local unitaries.



 §.§ Applications



In the following, we present some test cases to compare the different methods for estimating PE. For direct comparison, we restrict the analysis to cases where Eq. (<ref>) is applicable, first on the Werner state, then on generic states but using Hamiltonians with no local terms. 

Results for the first case are presented in Fig. <ref>, where a range of upper bounds–corresponding to global ergotropy, the SDP approximation of Eq. (<ref>) and the analytical bound of Eq. (<ref>)–is calculated for different visibilities of the two-qubit Werner state [Eq. (<ref>)] under the action of the Hamiltonian

    H = 1/2ω_a σ^z_a + 1/2ω_b σ^z_b + 1/2g (σ^x_a ⊗σ^x_b + σ^y_a ⊗σ^y_b),

with ω_a = 1.0, ω_b = 1.1, g = 0.33. Also a lower bound is computed through direct optimization of local unitaries. Coincidence of the latter with the bound of (<ref>) confirms that the estimate is exact for two qubits, as expected.

The SDP method of Eq. (<ref>) shows a clear improvement in comparison to upper bounding PE using global ergotropy. Increasing the level of the DPS hierarchy for testing separability however introduces no advantage, indicating that the PPT criterion is already a good and relatively low-cost approximation, also indicating that the main setback of the method is the approximation to unital channels.





Applying the techniques to higher-dimensional systems introduces computational complications since the dimensions of J_Λ grow as the square of the local dimensions. It is still possible, however, to perform tests for two-qutrit and two-ququart systems in the 0-th level of the DPS hierarchy [Eq. (<ref>)]. In Fig. <ref>, a comparison between the SDP approximation of Eq. (<ref>) and the bound of Eq. (<ref>) is presented for two-qutrit states in terms of the gap that each method produces with respect to the same lower bound for PE, obtained with direct optimization of the local unitaries (See Appendix <ref> for details). For each point in the graph, a random state and a random Hamiltonian are produced and all estimates are computed on the produced pair. States are obtained as ρ_ab = X X^†/tr[XX^†], with entries on X normally sampled <cit.>, and Hamiltonians are produced as H = v_ji σ^i_a ⊗σ^j_b, with v_ji normally distributed, σ^i_a/b corresponding to a generalized (orthonormal) Pauli basis. For the particular sampling used, the SDP method performs better in general, even closing the gap between the upper bound and lower bound for some points.



§ PARALLEL CAPACITY AS AN ENTANGLEMENT WITNESS
 



Now we shift the focus to another quantity characterizing the potential of quantum battery to store work, which has been introduced in Section <ref>, quantum battery capacity <cit.>. We analyze it under restriction to local unitary cycles, introducing thereby the quantity of parallel capacity, and demonstrate its role as an entanglement witness. 




 §.§ Parallel capacity


We start by formalizing the definition of parallel capacity. Similarly to the definition of PE (<ref>) of a multipartite quantum battery via restriction of work extracting unitaries to local ones, we define the parallel capacity of a N-partite quantum system equipped with a Hamiltonian H as the maximum energy gap that we can obtain between two states which are connected by parallel unitary operations,




    𝒞_ P(ρ, H):= max_U_P∈𝒰_P^N(   [U_Pρ U_P^† H]) - min_Ũ_P∈𝒰_P^N(   [ Ũ_PρŨ_P^† H] ),


where 𝒰_ P^N denotes the set of N-parallel unitaries, i.e., unitaries of form U_P = ⊗_i=1^N U_i, with U_i acting on i-th subsystem. Notice that the parallel capacity (<ref>) is bounded from below by the corresponding PE (<ref>), because max_U_P∈𝒰_P^N    [ H  U_Pρ U_P^†] ≥  [ H ρ]. On the other hand, it is naturally bounded from above by the maximal energy gap ‖ H ‖_∞ := E_ max - E_0, which is the difference of the highest E_ max and lowest E_0 eigenvalues of H. Therefore,

    ℰ_ P(ρ, H) ≤𝒞_ P(ρ, H) ≤‖ H ‖_∞.



In what follows, we focus on bipartite interacting quantum batteries equipped with the Hamiltonian (<ref>) and described by the state ρ_ab∈ D(ℋ_a ⊗ℋ_b). In particular, for a class of bipartite setups characterized by a Hamiltonian without local terms and/or LMM states, which have been analyzed in Section <ref> in the context of PE, parallel capacity is bounded from above by


    𝒞_ P(ρ_ab, H_ab) ≤
          2((λ⃗^|𝐕|·λ⃗^|𝐓|) - λ_0^|𝐕|λ_0^|𝐓|)    if d is even.
    
          2 (λ^|𝐕|·λ^|𝐓|)    if d is odd and det(-𝐕𝐓) ≥ 0,
    
          2((λ⃗^|𝐕|·λ⃗^|𝐓|) - 2λ_0^|𝐕|λ_0^|𝐓|)    if d is odd and det(-𝐕𝐓) < 0,


where equality holds for d=2 (see Appendix <ref>).

Imai et al <cit.> showed that it is possible to detect entanglement in a bipartite quantum battery statistically by probing a large variance of average work extracted under Haar-random parallel unitaries. 

We proceed by demonstrating an alternative entanglement criterion via detection of a large parallel capacity (<ref>) of the working medium.







 §.§ Maximum parallel capacity of separable states and entanglement gap energy


It is known that entanglement of a pure bipartite quantum state |Ψ_ab⟩ can be captured by considering its Schmidt decomposition |ψ_ab⟩ = ∑_i=1^r_S(Ψ_ab)√(p_i)|ϕ_i,a⟩⊗|ψ_i,b⟩, where ∑_i p_i = 1 and {|ϕ_i,a⟩}_i and {|ψ_i,b⟩}_i are certain orthonormal bases in ℋ_a and ℋ_b, respectively. In turn, the state |Ψ_ab⟩ is entangled if and only if its Schmidt rank r_S(Ψ_ab) > 1. For a mixed state ρ_ab, the Schmidt rank can be generalized to Schmidt number that captures the pure states with the highest Schmidt rank from all its possible ensemble realizations ρ_ab = ∑_i p_i |Ψ_i, ab⟩⟨Ψ_i, ab | <cit.>,

    r̃_S(ρ_ab) = inf_𝒟(ρ_ab)max_Ψ_i, ab r_S(Ψ_i, ab),

where 𝒟(ρ_ab) = { p_i, |Ψ_i, ab⟩} is the set of ensemble realizations of ρ_ab.

Bipartite density matrices ρ_ab with Schmidt number r̃_S(ρ_ab) = k establish a hierarchy of subsets 𝒮^k ⊂ D(ℋ_a ⊗ℋ_b) that are convex and satisfy 𝒮^l⊆𝒮^k l≤ k. Conversely, we say that ρ_ab has Schmidt number r̃_S(ρ_ab)=k if ρ_ab∈𝒮^k\𝒮^k-1. Notice that 𝒮^1 coincides with the set of separable states. Therefore, r̃_S(ρ_ab)> 1 implies that ρ_ab is entangled, whereas higher Schmidt number r̃_S(ρ_ab) implies stronger entanglement in ρ_ab <cit.>. In order to connect this observation to parallel capacity of ρ_ab, we define the maximum parallel capacity of states belonging to 𝒮^k as


    𝒞_ P^k(H_ab):=max_ρ_ab∈𝒮^k𝒞_ P(ρ_ab,H_ab) .
 

Consequently, any state ρ_ab such that 𝒞_ P(ρ_ab, H_ab) > 𝒞_ P^k(H_ab) necessarily has Schmidt number higher than k. This suggests that 𝒞_ P^1 can be used to check whether a state has Schmidt number k > 1 and, hence, is entangled. In the following theorem, we show that it is indeed strictly connected to the entanglement gap energy <cit.>, which is a known entanglement witness and defined as the energy difference between the separable state with minimum energy and the ground state energy,

    Δ E_Sep(H_ab) = min_ρ∈𝒮^1Tr[ρ H_ab] - E_0,

where E_0 is the energy of the ground state of H_ab.









Given a bipartite Hamiltonian H_ab, the corresponding maximum parallel capacity of separable states is given by

    𝒞_ P^1(H_ab)=‖ H_ab‖_∞-(Δ E_Sep(H_ab)+Δ E_Sep(-H_ab)).




First, it is immediate to verify that the parallel capacity (<ref>) (hence, also the maximal parallel capacities 𝒞_ P^k(H_ab)) is convex in ρ_ab. Therefore, the maximum in 𝒞_ P^1(H_ab) can be always achieved by a pure separable state, and the optimization can be performed over product states |Ψ⟩ = |ϕ⟩_a ⊗|ψ⟩_b,


    𝒞_ P^1(H_ab) = max_|Ψ⟩∈𝒮^1𝒞_ P(|Ψ⟩⟨Ψ|, H_ab) ,

On the other hand, for any pair of product states |Ψ⟩ and |Ψ̃⟩, it is possible to find local unitaries U_a ⊗ U_b that transform them into each other, so that |Ψ̃⟩ = (U_a ⊗ U_b) |Ψ⟩. Therefore, the maximization over local unitaries in the parallel capacity 𝒞_ P(|Ψ⟩⟨Ψ|, H_ab) can be substituted by maximization over the set of pure separable states. Moreover, any product state |Ψ⟩ has the same parallel capacity 𝒞_ P(|Ψ⟩⟨Ψ|, H_ab), therefore,

    𝒞_ P^1(H_ab) = max_|Ψ⟩∈𝒮^1⟨Ψ|H_ab|Ψ⟩ - min_|Ψ⟩∈𝒮^1⟨Ψ|H_ab|Ψ⟩.
 
Taking into account that entanglement gap energy (<ref>) can be always achieved by a pure separable state as well, we can connect the latter term in (<ref>) to it as

    min_|Ψ⟩∈𝒮^1 ⟨Ψ|H_ab|Ψ⟩ = Δ E_Sep(H_ab) + E_0.

On the other hand, the maximization in (<ref>) can be substituted by minimization with respect to the Hamiltonian with the opposite sign,

    max_|Ψ⟩∈𝒮^1  ⟨Ψ|H_ab|Ψ⟩ = - min_|Ψ⟩∈𝒮^1  ⟨Ψ|(-H_ab)|Ψ⟩,

and, hence, connected to the corresponding entanglement gap energy as

    max_|Ψ⟩∈𝒮^1  ⟨Ψ|H_ab|Ψ⟩ = - Δ E_Sep(-H_ab) + E_max,

where it is taken into account that the ground state energy of -H_ab is equal to -E_max, the maximal eigenvalue of H_ab taken with the opposite sign. Plugging in (<ref>) and (<ref>) into (<ref>), we obtain (<ref>), hence, proving the thesis.




 
    





We remark that there exist efficient numerical techniques to compute Δ E_Sep  <cit.>, making the computation of 𝒞_ P^1(H_ab) efficient too. 


As a corollary, we get that 𝒞_ P^1(H_ab)=‖ H_ab‖_∞
iff both the ground state energy manifold and the maximum energy manifold contain at least a separable state. 




 §.§ Entanglement witness
 

Theorem <ref> reveals that parallel capacity of a state provides an entanglement witness under suitable choice of Hamiltonian H_ab. Performing optimization of the parallel capacity over all states of the battery,



    𝒞_ P(H_ab):= max_ρ_ab  𝒞_ P(ρ_ab,H_ab) ,

we formulate the following criterion.

Given a Hamiltonian H_ab such that

    𝒞_ P(H_ab) > ‖ H_ab‖_∞- (Δ E_Sep(H_ab)+Δ E_Sep(-H_ab) ),
 
fulfillment of condition

    𝒞_ P(H_ab, ρ_ab) > ‖ H_ab‖_∞- (Δ E_Sep(H_ab)+Δ E_Sep(-H_ab) ),

witnesses entanglement of ρ_ab.








As a paradigmatic example, we consider the antiferromagnetic Hamiltonian (<ref>) for 2 qubits, which can be given as

    H_Ant = ω(  1_ab-4|ψ^-⟩⟨ψ^-|),

where |ψ^-⟩ := 1/√(2)(|01⟩-|10⟩) is the singlet state. First, let us show that H_Ant is a suitable Hamiltonian that fulfills (<ref>). In order to calculate 𝒞_ P(H_Ant), we recall that, as a parallel capacity, it features the bounds (<ref>), where the lower bound is true for any state ρ_ab. For the upper bound ‖ H_Ant‖_∞, we find that |ψ^-⟩ is the ground state of H_Ant with associated energy E_0=-3 ω, while the orthogonal subspace to |ψ^-⟩ is the maximum energy manifold with associated energy E_max= ω. Therefore,

    ‖ H_Ant‖_∞ = 4 ω.


On the other hand, taking ρ_ab = |Ψ^+⟩⟨Ψ^+|, it is straightforward to show that its ergotropy

    ℰ_P(|Ψ^+⟩⟨Ψ^+|, H_ Ant) = 4ω,

providing the lower bound on the maximal parallel capacity. Hence, we conclude that 𝒞_ P(H_Ant) = 4ω. 





In order to check that it satisfies (<ref>), we notice that the maximum energy manifold can be spanned by the basis {|00⟩, |11⟩, |Ψ^+⟩}. This implies Δ E_Sep(-H_Ant)=0, because, e.g., the separable state |00⟩ is a ground state for -H_Ant. On the other hand, the antiferromagnetic Hamiltonian is known to provide the largest possible entanglement gap for 2 qubits given by <cit.>

    Δ E_Sep(H_Ant) = (1 - 1/d) ‖ H_Ant‖_∞ = 2  ω.

In turn, applying Theorem <ref>, we obtain 𝒞_ P^1(H_Ant) = 2  ω, which is strictly smaller than 𝒞_ P(H_Ant). Therefore, H_Ant fulfills the condition (<ref>), and the corresponding parallel capacity can be used to detect entanglement.


Now, let us consider 2-qubit Werner states (<ref>), which are known to be entangled iff p > 1/3. First, we take into account that the representation (<ref>) of H_Ant suggests that 𝐕 is proportional to identity, and λ^|𝐕| = ω[ 1; 1; 1 ], while Werner states are described by λ^|𝐓| = p [ 1; 1; 1 ]. Applying (<ref>), where the equality holds, since we consider qubit subsystems, we obtain 

    𝒞_ P(W_p,H_Ant)=4 p  ω .


Therefore, the parallel capacity witnesses entanglement if and only if 

    p > 1/2.




 §.§ Parallel capacity and work fluctuations



While entanglement in quantum battery can be verified by its high capacity under local unitaries, extraction of work via randomly chosen local unitaries provides an alternative statistical entanglement witness. Indeed, entanglement in a bipartite d × d quantum battery can be verified by detecting large fluctuations in the corresponding work statistics <cit.>. Given a Hamiltonian H_ab, fulfillment of condition

    (ΔW)^2    >   1/d^2 - 1( d^2/4( |R_a|^2 |h_a|^2 + |R_b|^2 |h_b|^2 ) 
       +   s_1(d, R_a, R_b) ‖𝐕‖_2/d^2 - 1),

where s_1(d, R_a, R_b) = d - 1 + d^2/4(d-2/2 (|R_a|^2 + |R_b|^2 ) - d/2| |R_a|^2 - |R_b|^2 | ) and ‖𝐕‖_2 = tr[𝐕^T 𝐕] = ∑_i=0^d^2-2 (λ_i^|𝐕|)^2, witnesses entanglement in ρ_ab. Taking into account decomposition of the work variance into generalized Pauli operators <cit.>,

 
  

the criterion (<ref>) can be given in a more simple form, which depends only on the structure of the state of quantum battery,

    d^4/16‖𝐓‖_2 > s_1(d, R_a, R_b).

In order to compare it with the parallel capacity criterion (<ref>), following Section <ref>, we focus on a class of 2-qubit setups characterized by a Hamiltonian without local terms and/or locally maximally mixed states. In this case, for a given Hamiltonian H_ab, it is always possible to find states that do not fulfill (<ref>), hence, do not lead to fluctuations of work high enough to detect entanglement via (<ref>), yet have a parallel capacity that witnesses entanglement with respect to the criterion (<ref>) (see Appendix <ref>).

As an example, let us turn back to the case of antiferromagnetic Hamiltonian (<ref>) and 2-qubit Werner states (<ref>). For the latter, s_1 = 1, and the work fluctuation criterion (<ref>) witnesses their entanglement if and only if

    p > 1/√(3).

Comparing it with the parallel capacity criterion (<ref>), we conclude that there exists a subset of Werner states with 1/2 < p ≤1/√(3) that are entangled due to their large parallel capacity despite low work fluctuations.














§ EXTENDED PARALLEL ERGOTROPY
 

We conclude our treatment by discussing the concept of extended parallel ergotropy (EPE).

In close analogy with the recently introduced extended local ergotropy (ELE) <cit.>, 
EPE is defined by setting as allowed operations free time evolutions and parallel unitaries.
The set of extended parallel unitaries is then


    𝒰_ Pex:={
        U   s.t.  U= 𝒯exp [-i∫_0^TH_ab+H_a(t)+H_b(t)dt  ]} .

 
Interestingly, a seminal result of quantum computing states that <cit.> for any interacting Hamiltonian H_ab this set includes all unitary operations on the joint AB system <cit.>. 

This means that 𝒰_ Pex coincides with the set of unitaries on the joint AB system <cit.>, implying


    ∀ρ  ,  ℰ_ Pex(ρ,H)= ℰ(ρ,H) .
 

Notably, a series of parallel unitaries together with the entangling unitaries provided by internal time evolution enable any unitary transformation on the joint system. 
This comes as a result of quantum control theory <cit.> providing at least in some cases a practical way of implementing the desired unitary. 

More specifically, since 𝚄(d_ad_b) is compact, there exist a number N and a time T (both finite) such that any U ∈𝚄(d_ad_b)

 can be implemented through a sequence of N parallel unitaries and free time evolution,
with a total free evolution non greater than T. However, such N and T depend on the specific model and might be large <cit.>.





§ CONCLUSIONS



Maximum quantum work extraction is generally defined in terms of the ergotropy functional, which does not capture potential difficulties in experimental implementation of the optimal unitary allowing for it, in particular, for multipartite systems.

In this framework, we consider a quantum battery consisting of many interacting sub-systems and study the maximum extractable work via concurrent local unitary operations on each subsystem. It is defined in terms of a functional that we call parallel ergotropy, a quantity that in many cases is expected to be experimentally more accessible than the ergotropy of the entire battery. Thanks to its properties, we have derived useful bounds and exact analytical results for specific classes of states and/or Hamiltonians and provided receipts for numerical estimation of upper bounds via semi-definite programming techniques in the generic case.


In particular, we provided exact expressions of parallel ergotropy of qubit bipartite batteries in a locally maximally mixed state and equipped with an arbitrary Hamiltonian or in arbitrary state and equipped with a locally null Hamiltonian. 


We have also observed that parallel ergotropy outperforms work extraction via egoistic strategies, in which the first agent A extracts locally the maximum work, followed by the second agent B that extracts locally the remaining work. This indicates that cooperation between the agents in work extraction is necessary for an overall benefit. 




Apart from ergotropy, we have studied another quantity characterizing capability of a quantum system to accumulate and supply work, quantum battery capacity. Restriction to the concurrent local unitary operations has revealed that its counterpart, parallel capacity, can serve as a witness of entanglement. We have compared it with another entanglement witness based on statistical properties of work extraction from a multipartite quantum battery, namely revealing entanglement under large variance of the average work with respect to random Haar parallel unitaries. We found that a high parallel capacity that detects entanglement does not necessarily imply fluctuations of work high enough to detect entanglement via its large variance. 

Finally, we showed that if free time evolutions and parallel unitaries are allowed, one can always extract the maximum work stored in the quantum battery, i.e., saturate the gap between the parallel ergotropy and ergotropy of the entire system.


Our work leaves several open research problems. Although we introduced the parallel ergotropy and quantum battery capacity for a system composed of an arbitrary number N of subsystems, our study has been focused mainly on the case N=2. 
 Therefore, analysis of these quantities beyond the bipartite case would be a natural development of the present work. In particular, their relation to multipartite entanglement and behavior in the limit N →∞ have to be examined. 




  
*Acknoledgements.—
We thank Antonio Acín and Vittorio Giovannetti for fruitful discussions.
DF acknowledges financial support from PNRR MUR Project No. PE0000023-NQSTI. RN acknowledges support from the Government of Spain (Severo Ochoa CEX2019-000910-S and TRANQI), Fundació Cellex, Fundació Mir-Puig, Generalitat de Catalunya (CERCA program) and support from the Quantera project Veriqtas. This research was funded in whole or in part by the Austrian Science Fund (FWF) 10.55776/PAT4559623. For open access purposes, the author has applied a CC BY public copyright license to any author-accepted manuscript version arising from this submission.
















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§ GENERALIZED PAULI OPERATOR  FORMALISM FOR PARALLEL ERGOTROPY
 

In order to facilitate calculation of PE (<ref>) for bipartite quantum batteries of arbitrary dimension, we treat the corresponding states and Hamiltonians exploiting the generalized Pauli operators (GPO) expansion formalism <cit.>. Given a quantum system, let d be the dimension of the underlying Hilbert space. A GPO set is then a collection of d^2-1 operators σ^j such that 

    [σ^j ]=0 ,     [σ^iσ^j ]=2δ_ij.


For d=2, this set can be reduced to standard Pauli operators, while for standard form of the GPO set for d>3 we refer the reader to <cit.>. Equipped with the identity operator, GPO form an orthogonal basis on the real vector space of self-adjoint operators acting on the d-dimensional Hilbert space.
 
Let us consider a bipartite (d_a × d_b)-dimensional quantum system in state ρ_ab∈ D(ℋ_a ⊗ℋ_b) and equipped with the Hamiltonian (<ref>), where, without loss of generality, we assume

    _a[V_ab] = _b[V_ab] = 0.




Then we can represent the state and the local and non-local components of the Hamiltonian in the following way using the GPO,

    ρ_ab   =   1_ab/d_ad_b + 1/2((R⃗_a ·σ⃗_a) ⊗1_b/d_b) + 1/2( 1_a/d_a⊗ (R⃗_b ·σ⃗_b)) + 1/4∑_i=1^d_a^2-1∑_j=1^d_b^2-1𝐓_i,j (σ_a^i ⊗σ_b^j),
    
    H_a    =    (h⃗_a ·σ⃗_a  ) ⊗1_b
    ,          H_ b  =  1_a ⊗ (h⃗_b ·σ⃗_b  ),         V_ab  =  ∑_i=1^d_a^2-1∑_j=1^d_b^2-1𝐕_j,i (σ_a^i ⊗σ_b^j),

where

    R⃗_a    =   [(σ⃗_a⊗1_b) ρ_ab],         R⃗_b   =  [(1_a ⊗σ⃗_b) ρ_ab],        𝐓_i,j  =  [(σ_a^i ⊗σ_b^j)ρ_ab],

and

    h⃗_a    =   1/2d_b[(σ⃗_a⊗1_b) H_ab],         h⃗_b   =  1/2d_a[(1_a ⊗σ⃗_b) H_ab],         𝐕_j,i  =  1/4[V_ab(σ_a^i ⊗σ_b^j)].

Notice the swapped indices in the definition 𝐕_j,i, so that introduction of transpose of 𝐕 into the energy functional can be omitted,
 

    E(ρ_ab)    :=   [H_abρ_ab] 
       =    (h⃗_a ·R⃗_a) + (h⃗_b ·R⃗_b) + [𝐕𝐓]  .

GPO expansion of ρ_ab and H_ab in (<ref>) and (<ref>) suggests the following simplification of optimization over unitaries in the PE functional (<ref>). Since the adjoint representation of 𝚂𝚄(n) (hence, the representation of 𝚄(n) on 𝔰𝔲(n) as well) is a subgroup 𝒮𝒪(n^2-1) ⊆𝚂𝙾(n^2-1) <cit.>, for any unitaries U_a∈𝚄(d_a), U_b∈𝚄(d_b) acting on ℋ_a and ℋ_b, respectively, there exists a pair of associated rotations 𝐎_a ∈𝚂𝙾(d_a^2-1) and 𝐎_b ∈𝚂𝙾(d_b^2-1) transforming R⃗_a/b and 𝐓,

    (U_a ⊗U_b)[ρ_ab] = 1_ab/d_ad_b + 1/2((𝐎_aR⃗_a ·σ⃗_a) ⊗1_b/d_b) + 1/2( 1_a/d_a⊗ (𝐎_b R⃗_b ·σ⃗_b)) + 1/4∑_i=1^d_a^2-1∑_j=1^d_b^2-1𝐎_a 𝐓_i,j𝐎_b^T (σ_a^i ⊗σ_b^j).

Therefore, the optimization over 𝚄(d_a) and 𝚄(d_b) in (<ref>) can be substituted by optimization over 𝒮𝒪(d_a^2-1) and 𝒮𝒪(d_b^2-1), respectively,

    ℰ_ P(ρ_ab, H_ab) = E(ρ_ab)-𝐎_a ∈𝒮𝒪(d_a^2-1) 
    𝐎_b ∈𝒮𝒪(d_b^2-1) min  ( (h⃗_a ·𝐎_a R⃗_a) + (h⃗_b·𝐎_b R⃗_b) + [𝐕𝐎_a𝐓𝐎_b^T]  )

Since 𝒮𝒪(n^2-1) is a proper subgroup of 𝚂𝙾(n^2-1) (unless n = 2, when they coincide), we can establish upper bound on PE by performing the optimization over the entire rotation group,





    ℰ_ P(ρ_ab, H_ab) ≤  E(ρ_ab)-𝐎_a ∈𝚂𝙾(d_a^2-1) 
    𝐎_b ∈𝚂𝙾(d_b^2-1) min  ( (h⃗_a ·𝐎_a R⃗_a) + (h⃗_b·𝐎_b R⃗_b) + [𝐕𝐎_a𝐓𝐎_b^T]  ),

which is saturated for d_a = d_b = 2, i.e., bipartite qubit systems. 



§ EGOISTIC STRATEGIES FOR QUASI-GENERIC HAMILTONIANS



Let us consider a bipartite quantum battery composed of two qubits. Using the decomposition of its state (<ref>) and Hamiltonian (<ref>) via Pauli operators, we assume that the latter is characterized by 𝐕 with 3 distinct eigenvalues and det[𝐕] > 0. In turn, the state ρ_ab is chosen to satisfy the following conditions,

    𝐓   =    -η1, 
    R_a    =    0, 
    
        (R_b ·h⃗_b)    ≠    -|R⃗_b|/|h⃗_b|, 
    
        |R_b|    =    1/2.

Before to proceed with comparison of cooperative and egoistic strategies for this setup, let us verify of existence of physical states defined by these conditions. The positivity and normalization of a bipartite qubit state can be guaranteed by the following criterion <cit.>,

    3-(‖𝐓‖^2_2 + |R⃗_a|^2 + |R⃗_b|^2) ≥ 0, 
        2(R⃗_a^T 𝐓R⃗_b - det[𝐓]) - (‖𝐓‖^2_2 + |R⃗_a|^2 + |R⃗_b|^2 - 1 ) ≥ 0,  
        8(R⃗_a^T (𝐓 + 𝐓) R⃗_b - det[𝐓]) + (‖𝐓‖^2_2 + |R⃗_a|^2 + |R⃗_b|^2 - 1)^2 - 4(| R⃗_a|^2 |R⃗_b|^2 + ‖𝐓‖^2_2 (|R⃗_a|^2 + |R⃗_b|^2) + ‖𝐓‖^2_2) ≥ 0,

where ‖ X ‖_2 = √(Tr[X^T X]), and 𝐓 is the cofactor matrix of 𝐓 defined by the identity 𝐓𝐓^T = 𝐓^T 𝐓 = det[𝐓]1. The chosen state ρ_ab is straightforwardly characterized by det[𝐓] = -η^3 and ‖𝐓‖_2 = √(3)η. Therefore, the corresponding cofactor matrix is given by 𝐓 = η^2 1 with ‖𝐓‖_2 = √(3)η^2. Therefore, the criterion is reduced to



    η^2    ≤   11/12, 
    η^2 (3 - 2η)    ≤   3/4, 
    η^2 (15 - 16η + 6η^2)    ≤   9/8.

The last inequality appears to be the most restrictive one, and the entire set of conditions is fulfilled if η∈ [-|η_ min|, η_ max], where η_ min≈ -0.241 and η_ max≈ 0.329, proving the existence of physical states in this range of η.


Assuming η > 0, first, we calculate PE for the system satisfying (<ref>)–(<ref>),




    ℰ_ P(ρ_ab, H_ab) = (h⃗_b ·R⃗_b) + ηtr[(-𝐕)] - min_𝐎_a, 𝐎_b ∈𝚂𝙾(3)( (h⃗_b·𝐎_b R⃗_b ) + η [(-𝐕)𝐎_a𝐎_b^T] ),

where we have taken into account that the upper bound (<ref>) is saturated for a bipartite quantum battery. Since minimization over both 𝐎_a and 𝐎_b in (<ref>) is performed simultaneously, we can substitute it by minimization over 𝐎_1 := 𝐎_a and 𝐎_2 := 𝐎_a 𝐎_b^T,

    ℰ_ P(ρ_ab, H_ab) = (h⃗_b ·R⃗_b) + ηtr[(-𝐕)] - ( min_𝐎_1 ∈𝚂𝙾(3) (h⃗_b·𝐎_b R⃗_b ) + ηmin_𝐎_2 ∈𝚂𝙾(3)[(-𝐕)𝐎_2] ),

Straightforwardly, the optimal 𝐎_1 is such that 𝐎_1 R⃗_b = -|R_b|/|h_b|h⃗_b. The second minimization is achieved by a unique (since 𝐕 is non-degenerate) optimal 𝐎_2 such that tr[(-𝐕)𝐎_2] = -tr[|𝐕|] = -∑_i=0^d^2-1λ_i^|𝐕|, where {λ_i^|𝐕|}_i are singular values of 𝐕, i.e., eigenvalues of |𝐕| = √(𝐕^T 𝐕). Therefore, we obtain the following PE,




    ℰ_ P(ρ_ab,H_ab) = (h⃗_b ·R⃗_b) + |h_b||R_b| + η[|𝐕| - 𝐕].

Notice that the local unitary operation performed by A may require energy. In this case, some energy has to be pumped by A into the system, so that B is able to extract the maximal work globally.

Now let us calculate the ergotropy ℰ_ La(ρ, H) + ℰ_ Lb(𝖴_a[ρ], H) under egoistic strategies, as introduced in the hierarchy (<ref>). The maximal work that can be extracted by the agent A is given by

    ℰ_ La(ρ_ab, H_ab)    =    (h⃗_b ·R⃗_b) + ηtr[(-𝐕)] - min_𝐎_a ∈𝚂𝙾(3)( (h⃗_b·R⃗_b ) + η [(-𝐕)𝐎_a] ) 
       =   η [|𝐕| - 𝐕],

where we have taken into account that the optimal unitary U_a of A is associated to 𝐎_2. In turn, the maximal work that B can extract after A is

    ℰ_ Lb(𝖴_a[ρ_ab], H_ab)    =    (h⃗_b ·R⃗_b) + ηtr[(-𝐕)𝐎_2] - min_𝐎_b ∈𝚂𝙾(3)( (h⃗_b·𝐎_b R⃗_b ) + η [(-𝐕)𝐎_2 𝐎_b^T] ) 
       =    (h⃗_b ·R⃗_b) - ηtr[|𝐕|] - min_𝐎_b ∈𝚂𝙾(3)( (h⃗_b·𝐎_b R⃗_b ) - η [|𝐕| 𝐎_b^T] ).



 



 

Minimization of the last term in (<ref>) requires 𝐎_b = 1. On the other hand, in order to minimize the term (h⃗_b·𝐎_b R⃗_b ) it is necessary to apply a unitary associated to 𝐎_1. Therefore, the optimal values of both terms cannot be achieved simultaneously, and the maximum work that can extracted by B is strictly upper-bounded by

    ℰ_ Lb(𝖴_a[ρ_ab], H_ab) < (h⃗_b ·R⃗_b) - ηtr[|𝐕|] + |h⃗_b||R⃗_b| + η [|𝐕|] = (h⃗_b ·R⃗_b) + |h⃗_b||R⃗_b|,

and we obtain a strict inequality for PE and the ergotropy under egoistic strategies with respect to A,

    ℰ_ La(ρ_ab, H_ab) + ℰ_ Lb(𝖴_a[ρ_ab], H_ab) < ℰ_ P(ρ_ab, H_ab).


Following the same steps, it is straightforward to demonstrate that egoistic work extracting strategies in the swapped order of the agents does not allow to achieve PE as well. Indeed, if agent B acts first, it cannot find a unitary that minimizes both last terms simultaneously since 𝐎_1 ≠𝐎_2. Therefore, it can extract maximal work

    ℰ_ Lb(ρ_ab, H_ab)    =    (h⃗_b ·R⃗_b) + ηtr[(-𝐕)] - min_𝐎_b ∈𝚂𝙾(3)( (h⃗_b·𝐎_b R⃗_b ) + η [(-𝐕)𝐎_b^T] ) 
       =    (h⃗_b ·R⃗_b) + ηtr[(-𝐕)] - (h⃗_b·𝐎_bR⃗_b ) + η [𝐕𝐎_b^T],

where 𝐎_b is associated to the optimal unitary. After that, A can extract maximal work
 
    ℰ_ La(𝖴_b[ρ_ab], H_ab)    =    (h⃗_b ·𝐎_bR⃗_b) + ηtr[(-𝐕)𝐎_b^T] - min_𝐎_a ∈𝚂𝙾(3)( (h⃗_b·𝐎_b R⃗_b ) + η [(-𝐕)𝐎_a𝐎_b^T] ) 
       =   ηtr[|𝐕| - 𝐕𝐎_b^T].

Summing both contributions up and taking into account that 𝐎_b ≠𝐎_1, we find that PE is a strict upper bound in this case as well,

    ℰ_ Lb(ρ_ab, H_ab)+ℰ_ La(𝖴_b[ρ_ab], H_ab) = (h⃗_b ·R⃗_b) + (h⃗_b·𝐎_bR⃗_b ) + η[|𝐕| - 𝐕] < ℰ_ P(ρ_ab, H_ab).


For det[𝐕] < 0, choosing η < 0 leads to the same results. Therefore, we conclude that except for particular Hamiltonians characterized by det[𝐕]=0 or degenerate spectrum, there exist states that open the gap between cooperative and egoistic work extracting strategies.






§ UPPER BOUND ON PARALLEL ERGOTROPY IN SPECIFIC CASES
 

Considering bipartite systems with d_a = d_b = d, we assume that one of hypotheses

    
  * R_a = R_b = 0    ⇒   _a [ρ_ab] = 1_b/d,  _b [ρ_ab] = 1_a/d,
    
  * h_a = h_b = 0     ⇒    H_a = H_b = 0.







 


Hypothesis (i) refers to the assumption of a bipartite system being in a locally maximally mixed state, while the Hamiltonian is left generic. Hypothesis (ii), instead, assumes a specific non-local Hamiltonian (e.g., the antiferromagnetic Hamiltonian (<ref>)), while the state is left generic. 


Although requiring both hypotheses to be true simultaneously is natural from the mathematical point of view, it lacks physical relevance. Therefore, in what follows, we require only a single hypothesis to be true. This cancels out the local terms in (<ref>), and the PE functional is upper-bounded as follows,

    ℰ_P(ρ_ab, H_ab)    ≤    E(ρ_ab) - 𝐎_a,𝐎_b ∈𝚂𝙾(d^2 - 1) min( [𝐕𝐎_a𝐓𝐎_b^T] ) 
       =    E(ρ_ab) + 𝐎_a,𝐎_b ∈𝚂𝙾(d^2 - 1) max( [(-𝐕)𝐎_a𝐓𝐎_b^T] ).

Therein, it is necessary to distinguish between two cases depending on signs of eigenvalues of -𝐕𝐓.



 §.§ Case (-𝐕𝐓) ≥ 0

In this case, either both -𝐕 and 𝐓 have an even number of negative eigenvalues or both have an odd number of them. In the first case, 







we can decompose both matrices as -𝐕=𝐎_𝐕^'Σ_𝐕𝐎_𝐕 and  𝐓=𝐎_𝐓^'Σ_𝐓𝐎_𝐓, where 𝐎_𝐕, 𝐎_𝐕', 𝐎_𝐓, 𝐎_𝐓'∈𝚂𝙾(d^2 - 1), and Σ_𝐕 and Σ_𝐓 are diagonal, positive, and having eigenvalues of |𝐕| and |𝐓|, respectively, in non-decreasing order. 
In the second case, we use a similar decomposition, however, the value of the smallest eigenvalue is left negative, so that Σ_𝐕= diag(-λ^|𝐕|_0, λ^|𝐕|_1, …, λ^|𝐕|_d^2-2) and Σ_𝐓= diag(-λ^|𝐓|_0, λ^|𝐓|_1,…,λ^|𝐓|_d^2-2). In both cases, the optimization (<ref>) reduces to

    ℰ_P(ρ_ab, H_ab) ≤ E(ρ_ab) + 𝐎_1,𝐎_2 ∈𝚂𝙾(d^2 - 1)max[Σ_𝐕𝐎_1Σ_𝐓𝐎_2] ,


where 𝐎_1 = 𝐎_𝐕𝐎_a 𝐎_𝐓' and 𝐎_2 = 𝐎_𝐓𝐎_b 𝐎_𝐕'. Finally, the Von Neumann trace inequality <cit.> allows us to provide the following upper bound for any 𝐎_1, 𝐎_2 ∈𝚂𝙾(d^2 - 1), 



    [Σ_𝐕𝐎_1Σ_𝐓𝐎_2] ≤ (λ⃗^|𝐕|·λ⃗^|𝐓|),

where λ⃗^|𝐕| and λ⃗^|𝐓| are the vectors composed by {λ_i^|𝐕|}_i and {λ_i^|𝐓|}_i, respectively. Therefore,








    ℰ_ P(ρ_ab, H_ab) ≤[ρ_ab V_ab] + (λ⃗^|𝐕|·λ⃗^|𝐓|) ,
          if  (-𝐕𝐓)≥ 0 .





 §.§ Case (-𝐕𝐓)<0

In this case, the proof proceeds in the same way as above. Nevertheless, it is necessary to take into account that the decomposition of 𝐕 and 𝐓 via 𝚂𝙾(d^2-1)-matrices results in diagonal matrices Σ_𝐕 and Σ_𝐓 that have an odd and even number of negative eigenvalues, respectively, or vice versa. This means that, in their product not all negative signs cancel out, and the maximum is achieved thereby by leaving the smallest eigenvalue of |Σ_𝐕| (or |Σ_𝐓|) negative in Σ_𝐕 (or Σ_𝐓), leading to

    ℰ_ P(ρ_ab,H_ab) ≤[ρ_ab V_ab] + ( (λ⃗^|𝐕|·λ⃗^|𝐓|) - 2λ^|𝐕|_0λ^|𝐓|_0),    if  (-𝐕𝐓) < 0 ,

and concluding the proof of inequalities (<ref>).




§ UPPER BOUND ON PARALLEL CAPACITY IN SPECIFIC CASES
 

Similarly to parallel ergotropy, parallel capacity can be decomposed as follows,

    𝒞_ P(ρ_ab, H_ab) 
       = 𝐎_a, 𝐎_b ∈𝒮𝒪(d_a,b^2-1) max  ( h⃗_a ·𝐎_a R⃗_a +h⃗_b·𝐎_b R⃗_b + [𝐕𝐎_a𝐓𝐎_b^T]  ) 
       - 𝐎_a, 𝐎_b ∈𝒮𝒪(d_a,b^2-1) min  ( h⃗_a ·𝐎_a R⃗_a +h⃗_b·𝐎_b R⃗_b + [𝐕𝐎_a𝐓𝐎_b^T]  ) ,

and, by enlarging the domain from 𝒮𝒪(d^2-1) to 𝚂𝙾(d^2-1), bounded from above,


    𝒞_ P(ρ_ab, H_ab)     ≤𝐎_a, 𝐎_b ∈𝚂𝙾(d_a,b^2-1) max  ( h⃗_a ·𝐎_a R⃗_a +h⃗_b·𝐎_b R⃗_b + [𝐕𝐎_a𝐓𝐎_b^T]  ) 
       - 𝐎_a, 𝐎_b ∈𝚂𝙾(d_a,b^2-1) min  ( h⃗_a ·𝐎_a R⃗_a +h⃗_b·𝐎_b R⃗_b + [𝐕𝐎_a𝐓𝐎_b^T]  ) ,

where the equality holds iff d=2. Applying hypothesis (i) or (ii) of Appendix <ref> to the state and Hamiltonian, 

    𝒞_ P(ρ_ab, H_ab)  ≤𝐎_a, 𝐎_b ∈𝚂𝙾(d^2 - 1) max  [𝐕𝐎_a𝐓𝐎_b^T]  + 𝐎_a, 𝐎_b ∈𝚂𝙾(d^2 - 1) max  [(-𝐕)𝐎_a𝐓𝐎_b^T]  ,

Taking into account the linearity of the trace, so that tr(-X) = -tr(X), it is enough to consider optimization of one of the terms in (<ref>). Without loss of generality, we perform the optimization of the second term that coincides with (<ref>), hence, is given by (<ref>) for det(-𝐕𝐓) ≥ 0 and (<ref>) det(-𝐕𝐓) < 0. Then, we take into account multilinearity of determinant of a matrix, so that det(-X) = (-1)^ndet(X), where n=d^2 - 1 is dimension of 𝐕 and 𝐓. Therefore, for even d, the optimization of the first term in (<ref>) will be given by (<ref>) for det(-𝐕𝐓) < 0 and (<ref>) det(-𝐕𝐓) ≥ 0. This leads to

    𝒞_ P(ρ_ab, H_ab)     ≤    ( λ^|𝐕|·λ^|𝐓|) + (λ^|𝐕|·λ^|𝐓|) - 2 λ_0^|𝐕|λ_0^|𝐓|) 
       =    2( (λ^|𝐕|·λ^|𝐓|) - λ_0^|𝐕|λ_0^|𝐓|).

On the other hand, for odd d, the optimization of the first term in (<ref>) will be the same as for the second one, so that 

    𝒞_ P(ρ_ab, H_ab) ≤
          2 (λ^|𝐕|·λ^|𝐓|),    if det(-𝐕𝐓) ≥ 0,
    
          2((λ⃗^|𝐕|·λ⃗^|𝐓|) - 2λ_0^|𝐕|λ_0^|𝐓|)    if det(-𝐕𝐓) < 0.




§ COMPARISON OF ENTANGLEMENT CRITERIA IN SPECIFIC CASES FOR TWO QUBITS


First, we write down the parallel capacity criterion, taking into account the saturated upper bound (<ref>) for d=2,

    2(( λ^|𝐕|·λ^|𝐓|) - λ_0^|𝐕|λ_0^|𝐓|) > 𝒞_1(H_ab),

where we have taken into account decomposition (<ref>) in terms of eigenvalues of 𝐕 and 𝐓. In order to decompose in the same way the maximum parallel capacity of separable states, we recall that it is convex, so that it is enough to optimize the parallel capacity over pure separable states,

    𝒞_ P^1(H_ab) = max_|Ψ⟩∈𝒮^1𝒞_ P(|Ψ⟩⟨Ψ|, H_ab) .

Since |Ψ⟩ = |ϕ_a⟩⊗|ψ_b⟩ are product states, the corresponding matrix 𝐓 is a Kronecker product of the corresponding Bloch vectors n_ϕ_a and n_ψ_b. In turn, such a matrix has a single non-zero eigenvalue, which is equal to the scalar product (n_ϕ_a·n_ψ_b). Therefore, only a single term in the eigenvalue decomposition (<ref>) survives, and the optimal state corresponds to the case when the highest eigenvalue λ_2^|𝐕| of 𝐕 survives,

    𝒞_ P^1(H_ab) = 2 λ_2^|𝐕|max_n_ϕ_a, n_ψ_b (n_ϕ_a·n_ψ_b).

Since the optimization is performed over pure states, so that both Bloch vectors have magnitude 1, the maximum is achieved for n_ϕ_a = n_ψ_b, and we obtain the following form of the parallel capacity criterion,

    λ_1^|𝐕|λ_1^|𝐓| + λ_2^|𝐕|λ_2^|𝐓| > λ_2^|𝐕|.



 


On the other hand, the work fluctuation criterion for d=2 requires

    (λ_0^|𝐓|)^2 + (λ_1^|𝐓|)^2 + (λ_2^|𝐓|)^2 > 1 - | |R_a|^2 - |R_b|^2 |^2.

We question whether there exist entangled states that can be detected by the parallel capacity criterion but do not lead to large variance of stochastic work distribution. Inverting the previous inequality and taking into account that λ_0^|𝐓|≤λ_1^|𝐓|≤λ_2^|𝐓|, we find that entanglement in the states satisfying

 


    (λ_0^|𝐓| < √(1 - | |R_a|^2 - |R_b|^2 |^2/3)≤1/√(3)) ∪( λ_2^|𝐓| > λ_2^|𝐕|/λ_1^|𝐕| + λ_2^|𝐕|≥1/2)

can be detected by parallel capacity, yet not by the work fluctuations criterion.





    


 
 



  
 
 
 
    
     
    
    

 
 



§ CONSTRAINTS FOR LOCAL UNITALITY AND APPROXIMATE SEPARABILITY



The constraints that establish the relevant properties for J_Λ as an approximation to a product of unitary channels are given by the last two lines in Eq. (<ref>). In what follows, we prove that the condition for local unitality [Eq. (<ref>)] in terms of its Choi operator is equivalent to 

    Tr_𝚒𝚗^(i)[J_Λ] = 1/d_iTr_𝚒𝚗^(i), 𝚘𝚞𝚝^(i)[J_Λ]⊗1_𝚘𝚞𝚝^(i).

Further below, we comment on the remaining constraint, which approximates separability of the channel Λ.

In what follows, we restrict our discussion to the bipartite scenario to simplify notation, but the result can be straightforwardly adapted to the n-partite case.


Eq. (<ref>) is equivalent to Eq. (<ref>)



First, note that, in terms of Choi states, local unitality [Eq. (<ref>)] is equivalent to

    Tr_𝚒𝚗[J_Λ (ρ^T ⊗1_𝚒𝚗^(i)⊗1_𝚘𝚞𝚝)] = ρ' ⊗1_𝚘𝚞𝚝^(i).

By tracing subsystem 𝚘𝚞𝚝^(i) on both sides of the equation above, it can be deduced that ρ' = 1/d_i Tr_𝚒𝚗, 𝚘𝚞𝚝^(i)[J_Λ (ρ^T ⊗1_𝚒𝚗^(i)⊗1_𝚘𝚞𝚝)], and the condition for local unitality can be equivalently rewritten as

    Tr_𝚒𝚗   [J_Λ(ρ^T ⊗1_𝚒𝚗^(i)⊗1_𝚘𝚞𝚝)] = 
       1/d_i Tr_𝚒𝚗, 𝚘𝚞𝚝^(i)[J_Λ (ρ^T ⊗1_𝚒𝚗^(i)⊗1_𝚘𝚞𝚝)] ⊗1_𝚘𝚞𝚝^(i).



Assume all spaces involved to be finite-dimensional and let {B^1_μ_1⊗ B^2_μ_2}_μ_1,μ_2 be a orthogonal basis for operators acting on the input spaces, where B^i_0 = 1_𝚒𝚗^(i) and Tr[B^i †_μ B^i_ν ] = d_i δ_μ,ν. Let then J_Λ be decomposed as

    J_Λ = ∑_μ_1,μ_2 B^1_μ_1⊗ B^2_μ_2⊗ A_μ_1,μ_2,

where A_μ_1,μ_2 are the remaining part of the decomposition, acting on the output spaces. Extend the action of J_Λ to any bounded operator on the input spaces and use the basis elements (transposed) as inputs for the channel Λ, so that, e.g., for ρ = (B^1_μ_1)^T, it is obtained

    1/d_1 d_2Tr_𝚒𝚗[J_Λ (B^1_μ_1⊗1_𝚒𝚗^(2)⊗1_𝚘𝚞𝚝)] = A_μ_1,0,

which, combined with Eq. (<ref>), implies

    A_μ_1,0 = 1/d_2 Tr_𝚘𝚞𝚝^(2)[ A_μ_1,0] ⊗1_𝚘𝚞𝚝^(2).

These are equations relating the basis components of Tr_𝚒𝚗^(2)[J_Λ], adjoining the elements B^1_μ_1 in the corresponding input space and summing over the free index, the basis-independent expression is found

    Tr_𝚒𝚗^(2)[J_Λ] = 1/d_2 Tr_𝚒𝚗^(2), 𝚘𝚞𝚝^(2)[ J_Λ] ⊗1_𝚘𝚞𝚝^(2).

A similar reasoning can be applied for the identity on subsystem 1 and Eq. (<ref>) is obtained.


On the reverse direction, assuming Eq. (<ref>) valid, it suffices to multiply the resulting operators on both sides by ρ^T_S-S_i (meaning a state that acts on all subsystems except for 𝚒𝚗^(i)). Use that Tr_𝚒𝚗^(i)[J_Λ]·ρ^T_S-S_i = Tr_𝚒𝚗^(i)[J_Λ (ρ^T_S-S_i⊗1_𝚒𝚗^(i))], and trace out the remaining input subspaces. The resulting relation is precisely Eq. (<ref>), which is the local unitality condition.




 §.§ Approximate separability


The remaining constraint in Eq. (<ref>) specifies that J_Λ should be approximately fully separable, considering each party to be composed by the pair input and output subsystems. To simplify the discussion, a bipartite system is considered with subsystems A and B. Extending the results to channels that act on more subsystems is immediate.

A separable channel Λ is a channel with the structure

    Λ(∙) = ∑_i p_i Λ^i_A ⊗Λ^i_B(∙),

where p_i are the coefficients of the convex combination of the product channels Λ^i_A ⊗Λ^i_B, such that Λ^i_A ⊗Λ^i_B (ρ_A ⊗ρ_B) = Λ^i_A(ρ_A) ⊗Λ^i_B(ρ_B). Channel separability immediately translates into separability of the corresponding Choi state across the different pairs of input and output subsystems:

    J_Λ   = ∑ p_i |u_1, u_2⟩⟨ v_1, v_2| ⊗Λ^i_A(|u_1⟩⟨ v_1|)⊗Λ^i_B(|u_2⟩⟨ v_2|) 
       = ∑_i p_i  J_Λ^i_A⊗ J_Λ^i_B,

where the sum in the first equality is done on all variables i, u_1,u_2,v_1,v_2. On the second equality, J_Λ^i_X are the Choi states for each local channel (X=A,B): J_Λ^i_X = ∑_u,v |u⟩⟨ v| ⊗Λ^i_X(|u⟩⟨ v|).

Therefore, a test for separability on J_Λ similar to tests for separability on quantum states must be applied to ensure the ideal structure for the channel Λ. Since product states are extremal points in the set of separable states, it is expected that an approximation that converges to the set of separable channels should also converge to a pure product of unital channels. The hierarchy of tests introduced in <cit.> simplifies the problem of separability to linear restrictions on extended states, that represent a replication of the information used to produce the original state, a feat that can only be faithfully performed in an indefinite manner for separable states <cit.>.

Each level of the hirearchy includes the previous ones, with the “level zero" representing postivity of partial transpose (PPT), meaning that not only J_Λ must be positive semidefinite, but also the transposes over 𝚒𝚗^(i), 𝚘𝚞𝚝^(i), for any subsystem i. This is so because a full transpose of a channel preserves its positive-semidefiniteness, consequently, the partial transpose of a separable Choi state should correspond to the transposition of a local channel, mapping a valid separable channel into a different valid separable channel. Failure in complying to this criterion implies that the channel cannot be taken as separable.

Let 𝖣𝖯𝖲_k correspond to the set of Choi states that pass the k-th level of the test hierarchy, level 0 corresponding to operators J_Λ with PPT. For k ≥ 1, the set 𝖣𝖯𝖲_k is also characterized by the operators that possess a k-symmetric extension in all parties, i.e. operators J_Λ that admit a Choi state J^(k)_Λ≥ 0 acting on the extended space ℋ^⊗(k+1)_A ⊗ℋ_B, such that

    𝖲𝖶𝖠𝖯_a,b  J^(k)_Λ  𝖲𝖶𝖠𝖯_a,b   = J^(k)_Λ,  ∀ a,b ∈{0,…,k}, a<b 
    (J^(k)_Λ)^T_0… j   ≥ 0,  ∀ j ∈{0,…,k}, 
    Tr_A_1… A_k[J^(k)_Λ]    = J_Λ,

where 𝖲𝖶𝖠𝖯_a,b is the swap operator for swapping subsytems A_a and A_b (indexed from 0 to k) and T_0… j is partial transposition over all j+1 first copies of the subsystem A. 

The hierarchical structure of the test can be seen from Eq. (<ref>): a state that admits a (k+1)-symmetric extension necessarily admits a k-symmetric extension, since tracing J^(k+1)_Λ over any of the copy subsystems results in a Choi state that satisfies all the required properties for the k-th extension. Furthermore, a state is separable if and only if it passes the tests for all k ≥ 0 <cit.>, thereby ensuring completeness to the test. Any channel that is not exactly separable will fail the test for some given k. 

Although finding this certification could be costly if higher levels of the hierarchy become necessary, in practice it is observed that the 0-th level test already bounds well the separable set for bipartite, low-dimensional systems. With this, a convenient approximation to separable channels is attained.



§ DETAILS ON LOWER BOUNDING PE AND COMPARISON BETWEEN UPPER BOUNDS



For the results in section <ref>, a lower bound on PE was computed by direct optimization of the local unitaries applied on each subsystem. For qubits, a convenient parametrization is given by

    U_θ,ϕ,γ = [   e^i(γ-ϕ)/2cos(θ/2)  e^-i(γ+ϕ)/2sin(θ/2);   e^i(γ+ϕ)/2sin(θ/2) -e^-i(γ-ϕ)/2cos(θ/2) ],

with θ∈ [0,π] and ϕ, γ∈ [0, 2π]. This parametrization has been used for optimization using the Werner states (see Fig. <ref> in the main text), where direct nonlinear optimization using the Nelder-Mead algorithm was performed over U_θ,ϕ,γ. Given the symmetries of Werner states [Eq. (<ref>)], optimization only over subsystem A was considered, although, given the small number of parameters, good convergence can also be obtained in more general cases, with optimization over both parties. In Fig. <ref>, convergence is observed, as the lower bound from direct optimization matches the upper bound given by the analytical formula of Eq. (<ref>). For higher dimensions, an alternative parametrization in terms of generators of SU(N) has been used. For a given orthonormal basis of Hermitian N × N matrices, {σ^j}_1 ≤ j ≤ d^2-1, with traceless σ^j, unitaries are given by U = exp(-i ∑_j x_j σ^j), with x_j ∈ [0,π].  Lower bounds for double qutrit systems (Fig. <ref>) were obtained using the parametrization in terms of generators. Closure of the gap between estimates indicates that the approximation is effective for given pairs of states and Hamiltonians produced.





