Paper Title: Quantum Computing Universal Thermalization Dynamics in a (2+1)D Lattice Gauge Theory.



=5

IQuS@UW-21-085,UMD-PP-024-08




niklasmu@uw.edu
InQubator for Quantum Simulation (IQuS), Department of Physics,
University of Washington, Seattle, WA 98195, USA

Department of Physics, Duke University, Durham, NC 27708
Duke Quantum Center, Duke University, Durham, NC 27701
The NSF Institute for Robust Quantum Simulation, University of Maryland, College Park, Maryland 20742, USA

Duke Quantum Center, Duke University, Durham, NC 27701
Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708
School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853

Department of Physics and Maryland Center for Fundamental Physics, University of Maryland, College Park, MD 20742 USA
Joint Center for Quantum Information and Computer Science, NIST and University of Maryland, College Park, MD 20742 USA
The NSF Institute for Robust Quantum Simulation, University of Maryland, College Park, Maryland 20742, USA
National Quantum Laboratory (QLab), University of Maryland, College Park, MD 20742 USA

Department of Physics, Duke University, Durham, NC 27708
Duke Quantum Center, Duke University, Durham, NC 27701
Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708
The NSF Institute for Robust Quantum Simulation, University of Maryland, College Park, Maryland 20742, USA

§ ABSTRACT 
Simulating non-equilibrium phenomena in strongly-interacting quantum many-body systems, including thermalization, is a promising application of near-term and future quantum computation.
By performing experiments on a digital quantum computer consisting of fully-connected optically-controlled trapped ions, we study the role of entanglement in the thermalization dynamics of a Z_2 lattice gauge theory in 2+1 spacetime dimensions. 
Using randomized-measurement protocols, we efficiently learn a classical approximation of non-equilibrium states that yields the gap-ratio distribution and the spectral form factor of the entanglement Hamiltonian. 
These observables exhibit universal early-time signals for quantum chaos, a prerequisite for thermalization.
Our work, therefore, establishes quantum computers as robust tools for studying universal features of thermalization in complex many-body systems, including in gauge theories.


Quantum Computing Universal Thermalization Dynamics 


in a (2+1)D Lattice Gauge Theory 

    Marko Cetina
    February 2024
==========================================================================================








§ INTRODUCTION


Thermalization of isolated quantum many-body systems
in e.g., ultra-cold atomic gases <cit.>, trapped ions <cit.>, condensed matter physics <cit.>, cosmology <cit.>, and nuclear and high-energy physics <cit.>, remains a vibrant frontier.  
Most quantum systems thermalize according to the Eigenstate Thermalization Hypothesis (ETH) <cit.>, which
posits that observables eventually attain thermal values after long unitary evolution <cit.>. Yet, probing large quantum many-body systems undergoing thermalization is inherently challenging due to the non-equilibrium nature of the process, which precludes using first-principle classical computational techniques. Recent advancements in quantum information theory and experiment <cit.> have brought this topic within immediate experimental reach, potentially enabling, via quantum simulation <cit.>, verification of important thermalization paradigms, and discovery of novel principles grounded in quantum information science.


While it is commonly posited that quantum chaos and ergodicity are prerequisites for thermalization <cit.>, their demonstration in the context of quantum many-body systems remains somewhat elusive. Indicators of chaos and ergodicity involve measures associated with the eigenvalues of a given Hamiltonian <cit.>, or the properties of its eigenstates <cit.>. A connection has been drawn recently between 

the entanglement of quantum states and quantum chaos, via the so-called entanglement Hamiltonians (EHs) <cit.>. This connection can, in principle, be leveraged to experimentally probe thermalization via quantum simulation. Most studies to date, nonetheless, have stayed in the realm of theoretical exploration <cit.>.



Analog quantum simulation allows one to monitor a quantum system continuously in time. However, current analog quantum simulators have limited programmability and are restricted to probing specific physical models and a limited set of observables <cit.>. In contrast, universal digital quantum computers allow, in principle, the probing of the dynamics of any physical model. The universal control in digital quantum computers enables the use of tomographic techniques to extract a wide range of observables, including entanglement.  
Digitized time evolution, via 

Trotterization <cit.> or other schemes <cit.>, limits the near-term simulations to times that are short compared to those of thermalization. Nonetheless, there exist universal phenomena that are indicative of quantum chaos and ergodicity at earlier times. Probing these phenomena, therefore, is an attractive near-term opportunity for digital quantum computers.

Gauge theories and their lattice formulations are among prime physical models whose simulations will benefit from quantum-computing technology <cit.>. Gauge theories are key in high-energy and nuclear physics <cit.>, condensed and synthetic quantum matter <cit.>, local fermion-to-qubit mappings <cit.>, and quantum-error correction  <cit.>. Studying thermalization dynamics of gauge theories, e.g., in early universe and in high-energy particle collisions, remains challenging using first-principles simulation methods <cit.>. As a first step in experimentally probing thermalization dynamics of gauge theories, we study a Z_2 lattice gauge theory (LGT) in 2+1 spacetime dimensions <cit.> using a digital trapped-ion quantum computer <cit.>. 
We use a chain of fifteen ^171Yb^+ ions to realize a general-purpose fully-connected digital quantum computer with twelve qubits, and use this computer to natively and accurately encode system's initial state, evolve it in time, and measure observables.

Our analysis relies on EH tomography <cit.> in combination with randomized-measurement protocols <cit.> to learn representations approximating non-equilibrium states. The EH is defined as H_ E≡ -log(ρ_A), where ρ_A denotes the reduced density matrix of subsystem A formed by bipartitioning a (pure) quantum state. The utility of EH as a theoretical and experimental tool stems from the observation that, in many cases, it consists of approximately local operators <cit.>. This resemblance to conventional Hamiltonians, i.e., energy operators governing system dynamics, facilitates simple theoretical analysis: If an EH is (k-)local, it can be described by parameters whose number scales polynomially with the system size, unlike the matrix ρ_A which requires an exponential number of parameters. 

Randomized-measurement protocols <cit.> can be used to learn the EH: after repeatedly preparing and then measuring the quantum state in a randomly chosen basis, one can fit a  parameterized EH to the obtained measurement outcomes.

Because the spectrum of H_ E, known as the entanglement spectrum, scales logarithmically with the Schmidt eigenvalues of ρ_A, it is difficult to learn an EH and reproduce the entire entanglement spectrum with arbitrary accuracy. Randomized-measurement protocols may require resolving exponentially small probabilities even for relatively small systems, which is out of reach of current quantum devices and perhaps fault-tolerant devices, too. 
The present experiments can, therefore, only aim at approximating H_ E and accurately reproducing its low-energy spectrum. Applications of this approach include the verification of topological phases <cit.>.

Nonetheless, what if the precise quantitative structure of H_ E were not crucial, but only its statistical properties were? This viewpoint is that of random matrix theory <cit.>, where metrics like level distribution and spectral form factor differentiate between integrable and chaotic dynamics, while being indifferent to quantitative details, with the notable exception of symmetries. This statistical perspective underpins our study. Guided by physical insights regarding the expected operator content of a non-equilibrium EH, we ask whether one can learn, from experimental data, a classical representation of a state to answer a simpler question: does a quantum state exhibit universal signatures of quantum chaos evident in the statistical properties of its EH? Crucially, can we discern this scenario from one where the state lacks chaotic behavior?

To answer these questions, we focus on two observables indicative of quantum ergodic and chaotic behavior: the entanglement(-Hamiltonian) gap-ratio distribution (EGRD) <cit.> and the entanglement spectral form factor (ESFF) <cit.>. The EGRD is predicted to exhibit level repulsion for chaotic states, in contrast to uncorrelated levels in non-chaotic scenarios. Similarly, the ESFF is predicted to display a plateau-ramp structure in chaotic states
 [Strictly speaking, both indicate quantum ergodicity which usually implies quantum chaos.]. To experimentally test these predictions, we initialize our system in a product state and perform a quantum quench by digitally evolving the system for a variable length of time. At short evolution times, we observe an EGRD that is characteristic of uncorrelated states. With increasing evolution time, we observe the onset of level repulsion and a ramp-plateau structure in the ESFF, indicative of quantum chaos. Detailed analysis, including comparison with emulated data, suggests that the observed behavior primarily arises from the thermalization dynamics of the isolated quantum system under simulation, with experimental inaccuracies and shortcomings of our ansatz playing a small, though non-negligible, role. Our work, therefore, establishes present-time quantum computers as robust tools for studying universal feature of thermalization dynamics in complex many-body systems.

This manuscript is organized as follows. In Secs. <ref> and <ref>, we introduce, respectively, the model and the experimental setup, including state preparation, real-time evolution, and measurement. We discuss our tomography scheme and classical post-analysis in Sec. <ref>. Finally in Sec. <ref>, we present our main experimental results, reporting the observation of a behavior consistent with quantum chaos in the structure of the EH of time-evolved states. Several Appendices offer additional insights: Appendix
<ref> provides further details of 
the quantum circuits employed;
Appendix <ref> investigates finite-size, Trotterization, and late-time effects; and Appendix <ref> offers a comprehensive explanation of our EH-tomography protocol.







§ MODEL


We focus on a Z_2 LGT in 2+1 D with the Hamiltonian


    H = ∑_□ W_□ +g ∑_ℓσ_ℓ^z .


Here, W_□≡∏_ℓ∈□σ_ℓ^x is a magnetic-field operator, where □ denotes the elementary (square) plaquettes of the lattice [see fig:model(a)]. σ^x_ℓ and σ^z_ℓ are Pauli operators representing gauge link and electric fields on edge ℓ, respectively. These act on spin-1/2 hardcore bosons residing on the edges of a two-dimensional spatial square lattice. The first term in eq:Z2Hamiltonian corresponds to the magnetic energy, and the second to the electric energy, with coupling g controlling the relative strength of the two non-commuting contributions. In this formulation, the z basis, therefore, represents the electric basis, while the x basis corresponds to the magnetic basis. The Hamiltonian is expressed in dimensionless units and ħ is set to one throughout.

We consider the joint +1 eigenspace of the Gauss-law operators G_j≡∏_ℓ∈ +_jσ^z_ℓ as the physical Hilbert space, where ℓ∈ +_j denote the four links adjacent to a lattice site j. In the subsequent discussion, we consider a quasi-1D chain composed of L_x plaquettes along the x direction with periodic boundary conditions. Fixed boundary conditions are applied along the short side of the chain (with L_y=1 plaquettes) as illustrated in fig:model(b). In this configuration, all Gauss laws contain three links at each site. Consider a short-ribbon operator V_y≡σ^z_ασ^z_β, where α and β are the two opposing links along the inner and the outer circumference at an arbitrary position along the closed chain shown in fig:model(b) [Note that V_y is independent of α and β because, for Gauss law eigenstates, the operator can be acted on by Gauss law operators in such a way that it can be moved around the lattice.]. This operator commutes with the Hamiltonian and defines superselection sectors of the model; we work in the V_y=1 sector (i.e., short-ribbon operators possess eigenvalue one).

As initial conditions, we randomly select Gauss-law–respecting electric eigenstates (z-basis product states) of the form  | Ψ(t=0)⟩≡ | Ψ_0⟩ = ∏_ℓ | ℓ⟩ with | ℓ⟩∈{| ⟩ ,| ⟩}. We then define a subsystem A consisting of L_A ≤ L_x/2 plaquettes. The reduced density matrix ρ_A(t) of A after evolution time t is related to the density matrix ρ(t) of the whole system as


    ρ_A (t) ≡Tr_A̅[ ρ(t)],   ρ(t)≡ U(t) ρ_0 U(t)^†,


where A̅ is the complement of subsystem A shown in fig:model (c) and (d), ρ_0 ≡ |Ψ_0 ⟩⟨Ψ_0 |,  and U(t)≡ e^-iHt is the time-evolution operator.

To directly encode the degrees of freedom of the model described onto the qubits, one requires one qubit per link degree of freedom, or N_q=3L_x qubits in total. To adapt the simulation to the available hardware, we consider a dual formulation of the model that requires only N_q=L_x + 2 qubits. In the dual formulation, all operators, except those containing operators having support  at the boundaries between the subsystems, labeled ℓ_A̅A and ℓ_AA̅, are entirely represented using gauge-invariant (Ising dual) spin-1/2 variables which act solely within the physical Gauss-law subspace; they are denoted by μ_i^x,z Pauli matrices [see purple circles in fig:model(c)]. Concretely, μ_i^x → W_i= ∏_ℓ∈□_iσ_ℓ^x, while electric variables are represented by μ^z_i μ_i-1^z ≡σ_ℓ^z where ℓ is the link between plaquettes i and i-1. The corresponding “bulk” Hamiltonian terms for the subsystem A then reads


    H_A^ dual =  ∑_i∈A_ bulkμ^x_i  + g (∑_⟨ i,j ⟩∈ Aμ^z_iμ^z_j
    +κ∑_i∈ Aμ^z_i),


where κ≡ 1+V_y. Here, i ∈ A_bulk in the sum over μ_i^x indicates that the original plaquette □_i is entirely in A (i.e., does not touch the boundaries). ⟨ i,j ⟩∈ A denotes nearest-neighbor bulk-spin pairs i,j in A. Finally, the sum over μ_i^z runs over all bulk-spin indices i. The terms in the Hamiltonian acting on the complement are defined identically. Importantly, at the boundaries between the subsystems, ℓ_A̅A and ℓ_AA̅ [see the orange circles in Fig. <ref>(c)], the gauge-variant variables of the LGT, as described by eq:Z2Hamiltonian, are retained. Denoting the Ising-spin index at the A side of one of the boundaries to be k, the Hamiltonian terms coupling the subsystems at this boundary are


    H^ dual_A̅A≡μ^x_
        kσ^x_ℓ_A̅A + μ^x_
        k-1σ^x_ℓ_A̅A + gσ^z_ℓ_A̅A ,


with a similar definition for the other boundary. Note that on the plaquette k containing one boundary link, μ_k^xσ^x_ℓ_A̅A≡ W_k, where μ_k^x≡∏_ℓ∈□̃_kσ_ℓ^x, with □̃_i referring to all but non-boundary links of plaquette k, and similarly for the plaquette k-1. The Gauss laws at the boundaries are not eliminated by the duality. The two Gauss laws in the dual model that are independent (one at each boundary) are


    G^ dual_A̅A≡μ^z_
    
    kσ^z_ℓ_A̅Aμ^z_
    k-1 ,


and similarly for the boundary at ℓ _AA̅.
The dual Hamiltonian of the model is the sum of the terms above:


    H^ dual =H_A^ dual +H_A̅^ dual + H^ dual_A̅A+ H^ dual_AA̅ .



This formulation ensures that all gauge-invariant variables of the LGT and its dual have the same expectation values. Importantly, by maintaining the Gauss laws at the boundaries, the entanglement properties of the dual formulation are identical to those of the LGT <cit.>. This is a subtle but crucial distinction from the standard Ising duality <cit.>, which does not preserve the entanglement structure.

The entanglement structure depends on  the symmetries of the reduced density matrix, which stem from the remaining Gauss laws.
Specifically, [S_j,ρ_A(t)] =0,
where in LGT variables, S_j≡∏_ℓ_j ∈ A σ^x_ℓ_j,
with ℓ_j being the two links originating from a boundary site with lattice coordinates j=(j_x,j_y) within A. This is a direct consequence of the state being a Gauss-law eigenstate <cit.>. In addition, [V_x,A,ρ_A(t)] =0 where the long-ribbon operator within A is defined as V_x,A≡∏_ℓ∈𝒫σ^x_ℓ where σ^x_ℓ act on the original degrees of freedom along the path 𝒫 connecting both boundaries within A, see Fig. <ref>(d). Not all symmetry operators are independent.  For the quasi-1D chain of plaquettes shown in fig:model, only one of the two S_j is independent at each boundary. We place one of the boundaries at sites (0,0) and (0,1) and the other at sites (L_A,0) and (L_A,1). We then choose the two independent S_j operators to be S_(0,0) and S_L_A,0, henceforth referred to as  S_AA̅ and S_A̅A, respectively. Note that, when acting on a Gauss-law–respecting state, V_x,A =S_AA̅S_A̅A. Consequently, there are four independent symmetry blocks ρ_A(t)≡⊕_s=1,⋯,4ρ_A,s(t) that will play a role in our analysis.  In the dual formulation, these symmetry operators are simply S_AA̅^ dual≡μ_k^z σ^z_ℓ_A̅A and S_AA̅^ dual≡μ_k'^z σ^z_ℓ_A̅A where k and k' are the first and last plaquettes within A [This discussion is general, independent of the choice of a V_y eigensector.]. We denote the eigenvalue {1,1}, {1,-1}, {-1,1}, and {-1,-1} sectors of {S_A̅A^ dual,S_AA̅^ dual} operators with labels s=1,2,3,4, respectively. 






§ EXPERIMENTAL SETUP


We use a universal trapped-ion digital quantum computer, schematically shown in fig:introoverview(a), to simulate the time evolution of the dual formulation of the Z_2 LGT. We consider L_x=10 plaquettes, focusing on the entanglement structure of a subsystem consisting of L_A=4 plaquettes, as shown in fig:model(c). 

Our experimental system consists of a linear chain of fifteen ^171Yb^+ ions trapped along the 𝐱 axis of a microfabricated ion trap  <cit.> and spaced by approximately 3.7 μm. The twelve LGT spins are mapped to the ions as shown on fig:model(c). The | ⟩ and | ⟩ spin states are respectively mapped to the first-order magnetic-field–insensitive hyperfine | F=1, m_F=0⟩ and | F=0, m_F=0⟩ states of the ^2S_1/2 ground electronic manifold of the ions [Here, we use the quantum-information convention for the spin states, i.e., |↑⟩≡|0⟩ and |↓⟩≡|1⟩.]. The spin states are initialized in the |⟩ state by optical pumping and are measured by coupling to the excited ^2P_1/2, |F=0,m_F=0⟩ state using a 369-nm laser, wherein the presence (absence) of emitted photons differentiates between the bright state | ⟩ and the dark state | ⟩ with an average 0.3% infidelity, limited by off-resonant photon scattering.

The ions are individually addressed by an equispaced array of 1 μm-diameter, 355-nm, laser beams oriented perpendicular to the trap surface (𝐳 axis), together with a global 300 μm × 30 μm beam oriented along the 𝐲 axis, which is parallel to the trap surface (see fig:introoverview(a) and Ref. <cit.>). These beams drive stimulated Raman transition via the ^2P_1/2 and ^2P_3/2 excited electronic states, where a photon is absorbed from the global beam and emitted into an individual beam to flip the qubit from | ⟩ to | ⟩ and vice versa. All spin-manipulation light is produced by a single pulsed laser, modified to control its repetition rate to null 4-photon Stark shifts. The phase and amplitude of the 355-nm beams is controlled by single-channel and 32-channel acousto-optic modulators [Provided by L3 Harris Corporation.].




To minimize addressing errors, crosstalk, and stray coupling to axial motion of the ions <cit.>, single-qubit operations are realized using compound SK1 pulses <cit.> with Gaussian-shaped sub-segments, with typical infidelities of 0.2%. Entangling operations between any two qubits are realized via variable-angle pairwise Mølmer-Sørensen (MS) gates <cit.> employing laser-induced state-dependent forces on the ≈ 3-MHz motional modes of the ion chain oriented along the 𝐲+𝐳 axis. Robust decoupling from the ion motion at the end of each entangling gate is accomplished using amplitude-modulated, detuning-robust pulse waveforms <cit.>. Typical non-unitary errors of fully-entangling MS gates are 1% and consist predominantly of gate-angle errors and Z-flips of the individual qubits <cit.>.

Our ultimate goal is to study thermalization dynamics in real-time non-equilibrium conditions by learning system's entanglement structure. To this end, we first demonstrate the capability of our digital computer to compute the dynamics of the Z_2 LGT model, including initial-state preparation, time evolution, and subsequent measurements. We initialize the system in a randomly chosen electric eigenstates that respects Gauss law (|Ψ_0⟩=|↓↓↓↑↓↓↑↑↑↓↑↑⟩, see fig:model(c) for the qubit mapping, indicated in blue numbers, read from left to right) and simulate its time evolution for g=0.85 using four Trotter steps that separately implement the non-commuting terms of H^dual, i.e., those diagonal in electric (z) or magnetic (x) bases. Our choice of coupling is such that magnetic and electric terms are of similar magnitude, making the model sufficiently non-integrable. For each  evolution time t, we apply single-qubit gates to measure all qubits in either the Bloch x or the z bases and repeat each experiment N_shots=500 times. The measurement results are used to compute single- and two-qubit observables in the Bloch x and z bases and are plotted in Figure <ref> for select observables [For simplicity, the operators whose expectation values are measured are generally noted as Pauli σ operators. When acting on the bulk dual spins, they should be thought of as the Pauli μ operators, while when acting on the boundary spins, they are the Pauli σ operators in the original LGT.]. The statistical errors are determined by a standard bootstrap resampling analysis with N_boots=1000 bootstrap samples. Quantum circuits implementing Trotterized time evolution and measurements are provided in Appendix <ref>. Each Trotter step of evolution uses 12 variable-angle MS gates. 

For most evolution times, the expectation values of the physical operators (dark blue and red) show good agreement with the zero-parameter theory prediction (solid lines). We ascribe the ∼10% deviation in the evolution rate of the system to miscalibrations of the MS-gate angles. Importantly, the expectation values of gauge-invariance–violating operators (i.e., those that do not commute with the Gauss laws at the two boundaries), shown in light blue, are consistent with zero within measurement errors. Figure <ref> of Appendix <ref> provides a close-up of these observables, showing that gauge-invariance violation is about 5% throughout the time evolution. 

To establish the connection between these simulations and the signatures of chaos and thermalization, we now introduce a measurement and analysis protocol that allows the determination of the system's entanglement structure, going beyond low-order correlation functions.






§ ENTANGLEMENT-HAMILTONIAN SPECTRUM AND TOMOGRAPHY


Our main interest lies in the statistical properties of the Schmidt decomposition of the reduced density matrix of the subsystem A, given by ρ_A(t) ≡∑_λ p_λ(t) | λ(t) ⟩⟨λ(t)|, and its associated EH <cit.>


    H_ E(t) ≡ -log(ρ_A(t)) .


The EH's eigenspectrum, called the entanglement spectrum, is defined by {ξ_λ(t) ≡ -log(p_λ(t))}.





The statistical properties of the entanglement spectrum directly inform the thermalization dynamics of the system. One statistical probe is the distribution, P(r), of the gap ratios, r_λ, of the entanglement spectrum <cit.>,


    r_λ≡min(δ_λ , δ_λ-1)/max(δ_λ , δ_λ-1)  ,


where δ_λ≡ξ_λ - ξ_λ-1≥ 0 are the gaps between the eigenvalues ξ_λ of H_ E and r ≡{r_λ}. According to random matrix theory of Hermitian matrices, this quantity distinguishes chaotic from nonchaotic behavior depending on whether the distribution is centered away from zero (level repulsion) or centered around zero (uncorrelated levels) <cit.>. The gap ratios, nonetheless, only capture local correlations of the entanglement spectrum and, owing to their logarithmic nature, treat entanglement eigenvalues ξ_λ≡ -log(p_λ) associated with very small probabilities p_λ≪ 1 no differently than those where p_λ≈ O(1). 

A complementary view of the statistics of the entanglement spectrum that captures global correlations in the level distribution is afforded by the entanglement spectral form factor (ESFF) <cit.>,


    ℱ(θ ;t) ≡⟨1/ℛ^2_A(t)∑_λ,λ' e^iθ[ξ_λ(t) -ξ_λ'(t) ]⟩ ,


where ⟨·⟩ denotes the average over initial states |Ψ_0⟩. Here, ℛ_A ≡lim_α→ 0exp{1/1-αlog(∑_λ p_λ^α)} <cit.>  is the effective rank of H_ E, whose value depends on the state and lies in the range [1, 2^L_A] [In practice, both statistical measures are typically used to describe the properties of physical Hamiltonians, whose rank equals the Hilbert-space dimension. For EHs, we employ regularization suitable for non-full–rank matrices. To address levels with probabilities close to zero within machine precision, we define an effective rank. Only the ℛ_A lowest levels of the EH are included in the analysis. However, experimental constraints, such as finite statistics and device errors, typically lead to EHs that are reconstructed as nearly or exactly full rank, rendering any regularization obsolete. Nonetheless, non-full–rank EHs occur for states computed in exact diagonalization, particularly at early times, that we contrast our data with.].  Starting from an initial flat behavior, a ramp-plateau structure in the ESFF as function of the `Fourier time' θ is known to indicate ergodic behavior and imply quantum chaos <cit.>. 

While in classical computation of a tractably small system, the state and its entanglement are readily available, in a quantum-simulation experiment, these must be inferred from measurements. To tackle this challenge, we constrain the operator content of the EH to obtain an approximate H_E, and utilize an EH-tomography scheme based on randomized measurements <cit.>.   
Concretely, we perform a single-layer, single-qubit randomized-measurement <cit.> of the time-evolved state. 
At the end of the time evolution, this protocol applies one of N_𝒰 different gates 𝒰_j = ⊗_i=0,⋯,N_q-1 u_i,j for j=1,⋯,N_𝒰 consisting of independent single-qubit gates u_i,j sampled from a unitary 2-design <cit.>. For each 𝒰_j, N_shots experiments are performed, measuring all qubits in the Bloch z basis. 

We compare the relative frequencies of the different bitstrings b to the prediction based on an approximate EH inspired by the Bisognano-Wichmann (BW) theorem <cit.>. 
Explicitly, we assume that H_ E(t) is a linear combination of k-local terms 𝒪_i, i.e.


    H_ E (t ; {β_i} ) ≡∑_iβ_i(t) 𝒪_i .


To find a suitable set of operators { O_i }, we proceed as follows. Starting from the operator content of the physical Hamiltonian in eq:Z2Hamiltonian, containing at most 4-local terms (2-local in the dual formulation), new operators are generated iteratively by forming non-trivial commutators from the existing set. This process can be halted after two iterations (`commutators of commutators') resulting in a maximum of 7-local operators within the LGT framework (or 3-local in the dual formulation). We exclude operators that violate the Gauss laws and those that can be related to others upon application of the Gauss-laws. To ensure Hermiticity, we adjust prefactors and create linear combinations with conjugates [If this iteration were continued, eventually one would end up with an exponentially large set of all possible operators. Our approach is based on a purely heuristic observation that, in practice, only polynomial many (quasi)local operators are required to accurately reproduce the state ρ_A(t) with exponential accuracy <cit.>.]. To find the parameters β_i, a global optimization routine, based on standard Matlab libraries <cit.>, is used to determine the parameterization that most closely reproduces all measurements (refer to Appendix <ref> for details).






§ EXPERIMENTAL STUDY OF THERMALIZATION DYNAMICS OF Z_2 LGT


To study the evolution of the entanglement spectrum in the Z_2 LGT, we repeat the time-evolution experiments shown in Fig. <ref> starting from 6 randomly chosen initial electric eigenstates that satisfy the Gauss laws [We numerically checked that the results remain valid for other randomly chosen states in this basis]. Since these are not the eigenstates of the full Hamiltonian H, the system undergoes non-trivial time evolution when evolved under H. For each evolution time t, symmetry sector, and initial state, we subsequently perform randomized-measurement tomography with g=0.85, N_𝒰=24 bases, and N_ shots=750 bitstring measurements in each basis, and use these measurements to reconstruct the EH as described in the previous section. This procedure yields a set of gap ratios for each initial state, symmetry sector, and evolution time.

In fig:gap-ratio-exp-analysis(a), we plot in orange points the average gap ratio ⟨ r⟩≡∑_r rP̅(r) of the reconstructed EH  as a function of the scaled evolution time g t. The plotted gap ratios are averaged over both the symmetry sectors and the 6 randomly chosen initial states. The black lines correspond to the predicted exact distributions following the Trotterized time evolution. The cyan lines correspond to the EH obtained from an optimal BW-inspired ansatz, where we numerically minimize the
the relative entropy (Kullback–Leibler divergence) between the exact state and the ansatz, corresponding to the limit of infinitely many measurements. A buildup of the level repulsion is discernible as the observed average gap ratio ⟨ r ⟩ evolves from ≈ 0.4 predicted for a non-repulsive Poisson distribution (blue dashed line) towards ≈ 0.6 characteristic of repulsive level statistics of a Gaussian Unitary Ensemble (GUE) <cit.>. Three time regimes (I), (II), and (III) are identified that correspond, respectively, to the evolution of the predicted average gap ratio toward the Poisson-distribution value, toward the GUE value, and to saturation at the GUE value.

In fig:gap-ratio-exp-analysis(b) we plot, for the three ranges of evolution times, the corresponding normalized distribution of the gap ratios, P̅(r), combined over 6 random initial states and 4 symmetry sectors. We observe a clear transition from early-time non-repulsion (Poisson distribution, blue dotted line) in regime (I) to level repulsion (GUE, red) in regime (III). At intermediate times, a distribution is observed between the initial absence of level repulsion and the subsequent emergence of level repulsion.  Error bars and bands denote the variance resulting from averaging over symmetry sectors and initial states. The predicted exact distribution (black line) exhibits a sharper peak around zero at the earliest times due to the reduced density matrix not being full rank. Conversely, the BW-inspired ansatz (cyan line) generally parametrizes a full-rank matrix unless couplings are very finely tuned, resulting in an overestimation of level repulsion in the initial stages. Calculations indicate that the onset of the three regimes remains consistent for any randomly chosen initial states and depends only on the (subsystem-)size when measured in coupling-independent units gt, indicating the universality of thermalization dynamics.






In fig:ESFF-simulator-analysis
we plot the ESSF reconstructed from the data used in fig:gap-ratio-exp-analysis. The three panels of the plot correspond to the three time regimes discussed in relation to the EGRD evolution. The displayed theory curves are averaged over initial states, symmetry sectors, and each of the three time ranges. Starting from an initial flat behavior of the ESSR as a function of θ in panel (I), our data in panels (II) and (III) clearly show the buildup of a ramp-plateau structure, indicating ergodic behavior. In panel (III), we show a fit (purple dotted line) to the observed ramp, indicating a power-law behavior θ^κ where κ = 0.6 ± 0.2, with the fit error determined by changing the fit regime. This exponent is consistent with the results obtained from the numerical analysis of a significantly larger  system in Appendix <ref>. 


Our results demonstrate that, using observables that reveal universal statistical or global features of entanglement spectrum, the onset of chaotic behavior can be robustly traced in the entanglement dynamics of arbitrary initial states. Importantly, time digitization and experimental infidelities in current systems do not significantly impact qualitative features of thermalization dynamics. 






§ CONCLUSIONS


Using a digital trapped-ion quantum computer, we observe early-stage thermalization dynamics in a Z_2 lattice gauge theory in 2+1 dimensions by generating nonequilibrium quantum states and measuring their entanglement structure. As performing quantitative state tomography is extremely challenging and resource intensive, we focused on a simpler question: can a randomized-measurement–based entanglement-Hamiltonian
tomography recover universal properties of the EH that indicate quantum chaos? We experimentally determine the entanglement spectrum of time-dependent quantum states and use two properties of this spectrum, namely the entanglement gap-ratio distribution [Eq. (<ref>)] and the  entanglement spectral form factor [Eq. (<ref>)], as indicators of the emergence of quantum chaos. Our data indicate that both quantities behave as expected: the initially non-repulsive level distribution transitions to a repulsive one. Likewise, a ramp-plateau feature of the ESFF is initially absent but builds up with time. Our results indicate that the timescales for these entanglement-based phenomena mainly depend on subsystem size.

The key technique enabling our analysis is a Bisognano-Wichmann–inspired ansatz for the EH, which allows the parameterization of a nonequilibrium state using a polynomial number of parameters. While this classical parameterization optimally reproduces the observed (randomized) measurement results, in Appendix <ref>, we show that the higher-lying part of the EH spectrum is not quantitatively recovered. Indeed, the measurement cost for precise state tomography still scales exponentially with the subsystem size. Fortunately, to detect the presence of quantum chaos, one needs to only distinguish repulsion from non-repulsion in the EGRD  or identify a ramp-plateau structure in the ESFF. Our work indicates that selecting a sufficiently small subsystem of a much larger, potentially classically nonsimulable system, can sufficiently constrain these observables. 

Our randomized-measurement procedure remains classically simulable because the lattice sizes we considered are manageable with exact diagonalization. Our procedure is also heavily tailored at minimizing the computational load on the quantum computer, avoiding deep quantum circuits and the need for error mitigation, at the cost of classical post-processing. To extend our approach to larger systems, several steps can be implemented in future work:



  –
The employed single-qubit one-layer randomization strategy is symmetry ignorant, i.e., it randomizes regardless of the known symmetry structure of the subsystem density matrix ρ_A(t). Although our scheme is tomographically complete, and enables state reconstruction along with its symmetries, it is inefficient.  Symmetry-conscious protocols, for the LGT in this work and for other models, have been developed <cit.> and could be used in future work.


  –
A major methodological uncertainty is the BW-inspired parameterization of the EH, which is entirely heuristic. While there is research exploring the operator content of the EH of ground, excited, and thermal states <cit.>, further investigations are needed into the applicability and limitations of EH-based schemes for far-from-equilibrium states. Generally, the BW(-inspired) ansätze describe the low-energy regime of the EH well, and incorporating progressively more nonlocal terms  improves convergence into the bulk <cit.> which the EGRD, and to a lesser extent the ESFF, predominantly rely on.  Notably, our analysis reveals that despite the quantitative discrepancy in the bulk, the statistical distribution of the EH appears to be accurately reproduced. This suggests that it may be unnecessary to perform precise state tomography when one's interest is solely in statistical properties. 


  –
Trotterization corresponds to time evolution with an effective, rather than the desired, Hamiltonian. We observe that employing too few Trotter steps leads to poor convergence of our optimization procedure for the BW-inspired ansatz. In this case, more nonlocal operators must be included in the ansatz; constraining these becomes challenging, especially when relying on a limited number of measurements. Since using a large number of Trotter steps increases 
experimental errors, the optimal Trotterization and its interplay with the BW ansatz should be further investigated.


  –
Detailed comparison between the emulator and the experimental data (see Appendix <ref>) highlights the influence of device errors, primarily for the time evolution and tomography steps of our algorithm. The dominant errors in our study are Z-flip errors and over- and under-rotations due to mechanical motion of the ions <cit.>.
Our tomography protocol is especially susceptible to small single-qubit rotation errors during randomization <cit.>.  
Recently demonstrated sympathetic cooling during circuit execution <cit.> would allow the needed measurement circuits to be executed with higher fidelity. The same technique could extend our study to larger systems, later times, and more Trotter steps, e.g., allowing us to directly test the eigenstate thermalization hypothesis in a digital quantum-computing set-up. 



In summary, our results demonstrate that entanglement structure is a measurable quantity in present-day LGT quantum-simulation experiments, and illustrates the potential value of our approach to probe thermalization dynamics and its robust universal features in strongly coupled isolated quantum many-body systems. A compelling future direction is to extend the investigation to later times, to probe aspects such as pre-thermalization <cit.> or fluctuation-dissipation relations <cit.>, once experimental capabilities permit. 
This would also allow for probing the applicability of the Eigenstate Thermalization Hypothesis, e.g., in systems with non-Abelian symmetries <cit.> and other gauge theories <cit.>. Furthermore, the experimental and theoretical tools of this study can be applied in a number of other applications, including obtaining thermodynamic quantities such as work and heat exchanged during nonequilibrium processes <cit.>, and detecting phases of matter, including topological phases <cit.>, in quantum-simulation experiments.






§ ACKNOWLEDGMENT

N.M. thanks H. Froland, A. Polkovnikov, M. Savage, M. Srednicki, X. Yao, T. Zache, P. Zoller, and the participants of the InQubator for Quantum Simulation (IQuS) workshop “Thermalization, from Cold Atoms to Hot Quantum Chromodynamics ”(<https://iqus.uw.edu/events/iqus-workshop-thermalization/>) at the University of Washington in September 2023 for many valuable discussions leading to this work. M.C. thanks L. Feng for valuable discussions and help with the experimental setup leading to this work.
N.M. acknowledges funding by the Department of Energy (DOE), Office of Science, Office of Nuclear Physics, IQuS (<https://iqus.uw.edu>), via the program on Quantum Horizons: QIS Research and Innovation for Nuclear Science under Award DE-SC0020970. Z.D., M.C., and T.W. were supported by the National Science Foundation's Quantum Leap Challenge Institute for Robust Quantum Simulation under Award OMA-2120757. Z.D. further acknowledges support by the DOE, Office of Science, Early Career Award DE-SC0020271. This work is further supported by a collaboration between the US DOE and other Agencies. This material is based upon work supported by the DOE, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator. 






§ COMPETING INTERESTS

M.C. is a co-inventor on  patents that are licensed from the University of Maryland to IonQ, Inc.













§ CIRCUIT REPRESENTATION FOR STATE PREPARATION, EVOLUTION, AND MEASUREMENT



 


In this Appendix, we provide circuits employed to experimentally realize state preparation, time evolution, and randomized measurement in the Z_2 LGT in (2+1)D within the dual Isnig formulation.

We label qubits with indices q=0,…,L-1, with L=12. Qubit q=0 corresponds to the spin at ℓ_A̅A positioned at the interface between subsystem A and its complement, identifiable by an orange circle in fig:model(c). Qubits q=1,…,4 represent the succeeding dual variables arranged counterclockwise at the center of plaquettes denoted by purple circles in fig:model(c). Following this sequence, qubit q=5 resides at the opposite boundary marked as ℓ_AA̅, and so forth. 










Working in the electric eigenbasis and starting from an all-up spin state, we first perform single-qubit σ^x bit-flip operations to select a randomly chosen product state in the σ^z basis that is consistent with Gauss laws, i.e., G_j |ψ(0)⟩=|ψ(0)⟩ for all j. Recall that in the dual formulation, there are only two Gauss-law operators each located at one of the subsystem's boundaries: G^ dual_A̅A = μ^z_11σ^z_0 μ^z_1 and G^ dual_AA̅ = μ^z_4σ^z_5 μ^z_6. Importantly, any spin configuration in either bulk, A and A̅, is physical, as in the dual formulation, only gauge-invariant degrees of freedom are kept in the bulk.


For time evolution, we utilize a first-order Trotter scheme with time-evolution operator


    U(t)≡ e^-itH^ dual≡∏_δ t U(δ t).


We take δ t to be a variable time step while keeping the Trotter depth fixed. In the simulations conducted in this work, t/δ t= 4, resulting in states and observables closely resembling the exact time-evolved states at early times. However, for late times, Trotter effects become more pronounced.

U(δ t) in Eq. (<ref>) can be written as U(δ t)≡∏_a∈ X,Z,XX,ZZ U_a with U_a ≡  e^-iδ tH^ dual_a; H^ dual_a are the respective 1- and 2-local operators of the dual Hamiltonian in Eq. (<ref>), sorted in X and Z operations, as well as single- and two-qubit entangling gates. While XX, X, and Z are native operations, ZZ entangling operations are realized via basis transformation and usage of the native MS gate,


    R^zz_ij(α) ≡ R^y_i(-π/2) R^y_j(-π/2)    R^xx_ij(α)   R^y_i(π/2) R^y_j(π/2).


Here, R^x/y/z_i(α) ≡ e^-iα/2σ_i^x/y/z and R^xx_ij(α) ≡ e^-iασ_i^xσ_j^x. The experimental errors of our MS gates increase with the absolute value of the gate angle. To minimize the gate error, all MS gates are optimized as follows. First, all angles are mapped to the regime α∈ [-π,π]. Within this range, we make the following substitution


    R^xx_ij(α) →
             R^xx_ij(α)    if |α| ≤π/4
    
             R^xx_ij(α+π/2) R^x_i(π) R^x_j(π)    if |α| ∈ (π/4,3π/4]
        &  α<0 
    
             R^xx_ij(α-π/2) R^x_i(π) R^x_j(π)    if |α| ∈ (π/4,3π/4]
        &  α>0
    
              R^xx_ij(α+π)     if α < -3π/4
    
              R^xx_ij(α-π)     if α > 3π/4


so that all MS operations are restricted to the range |α| ≤π/4. 

An example study of the initial-state preparation, time evolution, and measurement in a fixed (Bloch x or z) basis is shown in fig:experimentbasis (b) and (c) of the main text, where we presented measurement of several gauge-invariant and non-gauge–invariant
one- and two-qubit observables. In addition, we present the measurement of several other non-gauge–invariant observables in fig:gauge_invariance. These observables are expected to be zero, but remain non-vanishing because of finite measurements and device errors. A main source of errors are likely coherent errors, related to the over- and under-rotation of single- and two-qubit gates. While the time-evolution circuit only contains gauge-invariant operations for any gate angle employed, the final single-qubit rotations R^y_i(-π/2) and R^x_i(π/2) (that transform all qubits i from the Bloch z basis into the Bloch x and y bases, respectively) can introduce gauge-invariance violation if their rotation angle is set inaccurately. Additionally, errors stemming from initial-state preparation and readout processes can contribute to this violation.






§ FINITE-SIZE DEPENDENCE, TROTTERIZATION, AND LATE-TIME BEHAVIOR


In this Appendix, we perform various tests aiming to extend our study toward larger systems, and to provide an outlook on the expected late-time behavior of non-equilibrium states, features that are inaccessible in current quantum computers. We present a detailed classical  computation of a fairly large system (L=22, L_A=8 with coupling g=1), evolving the state continuously, and computing its entanglement structure using exact diagonalization, hence avoiding a Trotterization error. Such an analysis is possible for the model under consideration, owing to its relative simplicity. This simplicity enables us to achieve results that are fairly insensitive to finite-size effects, even with moderately large systems.  A similar classical computation will be impossible for more involved models or in higher dimensions.

We compute the same entanglement-related observables indicative of quantum chaos as in the main text, starting from several randomly drawn initial product states. The EGRD, i.e., the (normalized) distribution of the gap ratios P̅(r) with r defined after Eq. (<ref>), is computed similarly as in the main text: we consider ten randomly chosen initial states, and combine the gap ratios that are separately computed for each symmetry sector of the respective reduced states and for all initial states. Additionally, we combine the distribution of gap ratios in three time regimes (I), (II), and (III). The result is shown fig:gap-ratio-theory-analysis(a). Further, the mean of the gap ratio, combining initial states and symmetry sectors but not times, ⟨ r ⟩, is shown in fig:gap-ratio-theory-analysis(b) as a function of scaled time gt. 
The distribution in fig:gap-ratio-theory-analysis(b) is peaked near zero first, closely resembling a Poisson distribution. In this regime, the rank ℛ_A≪d_A, where d_A is the Hilbert-space dimension of the subsystem. The effective rank ℛ_A is defined after Eq. (<ref>) and plotted for the present example in fig:gap-ratio-theory-analysis(c) starting from a product state with ℛ_A=1 at t=0. The largest ξ_λ values in this regime correspond to extremely small probabilities p_λ = exp(-ξ_λ), close to or at the level of machine precision. A regularization of the smallest p_λ is required: we manually cut off probabilities below 10^-15 before computing ℛ_A, varying this limit by two orders of magnitude in each direction to provide the blue bands in panel fig:gap-ratio-theory-analysis(c), then using only the ℛ_A lowest levels in the analysis of the EGRD and the ESFF. Gray bands represent the variance with respect to the randomly chosen initial states. In regime (II), ⟨ r ⟩ quickly jumps to a value associated with a Gaussian Orthogonal Ensemble (GOE), until continuing, albeit at much slower pace, towards a  Gaussian Unitary Ensemble (GUE) indicating quantum chaotic behavior. At the end of this stage, ℛ_A= d_A is maximal, and all probabilities p_λ are well above machine precision, eliminating the need for any regularization. Finally,  regime (III) sees the saturation of the gap ratio to GUE level statistics (up to minimal finite lattice-size effects). The beginning and end of the time regimes (I-III) depend on the subsystem size and are chosen as follows, (I): 0 ≤ gt <1.8, (II): 1.8 ≤ gt <5.0, and (III): 5.0 ≤ gt<10.0.





An analogous picture is evident in the time evolution of the ESFF defined in eq:defESFF, and shown in Figs. <ref> (d) and (e). Panel (e) shows the ESFF for various times 0 ≤ gt ≤ 10, starting from a flat distribution at earliest time and showing a plateau-ramp structure at late times. Similarly as in the main text, the ESFF is computed separately for every symmetry sector of the EH to avoid contamination from uncorrelated levels in different sectors, and then averaged over these sectors and over the randomly chosen initial states. Panel (d) shows the time average over regime (III). In this regime, the ramp-plateau is evident. A gray band marks the statistical deviation from the initial-state and symmetry-sector averages. We fit the ramp to the form ∼θ^0.6± 0.1 which is consistent with the experimental data shown in the main text for a much smaller system. The fit error is determined by varying the fit range.

Because the late-time behavior, i.e., g t≫ 1, is inaccessible to current Trotter-based digital quantum simulation, we extend our analysis towards this regime, aiming to elucidate the potential outcomes that future quantum simulators might uncover. A similar analysis has been performed previously in Ref. <cit.>. At late times, it is expected that the entanglement entropy of a subsystem becomes equal to the thermal entropy contained in that subsystem, corresponding to a global Gibbs state with a temperature corresponding to the average energy of the initial state. In fig:gap-ratio-theory-analysis (f), we plot the von Neumann entanglement entropy, for a variety of randomly chosen initial states. The individual curves are color coded relative to their initial average energy density ∼ E_0 = ⟨ψ(0)| H | ψ(0) ⟩, normalized relative to the energy bandwith Δ E,  i.e., the difference between the highest and lowest eigenvalues of the EH. Displayed are only initial states whose energies lie within the interquartile range, representing the central 50% energy spectrum of states. States highlighted in yellow denote the highest energies, while those in blue represent the lowest. We were not able to extend this study to even later times to observe the expected saturation of the von Neumann entanglement entropy, owing to the fact that eventually finite-size effects become large. These results demonstrate that entanglement-entropy saturation is not a practical measure of thermalization in present experiments, demanding long evolution times and exhibiting lack of initial-state insensitivity (i.e., universality).

Shown in fig:gap-ratio-theory-analysis(g) is a separation of the von Neumann entanglement entropy, S_vN, into components, S_vN,s, related to the symmetry structure of  ρ_A = ⊕_s ρ_A,s,


    S_vN = -∑_s p_s log(p_s) + ∑_s p_s S_vN,s,


where the first term is the symmetry component, and the second is the distillable entanglement <cit.>. Here, p_s≡Tr[ρ_A,s]≤ 1 with ∑_s p_s=1, and S_vN,s≡ - Tr[ρ̅_A,slog(ρ̅_A,s)] is the sector-wise entanglement entropy with ρ̅_A,s≡ρ_A,s / p_s. Dashed lines in the plot denote the distillable component, while dotted lines denote the symmetry part. It is evident that both components saturate on different time scales: the symmetry component saturates fairly quickly (albeit still later than the build-up of level repulsion) to its maximal value log(1/p_s) where p_s =1/4 corresponds to equal mixing of the four symmetry sectors of ρ_A. In contrast, the distillable components dominates the late-time behavior of the von Neumann entropy.




While experimental constraints prevent us from accessing late times, fig:vonNeumanExpt illustrates the von Neumann entanglement entropy derived from our experimental data, utilizing the same dataset as in the main text. In this figure, exact classically computed results are denoted by black lines, while cyan crosses represent the optimal BW-inspired parameterization assuming infinite measurements, and orange circles are the experimental data points. The right-hand side of the figure displays two panels: the top panel depicts the symmetry component, while the bottom panel depicts the distillable component. Notably, the observed behavior closely mirrors the (classically computed) findings of the much larger system in fig:gap-ratio-theory-analysis(g). Specifically, the symmetry component saturates at its maximum value of log(4), while the distillable entanglement demonstrates continued growth.  Generally,  the entanglement measured in our experiment via BW-inspired tomography overshoots the exact result. We attribute this discrepancy primarily to  over- or under-rotations within the single-qubit random-bases changes. 






§ ENTANGLEMENT-HAMILTONIAN TOMOGRAPHY


In this section, we outline the randomized-measurement and analysis strategy behind extracting approximate EHs from experiment. We also systematically investigate the performance of the approach with regard to the number of bases sampled, the number of shots performed, and the influence of device errors.




Our tomography protocol consists of two parts: randomizing the state via single-qubit random unitaries drawn from a circular unitary ensemble, followed by classical post-analysis of the extracted measurements to determine an EH that best describes all observables. We use single-qubit random circuits, 𝒰≡⊗_i u_i where u_i is the following single-qubit unitary 



width=0.36

u_i ^     

=

    R^z(γ^1_i)     R^y(γ^2_i)      R^z(γ^3_i)       ,




and for each qubit, the angles γ^1_i, γ^2_i, γ^3_i are drawn according to a circular unitary ensemble (an overall phase is ignored) following Refs. <cit.>. A drawback of this approach is that it does not maintain the symmetry structure of ρ_A, as it randomizes over the whole Hilbert space instead of each symmetry block of ρ_A. Such a symmetry-ignorant randomization mixes different symmetry sectors, resulting in an outcome that is inaccessible to any physical time-evolved quantum state. We choose it, nonetheless, to avoid the larger circuit depth associated with a symmetry-conscious randomization such as that proposed in Ref. <cit.>. We note that a symmetry-conscious scheme would significantly reduce the measurement cost, as the sampling cost would only scale as the size of the symmetry block instead of the Hilbert-space size. It would thus simplify the classical optimization significantly. However, because the single-qubit scheme is tomographically complete, one can still reconstruct, approximately, the symmetry structure from the data.

The ansatz for the EH is a sum of local operators 𝒪_i,


    H_A (t ; {β_i} ) ≡∑_iβ_i(t) 𝒪_i ,


which are either the physical-Hamiltonian terms or are chosen iteratively from commutators of those terms. β_i are real parameters to be constrained from data. While the BW theorem hints at an optimal choice of operators 𝒪_i for ground states, our ansatz is heuristic, as a systematic ansatz for non-equilibrium states is not known. Our criteria for the operators are that i) they are local, involving operators with support on at most two neighboring plaquettes, ii) they are compatible with the symmetries of the reduced state ρ_A (which will be discussed below), and iii) they are independent, meaning they cannot be transformed into each other through Gauss laws when acting on a physical state (although they are certainly not independent in the algebraic sense).




The operators we use are pictorially represented in Figs. <ref> plus all operators that are generated from those depicted upon translation. Here for the sake of generality, we represent them in terms of Z_2 variables, albeit, in practice, they are represented in the dual formulation in our algorithm. Figure <ref> shows a special operator; this operator (and related examples) relates operators within the subsystem to operators in the complement via Gauss laws. Further, it commutes with ρ_A, and is hence connected to the symmetry blocks of ρ_A. All terms in Figs. <ref> are obtained by considering the physical-Hamiltonian operators and by recursively commuting the physical-Hamiltonian operators, stopping at two recursions (i.e., commutators of commutators). Other selection strategies are also feasible.







Given the ansatz and following Ref. <cit.>, we then minimize the functional


    ⟨∑_b[P_𝒰(b) - Tr[ 𝒰^† |b ⟩⟨ b|  𝒰ρ_A(t ; {β_i}) ] ]^2 ⟩_𝒰


where P_𝒰(b) is the probability to measure a bitstring b in the basis determined by 𝒰 and


    ρ_A(t ; {β_i}) ≡ e^-H_A (t ; {β_i} )/Tr[e^-H_A (t ; {β_i} )] .


is the normalized reduced density matrix parameterized by H_A.
Here, ⟨·⟩_𝒰 is the average over random circuits.
Note that optimizing eq:optimziaton effectively weighs more favorably the largest Schmidt eigenvalues of ρ_A because they, on average, contribute the most to any random observable. Because of this, the optimization more accurately reproduces the low-energy part of H_A. 
The optimization is performed using Matlab's <cit.> non-linear fmincon optimization package with the `sqp' algorithm <cit.>. Convergence and uniqueness of the obtained minimum have been cross checked for several data sets using Matlab's GlobalSearch routine with default parameters <cit.>. All EH couplings are confined to the range β_i ∈ [-50,50]. We check explicitly that the routine does not come close to the boundary of the parameter regime. The EH that is obtained in this way is projected into symmetry sectors, which we then separately analyze. For this projection, the symmetry sector can be read off directly from the row and column numbers of the EH when they are interpreted as binary. The corresponding subblock H_ E,s is then selected for further analysis, where s labels one symmetry sector. For the EGRD analysis, we analyze the gap-ratio distribution separately for every sector, then combine the distributions and re-normalize the total distribution. For the ESFF analysis, we average over the sectors. Finally, EGRD and ESFF shown in the main text and appendices involve a combination/average over initial states. The error band and the error bars in the plots are the standard deviation for the symmetry-sector and initial-state averages. 

An example of the eigenvalue spectrum of the EH, comparing the exact values with those obtained from our procedure, is shown in fig:overviewES. Panel (a) shows the reconstructed entanglement spectrum for several times, comparing exact data (black lines), infinite-measurement results (cyan crosses), emulator data (blue squares), and quantum-computed data (orange circles), for one given initial state. Panel (b) shows the symmetry-resolved spectrum for gt=1.02. Here, s  are the symmetry sectors of ρ_A, see the discussion at the end of Sec. <ref> in the main text. While our analysis effectively describes the low-lying part of the entanglement spectrum, it fails to quantitatively reproduce the higher part associated with very small probabilities. Additionally, the reconstructed spectrum appears more mixed (more entangled) compared to e.g., the emulator data. This discrepancy primarily is due to finite-measurement statistics and device errors. Comparison with the infinite-(ideal-)measurement results also reveals that the BW-inspired parameterization falls short in capturing even quantitatively higher-lying components of the spectrum. It is noteworthy that the limitations of the parametrization become more pronounced at later times where the Trotter-step size is large and even the low-lying part deviates. This is because the effective Trotter Hamiltonian at late times deviates from the known target Hamiltonian from which the BW-inspired ansatz is derived.

In fig:BWgapratiobases, we study the accuracy of our tomography scheme concerning the number of random bases. Each basis is probed with N_ shots=750 samples, employing emulated data to eliminate the effect of device errors. Applying N_𝒰=4 to N_𝒰=128 random bases shows convergence in the lower-lying range of the entanglement spectrum relatively fast. However, the higher-lying part is not reconstructed even with a large sample size. The top panel displays the average EGRD over time, showing consistent behavior across the sample range. Error bars indicate the spread from combining initial states and symmetry sectors, which decrease with increasing N_𝒰. Concretely the error bars are obtained by computing the mean of the gap ratios for every sector and initial states separately, and then computing the standard deviation. 

Finally in fig:BWgapratioshots, we investigate the dependence on the number of measurement of probabilities P_𝒰(s) in each basis (i.e., the number of shots), maintaining a constant number of measurement bases, N_U=24. Once more, convergence is evident in the lower segment of the ES in the lower panel, while the higher portion remains beyond reach even with infinite shots (red octagons). The top panel illustrates the EGRD, indicating consistency, within error bars, with the optimal BW-inspired ansatz result (cyan band).

In principle, given that the scheme is tomographically complete, we anticipate the ability to precisely reconstruct the entire ES with exponential resources. However in practice, we were not able to do so because of the significant bias of the cost function eq:optimziaton towards the low-energy portion of the entanglement spectrum. This bias places considerable strain on the numerical minimization routine, pushing it beyond its numerical-accuracy threshold. Despite this, its accuracy is satisfactory for analyzing experimental data, and we abstained from further optimization attempts. Exploring advanced optimization routines, including machine-learning techniques <cit.>, holds promise for EH tomography in future studies.



 Paper Title: To reset, or not to reset -- that is the question.




 
Riverlane, Cambridge, CB2 3BZ, UK

Riverlane, Cambridge, CB2 3BZ, UK
Department of Physics and Astronomy, University College London, London WC1E 6BT, UK

Riverlane, Cambridge, CB2 3BZ, UK
Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, UK

Riverlane, Cambridge, CB2 3BZ, UK





§ ABSTRACT 
Whether to reset qubits, or not, during quantum error correction experiments is a question of both foundational and practical importance for quantum computing.  Text-book quantum error correction demands that qubits are reset after measurement.   However, fast qubit reset has proven challenging to execute at high fidelity.  Consequently, many cutting-edge quantum error correction experiments are opting for the no-reset approach, where physical reset is not performed.  It has recently been postulated that no-reset is functionally equivalent to reset procedures, as well as being faster and easier.  For memory experiments, we confirm numerically that resetting provides no benefit.  On the other hand, we identify a remarkable difference during logical operations. We find that unconditionally resetting qubits can reduce the duration of fault-tolerant logical operation by up to a factor of two as the number of measurement errors that can be tolerated is doubled. We support this with numerical simulations. However, our simulations also reveal that the no-reset performance is superior if the reset duration or infidelity exceeds a given threshold. Lastly, we introduce two novel syndrome extraction circuits that can reduce the time overhead of no-reset approaches. Our findings provide guidance on how experimentalists should design future experiments.



    

To reset, or not to reset – that is the question
    Ophelia Crawford
    August 5, 2024
================================================





Quantum error correction (QEC) is a vital tool for unlocking quantum applications far beyond the capabilities of classical computers. QEC works by periodically measuring a set of multi-qubit Pauli operators, called stabilisers, providing evidence of errors that are then corrected. Measuring a stabiliser commonly involves a so-called syndrome extraction circuit that uses auxiliary qubits to assist with measurements on the data qubits that store the logical information. Many syndrome extraction circuits start by resetting the auxiliary qubits to the |0⟩ state and end with single-qubit readout of the auxiliary qubits in the Z basis, thereby collapsing to either the |0⟩ or |1⟩ state <cit.>. As these circuits are repeatedly used, the collapsed states on the auxiliary qubits need to be reset back to |0⟩. Resetting can be achieved in various ways that seem, at first appearance, to be equivalent. However, in this paper, we reveal fundamental differences in the fault-tolerant properties of reset schemes previously thought to be similar. 

Let us survey the three broad classes of reset strategy. The first class is unconditional reset, where a non-unitary operation maps all qubit states into the |0⟩ state. The simplest option to achieve this is to wait until the qubits decay into their ground states <cit.>. However, this is too slow for mid-circuit reset.  By engineering a suitable interaction between a qubit and its environment, energy can be dissipated faster and with higher fidelity <cit.>. Unconditional reset is the conventional text-book approach and in the QEC literature is usually just called reset. The second class is conditional reset <cit.>, where a conditional bit-flip gate is applied on the auxiliary qubit if the previous measurement outcome was 1, otherwise no gate is applied.  The duration of conditional reset depends on both the bit-flip gate duration and the classical electronics latency, which increase the execution time of the circuit. However, alternatively, we may simply track the effect of the conditional bit-flip in software, which is exactly what is involved in the third class, no reset <cit.>.  Recent years <cit.> have seen QEC demonstrated with various qubit types <cit.>. Focusing on superconducting qubits, several QEC experiments used the no-reset scheme <cit.>, while others implemented unconditional reset <cit.>, making it timely to investigate the differences between these approaches.  



It has been previously claimed <cit.> that choosing the no-reset scheme “has an insignificant impact on code performance", but this has not been fully investigated.  When using no reset or conditional reset, misclassification of measurements has two consequences:  the wrong readout outcome is recorded, and an erroneous bit-flip (in software or on hardware) is applied to a qubit. This shows that a single failure event effectively causes a correlated pair of errors.  The central question is whether this correlated error significantly decreases the number of error events needed for an undetectable logical failure. On the one hand, we will confirm the claim from <cit.> that for quantum memory there is no significant impact on QEC performance. On the other hand, we find dramatic consequences when we perform a logical operation. As we scale quantum computers, we enter an era where it becomes possible to demonstrate small examples of fault-tolerant quantum computation (FTQC), for instance, by performing a joint Pauli measurement via lattice surgery <cit.>. During lattice surgery, classification errors on physical qubits alone can cause an overall misclassification of the logical Pauli measurement. When classical feedback is performed conditionally on this logical Pauli measurement, logical measurement misclassification transforms into a logical qubit error. In this context, we show that in the no-reset (and conditional-reset) schemes, these logical failure mechanisms (not present in quantum memory) may be formed by half as many errors as with unconditional reset.  Consequently, to tolerate the same number of errors, no-reset schemes require doubling the duration of lattice surgery; such slow-down is clearly undesirable.

Tolerating half as many errors will impact QEC performance, though the exact effect depends on specific details, including the noise model. We complement our aforementioned analytical insights with numerical results for a range of superconducting-inspired circuit-level Pauli noise models. Our simulations use the stability experiment <cit.> (see also <Ref>) as a proxy for lattice surgery and other experiments where similar logical failures occur. Our numerical results are consistent with a 2 × advantage of unconditional resets in the limit of fast, low-error resets. In particular, with a physical error rate of 10^-3, which can be expected in the relatively near future, we observe substantial advantage with instantaneous reset, some advantage with fast reset and no advantage with slow reset. However, for nearer-term experimental demonstrations of FTQC, when the physical error rate is higher, we observe the no-reset scheme to be, in general, superior, especially with slow resets.  We study this transition in detail, considering performance both in terms of numbers of QEC rounds and logical operation duration. 



Finally, we propose and numerically compare two alternative syndrome extraction circuits for the planar code that do not use unconditional resets but substantially decrease the aforementioned incurred time overhead. One of these novel syndrome extraction circuits uses an additional two-qubit gate per measurement round, transforming the classification error into a mixture of two more benign errors. This circuit gets close to recovering the performance of unconditional reset using the no-reset scheme. The other novel syndrome extraction circuit we propose uses two auxiliary qubits per stabiliser, effectively squeezing two measurement rounds into one. As a result, this second circuit's QEC performance is substantially better in the examined noise regime than previous circuits, although it requires ≈50% more qubits. Based on these results, we discuss when it is beneficial to use unconditional resets on superconducting hardware and when to use one of our alternative circuits.

The structure of our paper is as follows. In <Ref>, we describe how the decoding problem changes when we switch from the unconditional-reset to the conditional-reset or no-reset scheme. In <Ref>, we first describe a Pauli noise model, inspired by currently available realistic superconducting hardware, that we use throughout (<Ref>). Then, we present numerical results that verify the negligible difference between the three reset schemes for quantum memory with the rotated planar code (<Ref>). After that, we describe the stability experiment and why it is a useful proxy for estimating the time cost of FTQC. Furthermore, we demonstrate through numerical simulations with the stability experiment that FTQC incurs a substantial time overhead in the no-reset (and conditional-reset) scheme, provided resetting is fast enough and is of high enough fidelity (<Ref>). In <Ref>, we present two alternative syndrome extraction circuits in the no-reset scheme that overcome this time overhead problem. Even though we present circuits only for the planar code, they can also be applied straightforwardly to more general QEC codes. With <Ref>, we conclude the paper by discussing the obtained results and possible future research directions. 









§ THE STRUCTURE OF DETECTORS IN THE THREE RESET SCHEMES



A simple QEC experiment, like quantum memory or stability, with a Calderbank-Shor-Steane (CSS) code is composed of three steps: initialising the data qubits of a code in either the X or Z basis, measuring a set of stabilisers some number of times, and finally measuring the data qubits in the X or Z basis. More complicated QEC experiments, like lattice surgery, are composed of more steps. The stabilisers may be measured using standard syndrome extraction circuits shown in <Ref>. With standard circuits, the stabiliser measurements directly correspond to the outcomes of the auxiliary qubits. More complex circuits, such as circuits with flag qubits <cit.> or the circuit we discuss in <Ref>, obtain further information. 

In order to detect errors, we define detectors <cit.> that are measurement combinations with deterministic values in the absence of errors. We explain the detector definitions for simple CSS-code QEC experiments, as outlined in the previous paragraph, that use standard syndrome extraction with unconditional resets, like <Ref>(a). We may compose detectors for such experiments in the following way. For each stabiliser g, denote by m_g,j the outcome from the jth QEC round, where j∈{1,2,…, n}, and let (x,y) be the spatial coordinate of the auxiliary qubit of g. Between the first and last rounds of a QEC experiment, we define m_g,j⊕ m_g,j+1 as a detector at coordinate (x,y,j) for all stabilisers g and numbers of rounds j∈{1,2,…, n-1}. In the first round of QEC, if g is of the same type as the data qubit initialisation, then we assign m_g,1 as a detector with coordinate (x,y,0).  In the last round of QEC, for stabilisers of the same type as the data qubit measurement, we assign a detector at coordinate (x,y,n) defined as m_g,n⊕⊕_q∈supp(g)m_q, where m_q is the measurement outcome of data qubit q. The first two coordinates of detectors are called spatial coordinates, while the last coordinate is called a time coordinate.

Now, we analyse which detectors are triggered (change value from the noise-free case) by different errors. A Pauli error on a data qubit occurring during the initialisation of the auxiliary qubits triggers only detectors with the same time coordinate. We refer to errors with this property as “space-like”. On the other hand, a (classical or quantum) measurement error on the auxiliary qubit of g in the jth QEC round triggers at most two detectors, namely at coordinates (x,y,j-1) and/or (x,y,j). Such an error is described as “time-like”. Some Pauli errors that occur mid-syndrome-extraction-circuit trigger a set of detectors where neither the spatial nor the time coordinates coincide for all the detectors – these are known as hook errors <cit.>.

In order to evaluate the errors that have occurred during the experiment, the decoder takes in the triggered detectors along with a model of which errors trigger which detectors. For the planar code, in the case of a circuit-level Pauli noise model such as that in <Ref>, to a good approximation, this model can be captured in a graph and therefore graph-based decoders such as minimum-weight perfect matching (MWPM) <cit.> can be used. We declare success in a run of the experiment if the decoder correctly predicts whether some logical observable has changed due to errors. In the case of W-quantum memory, for W∈{X,Z}, this logical observable is given by ⊕_q∈supp(W) m_q, where W is the logical W operator of the code.

We now consider the case where we apply a bit-flip gate to each auxiliary qubit after measurement conditioned on the outcome (<Ref>(b-c)) instead of applying unconditional resets. We may apply these X gates in hardware, to give the conditional-reset scheme (<Ref>(b)), or track their effect in software, to give the no-reset scheme (<Ref>(c)). As we will see, in both schemes, one measurement classification error leads to a pair of correlated errors, unlike with unconditional reset – a previously overlooked phenomenon. In the conditional-reset case, we define the detectors as above; in the no-reset case, we determine the detectors as follows.

For the no-reset scheme, we calculate the effect of removing all conditional X gates by pushing them through the subsequent unitary Clifford gates and find that a conditional X can be accounted for by flipping the subsequent measurement <cit.>. Therefore, we can determine the stabiliser observable outcomes, m_g, j, from the no-reset measurement outcomes labeled n_g, j as

    m_g,j = n_g,j-1⊕ n_g,j  for  j=1,2,…,n,

where we define n_g,0=0; see <Ref>(c). Between the first and last rounds of QEC, the detectors at coordinate (x,y,j) are then

    m_g,j⊕ m_g,j+1    = n_g,j-1⊕ n_g,j+1  for  j=1,2,…,n-1,

where we have used n_g,j⊕ n_g,j=0 to simplify the expression.  Therefore, instead of comparing outcomes from consecutive rounds, we compare outcomes that are two rounds apart.  

Next, we consider the special case of the first and last QEC rounds. In the first QEC round, for each stabiliser g of the same type as the data qubit initialisation, we assign n_g,1 as a detector with coordinate (x,y,0). In the last round of QEC, for stabilisers of the same type as the data qubit measurement, we assign a detector at coordinate (x,y,n) for n_g,n-1⊕ n_g,n⊕⊕_q∈supp(g)n_q. 

This structural difference of detectors does not change the triggered detectors corresponding to any mid-circuit quantum Pauli error. However, misclassification of a measurement outcome results in a different combination of triggered detectors to the unconditional-reset case. To see this, let us assume we have a classification error on measurement result n_g,j, i.e., we read out n_g,j⊕ 1, even though the qubit collapsed into the n_g,j-eigenstate. This triggers two detectors at coordinates (x,y,j-1) and (x,y,j+1) (cf. <Ref>), so in the decoding graph it corresponds to a time-like edge with length 2. 

In <Ref>, we present a topological perspective on how vertical strings of classification errors can lead to an undetectable logical failure in lattice surgery and stability experiments, analogous to how horizontal strings of physical qubit errors do the same in quantum memory experiments.  Considering these vertical failure mechanisms, we see that the no-reset approach will tolerate only half as many classification errors as the unconditional-reset approach, since each error has twice the vertical length in the former scheme. This is not the case for quantum memory; however, as we will see in <Ref>, this difference significantly impacts the performance of the stability experiment, and hence FTQC.

In the conditional-reset scheme, a classification error on m_g, j further results in erroneously applying an X gate after measurement. As a result, the next measurement outcome m_g, j+1 is also flipped. Thus, two detectors are triggered at (x,y,j-1) and (x,y,j+1), introducing a time-like edge with length 2, as in the no-reset case.











§ QEC EXPERIMENTS WITH STANDARD SYNDROME EXTRACTION CIRCUITS




 §.§ Noise model

This work is motivated by the absence of unconditional resets in recent QEC experiments on superconducting devices <cit.>. Therefore, the numerical results presented in this paper are from simulating a quantum computer with superconducting-like properties. These properties are the qubit connectivity, the native gates and the noise model.

We assume that the device has fixed and uniform qubit connectivity, with a planar nearest-neighbour architecture connecting each (bulk) qubit to four others. This so-called square-grid connectivity enables implementation of the planar code and is found in several modern superconducting quantum computers <cit.>. In <Ref>, we will present a circuit that requires slightly different planar connectivity – some qubits need only be connected to three others instead of four. 




We construct our simulated circuits <cit.> in terms of the controlled-Z two-qubit gate (CZ) and single-qubit X- and Z-basis rotations, a common native gate set for superconducting qubits <cit.>. In particular, we use the π/2 rotations √(X) and S, given by

    √(X) = 1/√(2)(I - i X),    S = 1/√(2)(I - i Z).

We also use Z-basis measurement (MZ) and reset (RZ). Each of these operations has an associated duration, given in <Ref>. In particular, measurement and reset take an order of magnitude longer than one- and two-qubit gates (see also <cit.>). The CZ gate takes only twice as long as a one-qubit gate.

We parameterise our noise model with a single parameter, p. The p-dependent probabilities of different error mechanisms are given in <Ref>. These probabilities are based on an existing superconducting-inspired noise model <cit.>, with a few small differences. Firstly, we split the measurement noise into two – a quantum part, which is bit-flip noise applied to the qubit before the measurement, and a classical part, which is the misclassification of the outcome value, e.g., reading out 0 even though the qubit collapsed into the |1⟩ state. This distinction is not commonly made, as they have the same effect under the unconditional-reset scheme. However, with no reset this distinction becomes important as the two types of measurement noises have different effects. We assign 80% and 20% of the noise respectively to these two mechanisms, inspired by existing hardware properties <cit.>. Secondly, when a qubit is idling, we apply noise depending on the length of time for which it is idle, t. The noise is obtained by Pauli twirling the amplitude damping and dephasing channel to give a Pauli channel with associated probabilities

    p_X(t)    = p_Y(t) = 1/4(1 - e^-t/T_1), 
    
        p_Z(t)    = 1/2(1 - e^-t/T_2) - 1/4(1 - e^-t/T_1),

where p_W is the probability of a Pauli-W error occurring <cit.>. To parameterise the idle noise by p, we scale the T_1 and T_2 times so that

    T_1 = p/p^refT_1^ref,    T_2 = p/p^refT_2^ref

where T_1^ref and T_2^ref are given in <Ref> and p^ref = 0.01.

We note that our chosen noise model is not meant to capture the exact behaviour of any specific device. The analysis could be repeated with different device assumptions which may change the quantitative details of the results. However, it captures the key features of superconducting hardware and will be qualitatively representative of other qubit types.





 §.§ The impact of no-reset on memory




Using the noise model described in <Ref>, we performed quantum memory simulations for distance-d rotated planar code for d number of QEC rounds. <Ref> compares the unconditional-reset and no-reset schemes. We constructed circuits using the python library  <cit.> and took samples which were then decoded using the MWPM python library  <cit.>. The circuits are available here: <cit.>. The number of samples for each data point was the minimum required to observe 10^4 logical failures or reach 10^8 samples, whichever happened first. This ensures error bars that are too small to be visible in our plots. We calculated the sample logical failure probabilities p_X and p_Z, for X and Z-memory, respectively, and combined them into one quantity as p_L = p_X+p_Z-p_Xp_Z. Our results (<Ref>) confirm the statement of <cit.>, mentioned earlier, that there is negligible difference between the two reset schemes in the context of quantum memory. More precisely, the no-reset scheme's performance is slightly better, as the threshold is somewhat higher and the logical failure probabilities are slightly lower.

Note, however, that, as we will see in <Ref>, the time-like effective distance (i.e., the number of mid-circuit errors that cause an undetectable time-like logical failure) is halved in the no-reset scheme for the stability experiment. Therefore, in the no-reset scheme, it may be more reasonable to perform 2d-1 number of QEC rounds instead of d. Increasing the number of rounds for quantum memory degrades its performance, eroding the slight advantage of the no-reset scheme.






 §.§ The impact of no-reset on FTQC


During FTQC, there are purely time-like undetectable logical failures that would be affected by the choice of reset scheme. For instance, if we measure a joint logical Pauli product via lattice surgery between some logical qubits that are each encoded into a planar code, then this outcome is given as a joint parity of a set of stabilisers <cit.>; see also <Ref>(c). To make this fault-tolerant, we measure these stabilisers for n QEC rounds, with n sufficiently large. With the unconditional-reset scheme, the minimum number of errors that causes an undetectable logical failure is n, e.g., when one stabiliser is misclassified in all QEC rounds. However, with the no-reset scheme, this number becomes ⌈ n/2 ⌉, as now it is enough to misclassify this measurement in the first, third, fifth, etc. QEC rounds.

The stability experiment <cit.> captures this type of situation in a simplified proxy and thus can be used to assess the necessary number of QEC rounds during lattice surgery and other operations where time-like logical failures are relevant. We use a planar code patch that does not encode any logical qubits, but, instead, has one type of stabiliser that is over-determined. <Ref>(b) shows an example 4× 4 stability patch where the X-type stabilisers multiply into the identity operator and so are over-determined. This stability experiment is performed as follows: we initialise all data qubits in the |0⟩ state, then measure the stabilisers for n rounds, and finally we measure all data qubits in the Z basis. We assign detectors as in <Ref>, and define the logical observable as the product of all X-type stabilisers in the first QEC round. Mirroring lattice surgery, a minimum of n measurement errors amount to an undetectable logical failure in the unconditional-reset scheme, while ⌈ n/2 ⌉ classification errors are sufficient in the no-reset scheme, cf. <Ref>.

Based on this, we would expect the stability performance to be improved by using unconditional resets – assuming fast enough and high-enough fidelity reset gates. We performed simulations to assess this. We considered planar code stability patches of sizes w × w with w ∈{4,6,8} and physical error rates p∈{10^-2,10^-2.5,10^-3}. For each, we prepared  circuits <cit.>, sampled, and decoded until we reached either 10^9 total shots or 10^3 logical failures, whichever happened first. The obtained logical failure probabilities plotted against the number of QEC rounds, n=5, 7, 9, 11, 13, are shown in <Ref>(a) for the w=4 case. They clearly show that implementing unconditional reset improves the performance for p = 10^-2.5, 10^-3. Furthermore, this improvement increases as p decreases. 

We now quantify this improvement. In the stability experiment, we expect the logical failure probability p_L to satisfy

    log(p_L) = log(a) - γ n,
 where a and γ both depend on the width of the patch w <cit.>. 
Note that one QEC round takes t_nr = 840 ns with the no-reset scheme and, if the reset duration is t_res ns, one QEC round takes t_ur=(840+t_res) ns with the unconditional-reset scheme. These two times are equal only with instantaneous reset, i.e., t_res=0 ns. Therefore, as a fairer comparison, <Ref> may be expressed in terms of the time t the experiment requires, instead of the number of QEC rounds, so that

    log(p_nr, L)     = log(a_nr) - γ_nr t 
    log(p_ur, L)     = log(a_ur) - γ_ur t,
 
where subscripts nr and ur indicate the no- and unconditional-reset schemes, respectively. We define the time overhead of implementing unconditional resets as R_ur=γ_nr/γ_ur <cit.>. This can be interpreted as follows: if we perform a stability experiment for t_nr ns without unconditional resets then, in order to match this performance using unconditional resets, we need to spend approximately t_ur = R_urt_nr ns. If R_ur<1, then implementing unconditional resets decreases the time cost of FTQC, and hence is worth implementing; otherwise not. The time overhead is plotted in <Ref>(b), assuming instantaneous reset t_res = 0 ns, fast reset t_res = 100 ns, and slow reset t_res = 500 ns. We call p_br the break-even point, where R_ur=1. Clearly, as t_res increases, p_br decreases. Furthermore, it can be seen in <Ref>(b) that, as p decreases, the time overhead decreases too, indicating that when p is small enough, implementing unconditional reset improves FTQC performance. In all cases of instantaneous and fast reset, the break-even point satisfies 10^-2.5 < p_br < 10^-2. With slow reset, the break-even point is not visible in our plot, and we expect it to be at very low p. This indicates that, unless the reset gate error rate and duration are below certain values, implementing unconditional reset may not be beneficial for FTQC. However, if the break-even point is reached, either with decreased reset duration or higher-fidelity reset gates, the unconditional-reset scheme substantially improves FTQC performance.









§ RECOVERING TIME-LIKE DISTANCE BY SPREADING CLASSIFICATION ERRORS

We have seen that the absence of unconditional resets halves the time-like effective distance of FTQC protocols when using standard syndrome extraction circuits. In this section, we present an alternative syndrome extraction circuit that recovers the full time-like effective distance without the need for unconditional resets, and which also keeps the space-like effective distance as d. This circuit uses one additional two-qubit gate per QEC round and an additional O(d) qubits.







The idea is to spread the auxiliary qubit measurement classification errors to the data qubits, thereby triggering additional detectors and thus requiring more errors for an undetectable time-like logical failure. We achieve this by applying two conditional Pauli gates immediately after measuring the auxiliary qubit – one conditioned on the measurement outcome and the other conditioned on the auxiliary qubit state itself; see <Ref> for an X-stabiliser circuit. Note that the controlled gates are of Z-(X-)type for X-(Z-)type stabilisers, and that the classically-controlled gates do not have to be applied on the device – instead, their effect can be tracked in software. In the absence of a measurement classification error, the two conditional gates cancel so there is no overall effect on the data qubits. However, if there is a classification error, exactly one of the gates will have an effect, resulting in a Pauli gate being applied to the data qubit which then behaves like a Pauli error on the qubit. Therefore, misclassification of measurement m_g, j will now trigger four detectors – the two which directly result from the measurement classification error, at coordinates (x, y, j-1) and (x, y, j+1), and the two which arise from the resulting data qubit error, at coordinates (x, y, j) and (x', y', j). This pattern is shown in <Ref>, together with its decomposition into two graph-like errors. Reference <cit.> contains an example decoding graph for a 4× 4 stability experiment using the error-spreading circuit that we generated with . In <Ref>, we show a 4× 4 planar code stability patch and the qubit pairs which take part in error-spreading events. In the bulk of the code, we require no additional qubits. However, on the boundary, we require O(w) additional qubits for a w× w patch to ensure each auxiliary qubit has a qubit to which to spread its classification error; cf. <Ref>(b).

It is easy to see that an undetectable logical error will no longer occur if there are classification errors in only every other round because of the additional detectors now triggered by a classification error. 
Furthermore, by searching exhaustively with , we find that, at least for small examples, n errors are required for an undetectable logical failure in an n-round stability experiment. For quantum memory, it is straightforward to see that we need at least d errors for an undetectable logical failure, as each classification error only triggers detectors whose spatial coordinates are adjacent, as in the standard circuit case.

We simulated the error-spreading circuits in order to compare the performance to standard syndrome extraction circuits with no reset. We applied the same principles for the number of shots as in <Ref>. We present the quantum memory results in <Ref> and see that the performance is somewhat (although not significantly) diminished from the no-reset case. We attribute this to the error-spreading feature and the slightly higher depth of the error-spreading circuits. However, as we see from the stability time overhead plots <Ref>(b), we need fewer QEC rounds for the error-spreading circuit to avoid undetectable time-like logical failures and so we need fewer QEC rounds for quantum memory as well, making the gap in <Ref> smaller. For instance, in case of the 4× 4 patch for p=0.001, we have R_spr≈ 0.85, meaning we have to spend only 85% of the time on stabiliser measurements as with the standard no-reset circuits. Taking into account the differing execution times, we see that we need only ≈ 80% of the number of QEC rounds with the error-spreading circuit. Based on this, we can conclude that the error-spreading circuit improves the QEC performance over the standard no-reset circuits.

From comparing <Ref>(b) and <Ref>(c), we can further conclude that the error-spreading circuit is a good alternative to unconditional reset if the reset duration exceeds 100 ns for the error regimes we considered. However, if the reset duration is decreased further, then unconditional reset outperforms the error-spreading circuit.













§ RECOVERING TIME-LIKE DISTANCE BY SQUEEZING TWO QEC ROUNDS INTO ONE




In this section, we show that QEC performance can be improved without using unconditional resets by effectively squeezing two QEC rounds into one. This method, however, requires ≈ 50% additional qubits. We appoint two auxiliary qubits for each stabiliser, as shown in <Ref>(a). Note that the auxiliary qubits of the X-/Z-type stabilisers are arranged horizontally/vertically so that we avoid so-called “bad hook” errors. Therefore, the hardware needs the Cairo pentagonal connectivity <cit.>. The main idea is that while we measure the auxiliary qubits of one stabiliser type, we can execute unitary gates on the qubits of the other type and also have plenty of time to obtain two independent measurement outcomes for each stabiliser. <Ref>(b)–(c) illustrates this. For instance, the unitary parts of this circuit for a Z-type stabiliser are as follows. First, apply two layers of CZ gates between the top auxiliary and the top two data qubits and at the same time similarly for the bottom three qubits. Then, swap the auxiliary qubits, and finally repeat the two layers of CZ gates. In this way, we entangle the two auxiliary qubits with the data qubits independently and so measuring each provides independent stabiliser measurements. Since executing these unitary gates takes only 400 ns in total (when compiled to the gates of <Ref>), one QEC round with this circuit takes 1000 ns. This is only 160 ns longer than for the standard no-reset syndrome extraction; however, we obtain each stabiliser outcome twice.



We now show how this new circuit recovers the effective time-like distance. First, we define the detectors. For each stabiliser g, let (x^1,y^1) and (x^2,y^2) be the spatial coordinates of its two auxiliary qubits. For the sake of simplicity, we will consider outcomes in the conditional-reset scheme, which is equivalent to the no-reset scheme. Denote by m_g,j^k the outcome in the jth QEC round obtained on the auxiliary qubit at (x^k,y^k) (k=1,2). We set m_g,j^1 ⊕ m_g,j^2 as a detector for all j and assign the coordinates (x^1,y^1,j) to it. Furthermore, in the bulk of the experiment, we define m_g,j^1 ⊕ m_g,j+1^1 as a detector if j is even, and m_g,j^2 ⊕ m_g,j+1^2 when j is odd, and assign the coordinates (x^2,y^2,j+12) to it. The full decoding graph of a stability experiment on a 4× 4 patch with 5 rounds is available in <cit.>. <Ref> depicts the part of the decoding graph where only nodes (detectors) corresponding to one X-type stabiliser are shown. This restricted graph has edges of two types: diagonal and vertical edges. If m_g,j^k (k∈{1,2}) is flipped due to a quantum bit-flip measurement error, then at most two detectors are triggered, namely at coordinates (x^1,y^1,j) and (x^2,y^2,j±12), corresponding to a diagonal edge. If m_g,j^k (k∈{1,2}) is flipped due to misclassificiation, the most dangerous error for FTQC, then that will also flip m_g,j+1^3-k, hence triggers at most four detectors. However this can be decomposed into two parallel diagonal edges. To account for the vertical edges, consider an XX-error occurring after the SWAP gate in <Ref>(b). This error flips both outcomes of the jth QEC round, hence triggers at most two detectors at coordinates (x^2,y^2,j-12) and (x^2,y^2,j+12) – a vertical edge. We verified using 's functionality that the effective distance of stability experiments is recovered to be n. Furthermore, the effective distance for quantum memory remains the same as with standard syndrome extraction circuits: d.

We simulated these circuits in order to compare the performance to the previously discussed syndrome extraction circuits. We applied the same principles for the number of shots as in <Ref>. We present the quantum memory results in <Ref> and see that again the performance is slightly diminished from the no-reset case, though not as much as with the error-spreading circuits. We attribute this to the increased number of two-qubit gates used in these syndrome extraction circuits. However, as we see from the stability time overhead plots in <Ref>(b), we need fewer QEC rounds for the round-squeezing circuit to avoid undetectable time-like logical failures and so we need fewer QEC rounds for quantum memory as well, making the gap in <Ref> smaller. For instance, with p=0.001 we have R_sqz≈ 0.65. Since execution times per QEC round differ, we need only ≈ 55% of the number of QEC rounds with the round-squeezing circuit – slightly more than half as many as for the standard no-reset case. Based on this, we can conclude that this circuit improves on the QEC performance of both the standard and error-spreading no-reset circuits, though with the use of additional physical qubits.

From comparing <Ref>(b) and <Ref>(c), we can further conclude that this circuit is a good alternative to implementing unconditional resets, even with instantaneous reset, at least for the range of physical error parameters we consider. We expect that, with much lower physical error probability, this would no longer be the case.

Whilst no device with the connectivity shown in <Ref>(a) currently exists, it has lower connectivity than many current devices suggesting it should be easier to build. However, we point out that using an additional SWAP layer on one stabiliser type, it is possible to align the auxiliary qubits in a parallel direction without introducing “bad hooks” <cit.>. In this case, the required connectivity is a sub-graph of the popular square-grid connectivity.













§ CONCLUSION


We investigated the QEC consequences of performing mid-circuit reset, or not, during QEC experiments using standard syndrome extraction circuits.  Our main conclusion is that unconditionally resetting qubits, for instance by driving interactions with a dissipative environment, fundamentally improves the fault-tolerance properties of logical operations offering up-to 2× speedup of fault-tolerant quantum computers. Access to fast, high-fidelity unconditional resets would make this approach a clear winner.

Nevertheless, with current best-in-class devices operating close to the QEC threshold and focused more on memory experiments, our simulations indicated that not resetting qubits can result in better performance. This justifies the approach taken in recent proof-of-principle experiments <cit.> and offers encouragement for near-term experiments to continue this practice. For those keen to avoid developing unconditional-reset technology, we presented two alternative syndrome extraction circuits that provide additional protection against measurement errors. These alternative circuits are interesting near-term experiments and potential long-term solutions. Whether our alternative circuits are genuine contenders for the best approach to sub-threshold FTQC will depend on additional effects, such as leakage errors, not considered in this paper. Such investigation would be interesting future work.

Furthermore, we emphasise again that our results are based on a particular noise model which assumes, for example, the measurement classification and qubit bit-flip error rates are 20% and 80% of the total measurement error rate, respectively. A direction for future work could be to repeat the analysis with different noise models or obtain results from real quantum computers. In particular, varying the measurement time on a particular device could affect results, not just by changing the amount of idling error on other qubits, but also by altering the total measurement error and the above split into classification and qubit bit-flip error.

As for the third class of reset strategy, conditional reset, our analysis shows that it will never compete with other strategies, and so should not be used in any of the QEC experiments considered here. This observation debunks folklore claims that control systems for QEC require fast conditional reset. 








 Paper Title: Low energy excitations in bosonic quantum quasicrystals.






alejandro.mendoza@ufsc.br
Departamento de Física, Universidade Federal de Santa Catarina, 88040-900 Florianópolis, Brazil
Max Planck Institute for the Physics of Complex Systems, Nothnitzerstr. 38, 01187 Dresden, Germany


Max Planck Institute for the Physics of Complex Systems, Nothnitzerstr. 38, 01187 Dresden, Germany


Max Planck Institute for the Physics of Complex Systems, Nothnitzerstr. 38, 01187 Dresden, Germany
Theoretical Physics III, Center for Electronic Correlations and Magnetism,
Institute of Physics, University of Augsburg, 86135 Augsburg, Germany

§ ABSTRACT 
   We present the first principles construction of the low-energy effective action for bosonic self-organized quantum quasicrystals. Our generalized elasticity approach retains the appropriate number of phase- and corresponding conjugate density- degrees-of-freedom required for a proper description of the Goldstone modes. For the dodecagonal and decagonal quasicrystal structures we obtain collective longitudinal and transversal excitations with an isotropic speed of sound. Meanwhile, for the octagonal structure, the coupling between phononic and phasonic degrees of freedom leads in turn to hybridization of the latter with the condensate sound mode, producing collective excitations with a longitudinal and transversal component, and an anisotropic speed of sound. Finally, we discuss the fate of each excitation mode at the low and high density phase transitions limiting the quantum quasicrystal phase.  
      
    
    
     

    


Low energy excitations in bosonic quantum quasicrystals
    F. Piazza
    
=======================================================



Introduction.–Quasicrystals (QCs) are an exotic state of matter where crystalline and disordered behaviour meet. They retain long range orientational order but lack any discrete translational invariance <cit.>.  Although they were initially discovered in metallic alloys <cit.>, it is known today that this kind of exotic state can be stabilized in many different contexts <cit.>, as long as an appropriate effective competition between two or more characteristic length scales is present in the system <cit.>. One of the most intriguing versions of quasi crystalline systems  are the so-called quantum quasicrystals <cit.>, which appear as ground states of quantum many-body Hamiltonians. Such is the case of QCs produced by spin-orbit interactions in dipolar bosons <cit.>, quasicrystalline phases induced in ultracold bosonic gases by quasiperiodic optical traps <cit.> and more recently, self-organized quasicrystals in Bose-Einstein condensates produced by cavity mediated interactions <cit.>.
In this Letter, we present the first principles construction of the low energy elastic theory of bosonic quantum quasicrystals, having supersolids as a particular case. Our calculations maintain full connection with the microscopic properties of the system allowing us to make qualitative and quantitative predictions regarding the hybridization of the low momentum gapless energy modes, the corresponding propagation speed and the behavior of these excitations at the phase boundaries of the  quasicrystal phase. As particular examples we consider the three most relevant quasiperiodic structures in two dimensions, the octagonal, the decagonal and the dodecagonal quasicrystals. To our knowledge, this is the first microscopic elastic theory of bosonic quantum quasicrystals able to corroborate the symmetry arguments regarding the existence of five gapless excitation modes for these systems. Our results show that the hybridization process between phonons, phasons and the longitudinal condensate sound mode qualitatively depends on the kind of quasicrystal structure.


 Model and Method.–We consider a 2D bosonic gas at zero temperature, interacting through an isotropic non-local two-body interaction of the form U v(r), where U represents a dimensionless parameter characterizing the intensity of the non-local interaction with respect to the kinetic energy. The nature of the two-body interaction between particles is such that the system hosts a quasicrystalline pattern at high enough particle densities in the ground state. As discussed in the literature <cit.>, the key ingredient for the stabilization of such phases is the presence of competing minima in the interaction potential v̂(k). In dimensionless units, the path-integral formulation of the grand canonical partition function of the system 𝒵=∫ Dϕ Dϕ^*e^-S[ϕ,ϕ^*] <cit.> is characterized by the action: 

    S[ϕ,ϕ^*]      =∫_0^∞ dτ∫ d^2x  ϕ(x,τ)^*(∂_τ-1/2∇^2-μ.
          .+U/2∫ d^2x'v(x-x')|ϕ(x',τ)|^2 )ϕ(x,τ).


To access the low-energy properties of the system we should allow long-wavelength fluctuations of the global phase, as well as of the phase of the modulated pattern of the quasicrystal ground-state wave-function which can be expanded in a Fourier basis as ϕ_0(x)=√(ρ)∑_nc_nexp(iG_n·x). Here ρ stand for the average particle density of the system and c_n and G_n corresponds to the Fourier amplitudes and wave vectors of the pattern respectively. We thus propose a perturbed ground state wave function of the form:  

    ϕ(x,τ)=√(ρ)c_0√(1+δ n_0(x,τ)/ρ c_0^2)exp(iθ(x,τ))
       +   ∑_n≠0√(ρ)c_n√(1+G_n·Π(x,τ)/Z+G_n,⊥·Π_⊥(x,τ)/Z_⊥)
          exp(i G_n·x+i G_n·u(x,τ)
        + i G_n,⊥·w(x,τ).
       +   .iθ(x,τ)),

where the fluctuation fields u(x,τ) and w(x,τ) represent the phonon and phason fields introduced as in the classical case <cit.>. Moreover, due to the quantum nature of the system we also have to admit fluctuations in the canonical-conjugate density fields associated to these phase variables, analogously to what was proposed by Pretko et al. <cit.> for supersolids. Accordingly, we have included the fields Π(x,τ) and Π_⊥(x,τ) as conjugate to u(x,τ) and w(x,τ), respectively. Finally, we added the overall phase fluctuation field θ(x,τ) of the wave function and its canonically conjugate zero momentum density fluctuation δ n_0(x,τ). It is understood that all fluctuation fields are slowly varying functions in space and imaginary time, with zero mean over the size of the system and, as expected, in the absence of fluctuations our ansatz for ϕ(x,τ) reduces to the ground state wave function ϕ_0(x). Furthermore, the set of vectors {G_n,⊥} is constructed from a permutation of the set {G_n}, such that the extended basis G̃_n=G_n⊕G_n,⊥ is orthogonal in the sense ∑_nG̃_n,αG̃_n,β∝δ_α,β <cit.>.

It is simple to show that the proposed ansatz satisfies the normalization condition of the boson field  ∫_A d^2x<|ϕ(x,τ)|^2>=N=ρ A, as long as ∑_nc_n^2=1. Such normalization constraint allows the calculation of the chemical potential μ for a system with a given particle density ρ. Lastly, the free parameters Z and Z_⊥ in Eq. (<ref>) are chosen to enforce the expected relation between canonically conjugate fields in the low energy effective action of the system <cit.>. By replacing the above ansatz for ϕ(x,τ) in Eq. (<ref>) and expanding up to quadratic order in the fluctuation fields we obtain the following long-wavelength effective action:
 
    δ S   =∫ dτ d^2x (i δ n_0(x,τ)∂_τθ(x,τ)+i Π(x,τ)·∂_τu(x,τ).
       +.i Π_⊥(x,τ)·∂_τw(x,τ))+ ∫ dτ d^2x 1/2[ρ(∇θ)^2.
       +γ_0δ n_0(x,τ)^2+γΠ(x,τ)^2+γ_⊥Π_⊥(x,τ)^2
       +2Π·∇θ+.E^T C̃ E],

where E stands for the extended phonon-phason strain field defined as E={∂_xu_x,∂_yu_y,(∂_xu_y+∂_yu_x)/2,(∂_yu_x+∂_xu_y)/2,∂_xw_x,∂_yw_y,∂_yw_x,∂_xw_y} and C̃ represents the phonon-phason elastic tensor. Closed formulae for the elements of this tensor as well as for the γ's couplings in terms of the ground state wave function are provided in the Supp. Matt. <cit.>. The expression in Eq. (<ref>) for the low-energy effective action of a bosonic quasicrystal is one of the main results of this work. It allows qualitative and quantitative predictions since all couplings are known in terms of the ground state wave function of the system. Moreover, it generalizes the well-known density-phase action of a homogeneous condensate <cit.>, not only to the present case of quantum quasicrystals, but also to supersolids if we eliminate the phasonic degrees of freedom <cit.>.



Elementary excitations for the dodecagonal QC– The effective action in Eq. (<ref>) can be recast as δ S=1/2∫dω/(2π)d^2q/(2π)^2η̂(q,ω)^†M(q,ω)η̂(q,ω), where η̂(q,ω)=(δn̂_0(q,ω),
θ̂(q,ω),
  Π̂_x(q,ω),
  û_x(q,ω),
  Π̂_y(q,ω),
  û_y(q,ω), Π̂_x,⊥(q,ω),
  ŵ_x(q,ω),
  Π̂_y,⊥(q,ω), ŵ_y(q,ω)), the energy-momentum dispersion relations ϵ(q)=ω(q) can be computed by solving the equation [M(q,iω(q))]=0 <cit.>. The dodecagonal quasicrystal (Fig. <ref>(a)) is the simplest case due to the absence of the coupling between phonons and phasons. The latter are deformations of the density pattern which are absent in ordinary crystals. We obtain analytical expressions for the dispersion relation of the five gapless modes <cit.>. In this case there are three longitudinal modes and two transversal modes, as schematically represented in Fig. <ref>(b). Two of the longitudinal modes result from the hybridization of the condensate sound mode (corresponding to a modulation of the overall phase θ) with the longitudinal phonon mode, while the transversal phonon as well as the phason mode remain decoupled. 
  
 
  


To deepen our understanding of the excitation modes we calculate the eigenvectors η_j(q) using the charateristic equation M(q,i ω_j(q))η̂_j(q)=0 for each excitation mode. Each eigenvector corresponds to a fluctuation field in real space and time given by η_j(x,t)=Re[η̂_j(q)exp(i(q·x-ω_j(q)t))]. By expanding Eq. (<ref>) to linear order in the fluctuation fields we obtain a general expression for the correction to the ground state wave function: δψ_j,q=[u_j,q(r)exp(i(q·r-ω_j(q)t))-v_j,q(r)^*exp(-i(q·r-ω_j(q)t))], allowing us to identify the Bogoliubov modes u_j,q(r) and v_j,q(r) <cit.>. 
The local phase and density fluctuations can be then written as Δρ_j,q=| u_j,q(r)-v_j,q(r)|^2 and Δφ_j,q=| u_j,q(r)+v_j,q(r)|^2 <cit.>. In Fig. <ref>(c-l) we show the typical behavior of these quantities. Each column corresponds to a mode in Fig. <ref>(b) in increasing order of excitations energy. In agreement with previous observations for supersolids <cit.>, the hybridized  longitudinal modes present significant (O(1)) phase and density fluctuations, while the transversal phonon mode presents significant density fluctuations and  O(q) phase fluctuations. On the other hand, while showing the same scaling (O(1) vs O(q)) of the transversal phonon modes, phasons feature density and phase fluctuations patterns which presents a different behavior around the center of the quasicrystal structure [The center of the quasicrystal here is being understood as the special point where all reflections symmetry planes of the quasicrystal pattern intersects. Given the form of our anzats for ϕ_0(x) it coincides with the origin of our coordinates system.] compare e.g. panel (e) and (i) of Fig.<ref>.  Mathematically, this is ultimately a consequence of the phason's property of producing a redistribution of the QC structure destroying the center of the quasicrystal without altering the average properties of the system <cit.>. Finally, we would like to remark that differently from the case of supersolids, here all density and phase fluctuations patterns are aperiodic. 


Excitations in decagonal and octagonal QCs.–In the case of ten- and eight-fold quasicrystals, the presence of the phonon-phason coupling leads to a characteristic equation [M(q,iω(q))]=0 with convoluted analytical solutions. However, some general properties can be obtained from the numerical solution of this problem. In the case of the decagonal quasicrystal, the longitudinal and transversal modes separately hybridize among themselves, leading to two groups of three and two modes, respectively, all with isotropic dispersion relations. In contrast, for the octagonal quasicrystal, the structure of the elasticity action characterized by an anisotropic phason contribution and a finite phonon-phason coupling produces anisotropic dispersion relations ϵ(q) with propagation speed c_s that depend on the orientation of the momentum q when q→0.  Additionally, unless the momentum of the excitation q is oriented along one of the symmetry axes of the quasicrystal, every mode has a longitudinal and a transversal component. As an example, we show in Fig. <ref>(a) the anisotropic behavior of the propagation speed and hybribization of modes for an octagonal QC. The numerical evaluation was carried out considering values of the couplings for which the system displays strong anisotropy and  hybridization effects <cit.>. The red to blue color gradient indicates the longitudinal and transversal character of each mode, using the angular orientation of the field u as a measure of this property. Interestingly, the propagation of excitations along non-equivalent symmetry axes of the octagonal quasicrystal produces an exchange of the longitudinal and transversal character for some modes. Moreover, the evaluation of the local density and phase fluctuations profile in this case shows that the scaling of the fluctuations when the propagation direction of the excitation do not coincide with a symmetry axes of the quasicrystal are all O(1). Meanwhile, in the opposite case, longitudinal modes have density and phase fluctuations of amplitude O(1) and transversal modes with density fluctuations O(1) and phase fluctuations O(q). The aperiodic patterns of phase and density fluctuations for each scenario discussed can be found in <cit.>.


Instabilities of the quantum quasicrystal phase.– A first analysis of the instabilities can be based on the behaviour of the propagation speed of the gapless modes. With this aim we consider the case of the decagonal QC structure and pursue the calculation of the couplings in the action (<ref>), considering small amplitudes of the QC modulation <cit.>. 
In order to make a numerical evaluation possible without defining a particular form of the interaction potential, we assume that the QC amplitude is a concave increasing  function of ρ U [Mathematically this means ∂ c_1(Uρ)/∂ (Uρ)>0 and ∂^2 c_1(Uρ)/∂ (Uρ)^2<0, where c_1(Uρ) stand for the modulation amplitude of ϕ_0.]. Although this is the expected behavior of this quantity, such ansatz does not intend to capture the actual quantitative behavior of a certain model <cit.>. The results for the propagation speed of each mode are shown in Fig. <ref>(b) as a functon of ρ U. One recognizes a discontinuous behavior of the propagation speed at the homogeneous-superfluid-to-quantum-quasicrystal phase transition, analogous to previous observations in supersolids <cit.>, and a continuous behavior at the quantum-to-classical quasicrystal transition at high enough densities where the propagation speed of the lowest mode vanishes <cit.>,[Although Mean Field treatments of our action consistent with the Gross-Pitaevskki description are not able to reach a strictly zero propagation speed for the lowest mode, beyond Mean Field descriptions have succeeded in observing rigorously the supersolid to normal solid transition <cit.>.]. As we shall see next, this transition corresponds to the breaking of quantum phase coherence. 


To further explore the nature of the instabilities we compute the second-order correlations of the fluctuation fields. At the lower density boundary, when the QC amplitude is small and the ground state wave function is dominated by the main Fourier amplitude of the pattern (c_1), we find that Z, Z_⊥ and all elastic couplings in C̃ approaches zero as c_1^2, while γ and γ_⊥ diverges as c_1^-2. This behavior lead us to the conclusion that in this regime ⟨δ n_0^2⟩ and ⟨θ^2⟩ remain finite, while ⟨Π^2⟩∝ c_1^2, ⟨Π_⊥^2⟩∝ c_1^2, ⟨u^2⟩∝ c_1^-2  and ⟨w^2⟩∝ c_1^-2, as c_1→0, with c_1≫ c_n>1. In this scenario the average wave function can be roughly approximated as ⟨ϕ(x,τ)⟩≈√(ρ)∑_nc_nexp(iG_n·x)×exp(-G_n^2⟨u^2⟩/4-G_n,⊥^2⟨w^2⟩/4-⟨θ^2⟩/2). This result confirms that close enough to the superfluid-quantum-quasicrystal stability boundary, where we have large phonon and phason fluctuations, there is a strong non-perturbative “renormalization" of the Fourier amplitudes of the quasicrystal profile [The strong renormalization process is signaled by the non-analytic behavior of ⟨ϕ⟩ in the limit c_1→ 0 upon inclusion of phonon and phason fluctuations ]. This phenomenology suggests that in cases in which the mean field superfluid to quantum quasicrystal phase transition is weak enough, the nature of the transition could be modified to a continuous one, even with the presence of intermediate hexatic-like phases. 

At the upper boundary of the quantum quasicrystal phase (γρ→ 1^+) all couplings of the effective action in Eq. (<ref>) remain finite, however in this limit the propagation speed of the lower hybridized mode approaches zero, as observed  before. This behavior  signals the disappearance of this mode, in the same way in which the lower energy excitation mode disappears at the supersolid to normal solid phase transition <cit.>. Calculating the correlations of the fluctuation fields in this regime, we obtain to leading order that ⟨δ n_0^2⟩ ⟨u^2⟩, ⟨w^2⟩ and ⟨Π_⊥^2⟩ remain finite as the stability boundary is approached, while ⟨θ^2⟩∝(γρ-1)^-1/2 and ⟨Π^2⟩∝(γρ-1)^-1/2. This result is to be expected since in the high ρ U regime the localization of particles should eventually lead the system to the loss of quantum phase coherence. Interestingly, due to the effective Π·∇θ interaction the de-stabilization of the θ(x,τ) field also produces the destabilization of the Π(x,τ) field, indicating the natural breakdown of our theory at this phase boundary. All evidence available in this case indicates a second order phase transition in this limit, as observed in supersolid systems <cit.>.  






Conclusions.–We have developed the elastic theory of a self-organized quantum quasicrystal with 8, 10 or 12-fold rotational symmetry. We obtained the effective action of the system described in terms of the global wave function phase, the two-dimensional phonon and phason fields and their respective conjugate density fields. From here, we were able to prove from first principles the existence of five gapless extended low energy excitation modes [It is worth to mention, that due to the lack of periodicity quantum quasicrystals can also host particle-hole localized excitations with an energy arbitrarily close to zero. See Refs. <cit.>.] for the considered two-dimensional QC structures  <cit.>. We showed that the presence of the Π·∇θ coupling in our effective theory produces the hybridization of the condensate sound mode with phonon and phason modes, leading to a rich phenomenology depending on the QC structure. A particularly interesting finding was that, due to the phonon-phason coupling, the octagonal QC structure develops an anisotropic speed of sound that depends on direction of the propagation, as well as that all modes show a mixed longitudinal-transversal character. Finally, our results shed light on the behavior of the low energy fluctuation fields at the thermodynamic boundaries of the quantum quasicrystal phase, elucidating possible scenarios for phase transitions in two dimensions.            


Acknowledgments: A.M.C. acknowledge MPIPKS for financial support and hospitality.



 
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Supplemental Materials: Low energy excitations in super quasicrystals: a generalized elasticity approach 














§ I. CONSTRUCTION OF THE EFFECTIVE ACTION

In this section we present details about the  construction of the effective action for the low energy behavior of the octagonal, decagonal and dodecagonal cluster quasicrystalline structures. We assume that the pair interaction potential v(x) is such that the model self-stabilizes one of these phases as its ground state in the quantum regime of the system <cit.>. By quantum regime, we refer to the region of parameters in which particle delocalization is strong enough to produce a finite superfluid fraction in the system. As a consequence, we consider that the stationary configuration that minimizes the action in Eq. (1) can be expressed as a Fourier expansion of the form

    ϕ_0(x)=√(ρ)∑_n c_nexp(i G_n·x),

where c_n and G_n represent the amplitudes and wave vectors of the Fourier expansion of ϕ_0. Moreover, for the kind of QCs considered in this work we have G_n=k(n_1e_1+n_2e_2+n_3e_3+n_4e_4), where k stands for the characteristic wave vector of the quasicrystal and the basis vectors e_i's are specific for each kind of pattern. The sum in the Fourier expansion is performed over a four dimensional integer vector n whose components n_i's are all integer numbers. A possible definition of the vectors e_i's for each of the cases considered is presented in Table <ref>. 


The low energy excitation properties can be accessed considering the effects of long wave-length fluctuations in the Fourier amplitudes and in its corresponding conjugate phases, in the same spirit as presented in ref <cit.>. In our case we propose a general ansatz for the perturbed wave function field of the form 

    ϕ(x,τ)=√(ρ)c_0√(1+δ n_0(x,τ)/ρ c_0^2)exp(iθ(x,τ))
        +∑_n≠0√(ρ)c_n√(1+G_n·Π(x,τ)/Z+G_n,⊥·Π_⊥(x,τ)/Z_⊥)
          exp(i G_n·x+i G_n·u(x,τ)
        + i G_n,⊥·w(x,τ)
        +iθ(x,τ)),

where the set {G_n,⊥} is constructed as a permutation of the set {G_n} such that the extended four dimensional basis G̃_n=G_n⊕G_n,⊥, is orthogonal in the sense that ∑_nG̃_n,αG̃_n,β∝δ_α,β <cit.>. For the purpose of calculations we can write G_n,⊥=k(n_1e_1,⊥+n_2e_2,⊥+n_3e_3,⊥+n_4e_4,⊥), where the basis {e_i,⊥} corresponding to our particular choice of {e_i} is chosen as shown in Table <ref>. Moreover is important to notice that the index vector {n_i} for two corresponding G_n and G_n,⊥ coincide. 


Since we are interested in the low energy behavior of the system we can work in the hydrodynamic regime, in which the fluctuation fields introduced vary slowly in space and imaginary time. Mathematically, this argument allows us to neglect all components of the effective theory containing the product of slowly varying terms with spatially oscillatory contributions.

Now we can proceed with the evaluation of each term of the original action (Eq. (<ref>)) as a function of the fluctuation fields, keeping only up to quadratic contributions. The first contribution can be recast in the symmetric form 

    δ S_1=∫ dτ d^2x1/2(ϕ(x,τ)^*∂_τϕ(x,τ)-ϕ(x,τ)∂_τϕ(x,τ)^*),

which up to quadratic order in the fluctuation fields yields

    δ S_1=∫ dτ d^2x (i δ n_0(x,τ)∂_τθ(x,τ)+i (ρ∑_nc_n^2G_n^2/2Z)Π(x,τ)·∂_τu(x,τ)+i (ρ∑_nc_n^2G_n,⊥^2/2Z_⊥)Π_⊥(x,τ)·∂_τw(x,τ)).

In this way, by choosing Z=ρ∑_nc_n^2G_n^2/2 and Z_⊥=ρ∑_nc_n^2G_n,⊥^2/2 we obtain for δ S_1, the expected structure combining the phases and the corresponding conjugate density fluctuation fields. 

The second contribution of our action can be recast as

    δ S_2=∫ dτ d^2x1/2∇ϕ(x,τ)^*·∇ϕ(x,τ).

In this case, the use of our ansatz leads us to the expression 

    δ S_2   =   δ S_2^0+∫ dτ d^2x{1/8ρ c_0^2(∇δ n_0)^2+1/8Z((∇Π_x)^2+(∇Π_y)^2)+1/8Z_⊥((∇Π_x,⊥)^2+(∇Π_y,⊥)^2).
       +   .ρ c_0^2/2(∇θ)^2+ρ∑_n≠0c_n^2/2(∇θ)^2+Z/2((∇ u_x)^2+(∇ u_y)^2)+Z_⊥/2((∇ w_x)^2+(∇w_y)^2)+Π·∇θ},

where δ S_2^0, represents a fluctuation-independent contribution. The linear order contributions in the fluctuation fields are not presented in this and the rest of the terms of the effective action because they have a zero total contribution, once we are considering fluctuations around the global minimum of our action. At this point it is worth noticing that the particular form considered for our ansatz in Eq. (<ref>) as well as the appropriate selection of Z and Z_⊥ are responsible for producing a form of δ S_1 and δ S_2 preserving the expected relation between the couplings associated to conjugate density and phase fields. Other choices for this ansatz, for instance including  a dependence of Z and Z_⊥ with |G_n|, even when preserves the symmetries of the original expression will not be able to reproduce this property.  

The third contribution of the original microscopic action, δ S_3=∫ dτ d^2xμϕ(x,τ)^*ϕ(x,τ), does not provide any quadratic contribution to the total effective action in the long wave limit. Finally, we can proceed with the contribution resulting from the non-local interaction term 

    δ S_4=∫ dτ d^2xd^2x'1/2U v(x-x')|ϕ(x,τ)|^2|ϕ(x',τ)|^2.

This last term produces two kinds of contributions, one including only density-density interactions defined as δ S_4,1 and a second one corresponding to the phonons and phasons deformation fields defined as δ S_4,2. Terms of density-phase interactions cancel out exactly due to the reflection and rotational symmetries of the modulated patterns considered. Taking into account that in the determination of δ S_4,1 and δ S_4,2 we have to deal with a convolution with a non-local interaction kernel, it is easier to perform the evaluation of these terms in momentum space. After a lengthy but strightforward calculation we obtained that δ S_4,1 in the long wave limit is given as

    δ S_4,1   =   ∫ dτd^2q/(2π)^21/2(γ_0δn̂_0(q,τ)δn̂_0(-q,τ)+γΠ̂(q,τ)·Π̂(-q,τ)+γ_⊥Π̂_⊥(q,τ)·Π̂_⊥(-q,τ)
    ),

where 

    γ_0   =   -1/2∑_n1≠0,n3,n4Uv̂(G_n1)c_n1c_n3c_n4/c_0^3δ(G_n1+G_n3-G_n4,0)+∑_nUv̂(G_n)c_n^2/c_0^2,
    γ   =   -1/4∑_n1,n2,n3,n4Uv̂(G_n1-G_n2)ρ^2Π_i=1^4 c_ni/Z^2(G_n1-G_n2)_x^2δ(G_n1-G_n2+G_n3-G_n4,0),
    γ_⊥   =   -1/4∑_n1,n2,n3,n4Uv̂(G_n1-G_n2)ρ^2Π_i=1^4 c_ni/Z_⊥^2(G_n1,⊥-G_n2,⊥)_x^2δ(G_n1-G_n2+G_n3-G_n4,0).

The obtained expression for each γ reduces to the one well known for the homogeneous scenario (c_0=1 and c_n≠0=0), γ_0=Uv̂(0), γ=0 and γ_⊥=0. It is also important to remark that since V̂(k) has a negative part responsible for exciting the main Fourier modes of the modulated pattern, we will have γ_0, γ and γ_⊥ as positive parameters, which guarantees the stability of our effective theory. We should note that in the long wave limit the terms proportional to the gradient of the density fluctuation fields in δ S_2 are subleading respect to the contribution in δ S_4,1, which is why these terms do not appear in Eq. (<ref>) of the main text.     

The last contribution in δ S_4 corresponds to the phonon and phason interaction. This term can be written in general as 

    δ S_4,2   =   ∫ dτd^2q/(2π)^2U ρ^2/4∑_nΠ_i=1^4 c_ni δ(G_n1-G_n2+G_n3-G_n4,0)
       ×   (2v̂(q-(G_n1-G_n2))-2v̂(G_n1-G_n2))((G_n1-G_n2)·û(q,τ)+(G_n1,⊥-G_n2,⊥)·ŵ(q,τ) )
          ((G_n1-G_n2)·û(-q,τ)+(G_n1,⊥-G_n2,⊥)·ŵ(-q,τ) ).

It is possible to convince ourselves that unlike all previous terms in our effective action this contribution has a different form for each quasicrystal. As expected in the long wave limit δ S_4,2 reproduces the structure of the well-known classical elastic Hamiltonian for each kind of quasicrystal structure <cit.> yielding 

    δ S_4,2   =   1/2∫ dτd^2q/(2π)^2[((2μ+λ)q_x^2+μ q_y^2)û_x(q,τ)û_x(-q,τ)+((2μ+λ)q_y^2+μ q_x^2)û_y(q,τ)û_y(-q,τ).
       +   . (2μ+2λ)q_xq_yû_x(q,τ)û_y(-q,τ)+(K_1q_x^2+(K_1+K_2+K_3) q_y^2)ŵ_x(q,τ)ŵ_x(-q,τ).
       +   . (K_1q_y^2+(K_1+K_2+K_3) q_x^2)ŵ_y(q,τ)ŵ_y(-q,τ)+2(K_2+K_3)q_xq_yŵ_x(q,τ)ŵ_y(-q,τ).
       +   .(R q_x^2-R q_y^2)(û_x(q,τ)ŵ_x(-q,τ)+û_y(q,τ)ŵ_y(-q,τ))+2Rq_xq_y(û_x(q,τ)ŵ_y(-q,τ)-û_y(q,τ)ŵ_x(-q,τ))
    ].

The expression presented corresponds to the most general scenario, consistent with the octagonal quasi-crystalline pattern. In the case of the decagonal quasi-crystal we will have K_2=-K_3, which makes zero the off-diagonal phason-phason interaction proportional to ŵ_x(q,τ)ŵ_y(-q,τ) and finally, in the case of the dodecagonal quasicrystal we will have R=0, which makes zero all elastic phonon-phason interactions. It is important to mention that the particular form obtained for our elastic phonon-phason action is a direct consequence of the particular choice we made for G_n,⊥. Different but equivalent forms can be obtained for δ S_4,2 if we consider other permutations of G_n in the definition of G_n,⊥. Furthermore, from the comparison between Eqs. <ref> and <ref> it is possible to conclude that


    (2μ+λ)   =U ρ^2/2∑_nΠ_i=1^4 c_ni δ(G_n1-G_n2+G_n3-G_n4,0)(G_n1-G_n2)_x^2∂_qx^2v̂(q-(G_n1-G_n2))|_q=0
    μ   =U ρ^2/2∑_nΠ_i=1^4 c_ni δ(G_n1-G_n2+G_n3-G_n4,0)(G_n1-G_n2)_x^2∂_qy^2v̂(q-(G_n1-G_n2))|_q=0
    
    K_1   =U ρ^2/2∑_nΠ_i=1^4 c_ni δ(G_n1-G_n2+G_n3-G_n4,0)(G_n1,⊥-G_n2,⊥)_x^2∂_qx^2v̂(q-(G_n1-G_n2))|_q=0
    
    K_T   =U ρ^2/2∑_nΠ_i=1^4 c_ni δ(G_n1-G_n2+G_n3-G_n4,0)(G_n1,⊥-G_n2,⊥)_x^2∂_qy^2v̂(q-(G_n1-G_n2))|_q=0
    
    R   =U ρ^2∑_nΠ_i=1^4 c_ni δ(G_n1-G_n2+G_n3-G_n4,0)(G_n1,⊥-G_n2,⊥)_x(G_n1,⊥-G_n2,⊥)_x
    ×   ∂_qx^2v̂(q-(G_n1-G_n2))|_q=0,

where K_T=K_1+K_2+K_3. These expressions complete the determination of all couplings in our effective theory from the knowledge of the ground state wave function. Finally, we can notice that the gradient terms of the phonon and phason fields in Eq. (<ref>) can be absorbed in δ S_4,2 modifying the elastic coefficients as follows (2μ̃+λ̃)=(2μ+λ)+Z, μ̃=μ+Z, K̃_1=K_1+Z_⊥ and K̃_T=K+Z_⊥. This allows us to write our total effective action in the following more compact way 

    δ S   =∫ dτ d^2x (i δ n_0(x,τ)∂_τθ(x,τ)+i Π(x,τ)·∂_τu(x,τ)+i Π_⊥(x,τ)·∂_τw(x,τ))+ ∫ dτ d^2x 1/2[ρ(∇θ)^2.
       .+γ_0δn̂_0(x,τ)^2+γΠ(x,τ)^2+γ_⊥Π_⊥(x,τ)^2+2Π·∇θ+E^T C̃ E],

where E stands for the extended phonon-phason strain field defined as E={∂_xu_x,∂_yu_y,(∂_xu_y+∂_yu_x)/2,(∂_yu_x+∂_xu_y)/2,∂_xw_x,∂_yw_y,∂_yw_x,∂_xw_y} and C̃ represents the phonon-phason elastic tensor <cit.> given by


    C̃=
    [ λ̃ +2μ̃      λ̃       0       0       R       R       0       0;      λ̃  λ̃+2μ̃       0       0      -R      -R       0       0;       0       0      μ̃      μ̃       0       0      -R       R;       0       0      μ̃      μ̃       0       0      -R       R;       R      -R       0       0  K̃_̃1̃     K_2       0       0;       R      -R       0       0     K_2  K̃_̃1̃       0       0;       0       0      -R      -R       0       0    K̃_T     K_3;       0       0       R       R       0       0     K_3    K̃_T ].







§ II. EXCITATION PROPERTIES OF THE DODECAGONAL QUASICRYSTAL

As we can notice, the effective action in Eq. (<ref>) can be written in the general form

    δ S=1/2∫dω/(2π)d^2q/(2π)^2η̂(q,ω)^†M(q,ω)η̂(q⃗,ω),

where the following definition of the Fourier transform of f(x,τ) and its inverse have been used

    f̂(q,ω)   =   ∫d^2q dω/(2π)^3e^i(q·x-ωτ)f(x,τ)
    
    f(x,τ)   =   ∫ d^2x dτ e^-i(q·x-ωτ)f̂(q,ω).

Moreover, η̂(q,ω)={δn̂_0(q,ω),θ̂(q,ω),Π̂_x(q,ω),û_x(q,ω),Π̂_y(q,ω),û_y(q,ω),Π̂_x,⊥(q,ω),ŵ_x(q,ω),Π̂_y,⊥(q,ω),ŵ_y(q,ω)} and the interaction matrix M(q,ω) is written as

    M(q,ω)=
    [                   γ_0                    -ω                     0                     0                     0                     0                     0                     0                     0                     0;                     ω                 ρ q^2                 -iq_x                     0                 -iq_y                     0                     0                     0                     0                     0;                     0                  iq_x                     γ                    -ω                     0                     0                     0                     0                     0;                     0                     0                     ω (2μ̃+λ̃)q_x^2+μ̃q_y^2                     0         (μ̃+λ̃)q_xq_y                     0      R/2(q_x^2-q_y^2)                     0               Rq_xq_y;                     0                 i q_y                     0                     0                     γ                    -ω                     0                     0                     0                     0;                     0                     0                     0         (μ̃+λ̃)q_xq_y                     ω (2μ̃+λ̃)q_y^2+μ̃q_x^2                     0              -Rq_xq_y                     0      R/2(q_x^2-q_y^2);                     0                     0                     0                     0                     0                     0                   γ_⊥                    -ω                     0                     0;                     0                     0                     0      R/2(q_x^2-q_y^2)                     0              -Rq_xq_y                     ω   K̃_1q_x^2+K̃_Tq_y^2                     0       (K_2+K_3)q_xq_y;                     0                     0                     0                     0                     0                     0                     0                     0                   γ_⊥                    -ω;                     0                     0                     0               Rq_xq_y                     0      R/2(q_x^2-q_y^2)                     0      (K_2+K_3)q_x q_y                     ω  K̃_T q_x^2+K̃_1q_y^2 ].



As we can observe the interaction matrix 𝐌(q,ω) separates in two independent blocks of 6×6 and 4×4 when R=0, i.e. in the case of the dodecagonal quasicrystal pattern. This case corresponds to the simplest scenario, in which phasons and phonons fields do not interact with each other.

Now we can proceed with our study of the low energy behavior of the system. The dispersion relation ω(q) corresponding to each excitation mode can be obtained by solving the equation [M(q,iω(q))]=0. The solution of this equation leads to two groups of solutions corresponding to the two different blocks of the matrix M(q,ω) in this case. The solutions arising from the 6×6 block of M(q,ω) are given by

    ω_1(q)   =   √(A+√(A^2+B))q
    ω_2(q)   =   √(A-√(A^2+B))q
    ω_3(q)   =   √(γμ̃)q,

where A=(ργ_0+γ(2μ̃+λ̃))/2 and B=(2μ̃+λ̃)γ_0(1-γρ). The form of the solutions indicates a hybridization process of the modes with dispersion relation ω_1(q) and ω_2(q), the analysis of eigenvectors associated to these eigenvalues allow us to conclude that these two excitation modes correspond to a hybridization of the longitudinal Bogoliubov mode with the longitudinal phonon mode, while ω_3(q) corresponds to the transversal phonon mode. A similar procedure for the 4×4 block of M(q,ω) yields the dispersion relations 

    ω_4(q)   =   √(γ_⊥K̃_1)q
    ω_5(q)   =   √(γ_⊥K̃_T)q,

corresponding to the longitudinal and transversal phason modes, respectively. 



§ III. FLUCTUATION EIGENSTATES OF THE DODECAGONAL QUASICRYSTAL

The longitudinal or transversal character of the excitation modes in the dodecagonal quasicrystal refers to the way in which the vectorial fields Π(x,τ), u(x,τ), Π_⊥(x,τ) and w(x,τ) oscillate respect to the momentum of the excitation q. For a better understanding of the nature of these modes we calculated the corresponding eigenvectors η̂_j(q) in a situation in which q is oriented along the "x"-axis, yielding 


    η̂_1(q_x)   =   { (c_1^2-γ(2μ̃+λ̃))c_1 q_x, -iγ_0(c_1^2-γ(2μ̃+λ̃)), γ_0(2μ̃+λ̃)q_x, -iγ_0c_1,0,0,0,0,0,0}
    η̂_2(q_x)   =   { -(c_2^2-γ_0ρ)c_2 q_x, iγ_0(c_2^2-γ_0ρ), γ_0(2μ̃+λ̃)q_x, -iγ_0c_2,0,0,0,0,0,0}
    η̂_3(q_x)   =   {0, 0, 0, 0, c_3 q_x, -iγ, 0, 0, 0, 0}
    η̂_4(q_x)   =   {0, 0, 0, 0, 0, 0, c_4 q_x, -i γ_⊥, 0, 0}
    η̂_5(q_x)   =   { 0, 0, 0, 0, 0, 0, 0, 0, c_5 q_x, -iγ_⊥},

where c_i=∂_q ϵ_i(q), refers to the sound velocity of the corresponding excitation mode. From here, the fluctuation fields in real space and time writes as η_j(x,t)=Re[η̂_j(q)exp(i(q·x-ω_jt))] and the correction to the ground state wave function as 

    δψ   =   √(ρ)(δ n_0(x,t)/2ρ c_0^2+ic_0θ(x,t)+1/2∑_n≠0c_n(G_n·Π(x,t)/Z+G_n,⊥·Π_⊥(x,t)/Z).
          .×exp(iG_n·x)+i∑_n≠0c_nexp(iG_n·x)× (θ(x,t)+G_n·u(x,t))+G_n,⊥·w(x,t)).).

Comparing this expression with the usual ansatz for the solution of the linearized Gross-Pitaevskii equation <cit.>, δψ_j,q=u_j,q(r)exp(i(q·r-ω_j(q)t))-v_j,q(r)^*exp(-i(q·r-ω_j(q)t)), we identify in each case the  functions u_j,q(r) and v_j,q(r). This allows us to compute the phase and density fluctuations for the excitation modes as Δρ_j,q=| u_j,q(r)-v_j,q(r)|^2 and Δφ_j,q=| u_j,q(r)+v_j,q(r)|^2 <cit.>. In Fig.<ref>(c-l) we show the behavior of these quantities considering typical parameters for the ground state wave function and for the pair interaction potential. For the numerical evaluation, the effective couplings of the theory can be roughly estimated using a single mode expansion for the wave function of the dodecagonal QC. In this case, we can write ϕ(x)=√(ρ)(c_0+c_1∑_j=1^12exp(i ke_j·x)), where {e_i} is a set of unitary vectors forming regular dodecagon. In the small c_1 limit we can reach to 

    Z   ≈   Z_⊥≈6ρ k^2c_1^2
    γ   ≈   γ_⊥≈ Uv̂(k)/(6c_1^2k^2)
    γ_0   ≈   Uv̂(0)
    
    (2μ̃+λ̃)   ≈   K̃_T≈6ρ k^2c_1^2+3Uρ^2c_1^2(kv̂'(k)+3k^2v̂”(k))
    μ̃   ≈   K̃_1≈6ρ k^2c_1^2+3Uρ^2c_1^2(3 kv̂'(k)+k^2v̂”(k)).

Considering U=1, ρ=0.4, k=1,  v̂(0)=2, v̂(k)=-1, v̂'(k)=0, v̂”(k)=1.5, c_0=1, c_1=0.15 and a momentum of the excitation q=0.1(1,0) we obtain the density and phase fluctuations patterns shown in Fig.1 (c-l).





§ IV. NUMERICAL CONSIDERATIONS FOR FIG. 2

For the numerical evaluation associated to Fig.2 (a) we considered the following set of arbitrary parameters ρ=4, γ_0=4, γ=1 γ_⊥=1, 2μ̃+λ̃=52, μ̃=20, K̃_1=24, K̃_T=56 and R=32.  

Moreover, as explained in the main text for Fig.2 (b) we need more realistic estimates of the behavior of the couplings of the effective theory in Eq.(3), these can be obtained from a single mode expansion, which for the decagonal quasicrystal, yields 

    Z   ≈   Z_⊥≈5ρ k^2c_1^2
    γ   ≈   γ_⊥≈ Uv̂(k)/(5c_1^2k^2)
    γ_0   ≈   Uv̂(0)
    
    (2μ̃+λ̃)   ≈   5ρ k^2c_1^2+5/2Uρ^2c_1^2(kv̂'(k)+3k^2v̂”(k))
    μ̃   ≈   5ρ k^2c_1^2+5/2Uρ^2c_1^2(3 kv̂'(k)+k^2v̂”(k))
    K̃_1=K̃_T   ≈   5ρ k^2c_1^2+5Uρ^2c_1^2(kv̂'(k)+k^2v̂”(k))
    
    R   ≈   5/2Uρ^2c_1^2(-kv̂(k)+k^2v̂”(k)).

Additionally, for numerical evaluations we use the anzats  c_1 (Uρ)=0.035+0.51(Uρ-0.2)/(Uρ)^0.65. This ansatz is only intended to capture the two main features of the dominant Fourier amplitude of the QC pattern, i.e. c_1(Uρ) is a strictly growing function defined from the QC phase boundary (Uρ=0.2). Secondly, c_1(Uρ) should be concave reflecting the tendency of the system to avoid collapse as Uρ is increased. Furthermore, for the pair interaction parameters we consider v̂(k)=-1, v̂'(k)=0, v̂”(k)=1 and v̂(0)=1.     




§ BEHAVIOR OF THE DENSITY AND PHASE LOCAL FLUCTUATIONS IN THE OCTAGONAL QUASICRYSTAL PATTERN


In this section we present the typical behavior of the local density and phase fluctuations for the octagonal quasicrystal for different orientations α of the propagation direction of the excitation respect to one of the main directions of the quasicrystal. For numerical calculations all the couplings of our effective theory takes the same values considered for the construction of Fig.(<ref>). 

The first case presented corresponds to the most general scenario for which all low energy modes are hybridized. As a consequence the scaling of the density and phase fluctuations is O(1) for all modes. For the numerical evaluation in this case we select α=π/16, the results are presented in Fig.(<ref>). Next we consider separately the cases of α=0 in Fig.(<ref>) and α=π/8 in Fig.(<ref>). For these special directions, as they correspond to symmetry directions of the octagonal quasicrystal structure, modes are purely longitudinal or transversal and the hybridization takes place only between modes in each of these separate groups. Because of this feature the scaling of the density and phase fluctuations for the longitudinal and transversal modes remains different in these cases.  


  









 Paper Title: Transmission of quantum information through quantum fields in curved spacetimes.

 







mk7592@princeton.edu
Department of Physics, Princeton University, Princeton, New Jersey 08544, USA


erickson.tjoa@mpq.mpg.de
Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany


§ ABSTRACT 

We construct a relativistic quantum communication channel between two localized qubit systems, mediated by a relativistic quantum field, that can achieve the theoretical maximum for the quantum capacity in arbitrary curved spacetimes using the Unruh-DeWitt detector formalism. Using techniques from algebraic quantum field theory, we express the quantum capacity of the quantum communication channel purely in terms of the correlation functions of the field and the causal propagator for the wave equation. Consequently, the resulting quantum channel, and hence the quantum capacity, are by construction manifestly diffeomorphism-invariant, respect the causal structure of spacetime, and are independent of the details of the background geometry, topology, and the choice of Hilbert space (quasifree) representations of the field.  



Transmission of quantum information through quantum fields in curved spacetimes
    Erickson Tjoa
    August 5, 2024
===============================================================================







§ INTRODUCTION


In the spirit of it-from-(qu)bit, quantum information theory has been extensively used to understand generic physical phenomena from quantum many-body physics to high-energy physics, from applied to foundational physics. Relativistic quantum information (RQI) is one of the many attempts to realize this spirit, which thus far focuses on the role of relativity in quantum information-processing tasks, as well as a deeper understanding of fundamental theories such as relativistic quantum field theories (QFT). The role of quantum information theory in relativistic QFTs cannot be understated: a non-exhaustive list includes rigorous study of the entanglement structure, complexity, and measurement theory for QFTs <cit.>; operational approach to particle production due to non-inertial motion and gravitational fields <cit.>; connections between relativistic causality, information-theoretic causality and reference frames <cit.>;  quantum information protocols mediated by the relativistic quantum field <cit.>; etc. 

This work is motivated by a somewhat vague, ambitious, and fundamental question: 

    Problem: What is the information-carrying capacity of quantum fields?

One can think of several ways to answer this question by making the problem statement more precise. For our purposes, we will adopt an operational approach, namely, whether two parties Alice and Bob, each in possession of some quantum-mechanical probes (atoms, qudits, oscillators, etc.) are able to transmit (quantum) information using the relativistic quantum field as a medium — that is, as a relativistic quantum channel mediating the communication between Alice and Bob. The paradigmatic model of such a communication channel is through the Unruh-DeWitt (UDW) particle detector model and its generalizations <cit.>. Indeed, transmission of classical information has been analyzed by many authors both perturbatively and non-perturbatively and in flat and curved spacetimes <cit.>. Transmission of quantum information in the non-perturbative regime has been studied only in flat spacetimes in very specific examples <cit.>. Note that in all these cases, the communication is encoded through the field amplitude and not, say, through the polarization degrees of freedom of the electromagnetic field, since on its own the polarization is blind to the relativistic features and constraints. 

Here we present the most general formulation for the transmission of quantum information between Alice and Bob, each in possession of a two-level system (a `qubit detector'), by suitable coupling to a relativistic scalar field in arbitrary curved spacetime. Clearly, our work constitutes a natural extension of the flat spacetime result in <cit.>. However, we will do so by exploiting techniques from algebraic quantum field theory (AQFT) <cit.>. Crucially, what we gain from this will not be a simple rewriting of all the formulae in the literature: we will see that the quality of the relativistic quantum channel mediated by the quantum scalar field can be formulated purely in terms of smeared correlation functions of the field and the detector parameters. Our results make clear what features of the field-mediated communication channel are necessary for perfect transmission of quantum information, and they are by construction manifestly diffeomorphism-invariant, respect the causal structure of spacetime, and are independent of the details of the background geometry, topology, and (to some extent) the Hilbert space representations of the field. In particular, it automatically works for all quasifree representations that give rise to Fock representations <cit.>).

This paper is organized as follows. In Section <ref> we briefly review the scalar field theory in curved spacetimes using the AQFT framework. In Section <ref> we describe our UDW detector setup and review the quantum capacity of a quantum channel. In Section <ref> we give the protocol that allows the transmission of quantum information through the quantum field. In Section <ref> we discuss the validity of the channel and study the regimes in which it fails. In Section <ref> we conclude with some discussions of our results and outlook. We set c=ħ=1 and adopt mostly-plus signature for the metric.




§ SCALAR FIELD THEORY IN CURVED SPACETIME




In this section we review the scalar field theory in curved spacetimes within the AQFT framework, adapting the convention given in <cit.> (for more details and reviews, see also <cit.>). 



 §.§ AQFT for scalar field


Consider a Klein-Gordon (KG) scalar field ϕ in an (n+1)-dimensional globally hyperbolic Lorentzian spacetime ℳ with metric tensor g_ab, whose equation of motion is given by the KG equation: 

    (-∇_a∇^a + m^2 +ξ R)ϕ = 0 ,

where ∇_a is the covariant derivative with respect to the Levi-Civita connection, m is the mass parameter, R is the Ricci scalar curvature, and ξ≥ 0 controls the non-minimal coupling of ϕ to R. Global hyperbolicity implies that we have a foliation of spacetime Σ and the KG equation is well-posed with initial data given on Cauchy surfaces. 

The KG operator P̂ -∇_a∇^a + m^2 +ξ R admits a unique advanced and retarded Green's functions G_A(,),G_R(,) = G_A(,) which satisfies

    P̂_ G_R/A(,) = δ^n+1(-) ,

where the subscript on P̂_ means the KG operator is w.r.t. the variable . We can use these Green's functions to construct the antisymmetric causal propagator (also known as the Pauli-Jordan distribution)

    E(,)  G_R(,) - G_a(,) .

Canonical quantization of the field ϕ gives rise to a field operator ϕ̂ obeying the canonical commutation relations (CCR) 

    [ϕ̂(),ϕ̂()] = i E(,) .

As ϕ̂ is an operator-valued distribution, in the algebraic approach to QFT we consider smeared field operators, i.e., by viewing the field operator as an -linear map from the test space of smooth compactly supported functions  to a *-algebra (), i.e., as a map ϕ̂: C^∞_0 ()→𝒜() with

    ϕ̂(f)∫ V  f()ϕ̂() .

The algebra of observables () is defined to be the collection of sums of products of ϕ̂(f) over all functions in  obeying the smeared CCR relation (<ref>), i.e., a CCR algebra <cit.>. The KG equation is encoded in the fact that we identify ϕ̂(f) ≡ 0 if f=P̂h for some h∈. 



The dynamical content of the field theory is reflected in terms of the solution space of the KG equation. The solution space _() can be equipped with a symplectic form σ:_()×_()→, defined as

    σ(ϕ_1,ϕ_2) ∫_Σ_tΣ^a [ϕ_1∇_aϕ_2 - ϕ_2∇_aϕ_1] ,

where Σ^a = -t^a Σ, -t^a is the inward-directed unit normal to the Cauchy surface Σ_t, and Σ = √(h) ^n is the induced volume form on Σ_t <cit.>. As is well-known, this definition is independent of the choice of Cauchy surface. The field operator ϕ̂(f) can be expressed as symplectically smeared field operator  <cit.> 

    ϕ̂(f) ≡σ(Ef,ϕ̂) ,

and the CCR algebra can be written as 

    [σ(Ef,ϕ̂),σ(Eg,ϕ̂)] = iσ(Ef,Eg) = i E(f,g) ,

where σ(Ef,Eg) = E(f,g) in the second equality follows from Eq. (<ref>) and (<ref>). 

Since the smeared field operators are unbounded operators, for free fields it is more convenient technically to work the Weyl algebra (), whose elements are bounded operators constructed by “exponentiating” the field operators. The Weyl algebra () is a unital C^*-algebra generated by elements that formally take the form 

    W(Ef) ≡e^iϕ̂(f) ,   f∈ .

These elements satisfy Weyl relations:

    W(Ef)^†   = W(-Ef) , 
       
        W(E (P̂f) )    =  ,
    
        W(Ef)W(Eg)    = e^-i/2E(f,g) W(E(f+g))

where f,g∈. The third relation enforces relativistic causality (or microcausality).


In AQFT, a quantum state is defined as a -linear functional ω:()→ (similarly for ()) such that 

    ω() = 1 ,  ω(A^† A)≥ 0  ∀ A∈() .

The state ω is pure if it cannot be written as ω= αω_1 + (1-α)ω_2 for any α∈ (0,1) and any two algebraic states ω_1,ω_2; otherwise we say that the state is mixed. 


The way to pass from AQFT to the Hilbert space approach is through Gelfand-Naimark-Segal (GNS) reconstruction theorem <cit.>. The GNS theorem gives us, for a given algebra of observables () and a state ω, a GNS triple (ℋ_ω, π_ω,|Ω_ω⟩), where π_ω: ()→ℬ(ℋ_ω) is a Hilbert space representation with respect to state ω. In its GNS representation, any state ω is realized as a vector state |Ω_ω⟩∈ℋ_ω and  A∈() are represented as bounded operators Âπ_ω(A)∈ℬ(ℋ_ω), and we write ω(A) = ⟨Ω_ω|Â|Ω_ω|$⟩. Since there exists infinitely many unitarily inequivalent representations of the CCR algebra, the algebraic approach allows us work with all representations at once and only pick the physically relevant representation at the very end. 

For a fixed stateω, we can construct the Wightmann-point functions, defined by

    𝖶(f_1,...,f_n)ω(ϕ̂(f_1)...ϕ̂(f_n))

wheref_j∈and the RHS is computed within some GNS representation of(). The GNS representation of the Weyl algebra()allows us to calculate Eq. (<ref>) by differentiation[Strictly speaking, the state on () and on () are different, but they are in one-to-one correspondence whenever they are related by Eq. (<ref>) (see <cit.> for good exposition on this). ]: for example,  the smeared Wightman two-point function reads

    𝖶(f,g) ≡ -∂^2/∂ s∂ t|_s,t=0ω(e^iϕ̂(sf)e^iϕ̂(tg))

where the RHS is calculated in the GNS representation of()<cit.>. 

The consensus within the AQFT community is that physically reasonable states should be Hadamard states<cit.>. Very roughly speaking, these states respect local flatness and the expectation values of all observables (in particular, the renormalized stress-energy tensor) are finite <cit.>. A particularly nice subclass of Hadamard states is the family of quasifree states: for these states, all odd-point functions in the sense of (<ref>) vanish and all higher even-point functions can be written as in terms of just two-point functions. In modern terminology, the term quasifree state is synonymous with Gaussian state, where the one-point functions need not vanish and higher-point functions only depend on one- and two-point functions <cit.>.

The relevance of quasifree states lies in the fact that they are completely specified once we know the Wightman two-point functions. More precisely, any given quasifree stateωis associated with a real-bilinear inner productμ:_()×_()→that satisfies <cit.>
    |σ(Ef,Eg)|^2 ≤μ(Ef,Ef)μ(Eg,Eg) ,

where we recall thatEf,Eg∈_()for allf,g∈. The inequality is saturated ifω_μis a pure quasifree state. Any quasifree state can then defined by those that satisfy

    ω(W(Ef))  e^-μ(Ef,Ef)/2 .

However, this definition is not useful unless we can compute the induced norm||Ef|| √(μ(Ef, Ef)). 

It turns out that||Ef||is related to the Wightman two-point function as follows <cit.>: we complexify_()into the space of complex solutions_()and define the KG bilinear product by 

    ⟨φ_1,φ_2|_⟩ iσ(φ_1^*,φ_2)

forφ_1,φ_2∈_(). It can be shown that for any quasifree state, there exists a subspaceℋ⊂_()such that(ℋ,⟨·,·|_⟩)is a Hilbert space and an-linear mapK:_()→ℋsuch that for allφ_1,φ_2∈_()* K_()+i K_() is dense in ℋ;
    
  * μ(φ_1,φ_2) = ⟨Kφ_1,Kφ_2|_⟩;
    
  * σ(φ_1,φ_2) = 2⟨Kφ_1,Kφ_2|_⟩;
    
  * _()≅ℋ⊕ℋ, where ℋ is the complex-conjugate Hilbert space of ℋ and ⟨u,v|_⟩=0 for all u∈ℋ and v∈ℋ.

In a more familiar language of canonical quantization, the mapKprojects to the “positive-frequency part” of a real solution to the KG equation. The Wightman two-point function is then given by <cit.>
    𝖶(f,g)    = ⟨KEf,KEg|_⟩
       = μ(Ef,Eg) + i/2E(f,g) ,

where we have used the fact thatσ(Ef,Eg) = E(f,g). SinceE(f,g)is antisymmetric, it follows that||Ef||^2 = μ(Ef,Ef) ≡𝖶(f,f)and hence

    ω(W(Ef)) =   e^-1/2𝖶(f,f) .

Indeed, we may very well take Eq. (<ref>) as the definition of quasifree states. 




The most important example of a quasifree state is the “vacuum state”ω, where we can write the (unsmeared) vacuum Wightman function as 

    ω(ϕ̂()ϕ̂()) ≡𝖶(,)    = ∫^n  u^*_() u^*_() ,

whereu_()are called the positive-frequency modes of KG operator with respect to the KG inner product

    (ϕ_1,ϕ_2)_kg iσ(ϕ_1^*,ϕ_2) ,

whereϕ_j∈_()are complexified solutions to Eq. (<ref>). The vacuum stateωis not unique and there is a sense in which every GNS vector is a “vacuum state” in that particular representation. However, once a particular basis of modes{u_}are chosen, it fixes the vacuum state as one whose Wightman function satisfies (<ref>). In flat spacetime, the standard plane-wave basis in the inertial coordinates defines the Minkowski vacuum, which is the only state that respects the full Poincaré symmetry and minimizes the expectation value of the (renormalized) stress-energy tensor <cit.>. In curved spacetimes, there are multiple inequivalent states that qualify as a vacuum state <cit.>.




 §.§ Relationship with canonical quantization
 

The usual Fock space in canonical quantization arises from the GNS reconstruction applied to()with respect to some vacuum stateω. Technically speaking, any pure quasifree (Hadamard) state qualifies as a vacuum state, as their GNS representation gives rise to the Fock vacuum in the standard sense <cit.> (see also <cit.> for a different definition in Minkowski spacetime). This is the best one can do in generic curved spacetimes as there is no preferred vacua without further assumptions such as time translation symmetry. 

In more detail, the GNS Hilbert spaceℋ_ωfor the pure quasifree representation is the Fock space over the one-particle Hilbert spaceℋℋ_ω≡ = ⊕_n=0^∞ℋ^⊙ n ,

whereℋ^⊙0≅andℋ^⊙nmeans symmetrized direct sum (then-particle sector of the Fock space). In this representation, we can write

    π_ω(ϕ̂(f)) = â((KEf)^*) + â^†(KEf) .

In what follows we dropπ_ωif it is clear from the context that we are using a Fock representation. Note that if we consider complex smearing functionsf:→, we can write

    ϕ̂(f) ≡ϕ̂( f) + i ϕ̂( f) .

The operatorsâ(u^*),â^†(v)are smeared ladder operators[There are various conventions on how to label the smeared ladder operators (see <cit.>). We follow <cit.> in that it views â,â^† as being linear in their arguments. The convention in <cit.> does not include complex conjugation in the argument for â(·) and views â as an anti-linear map while â^† is linear.] obeying the CCR

    [â(u^*),â^†(v)] = ⟨u,v|_⟩

on a suitable dense domain of the Fock space.

The standard canonical approach is recovered by working with the unsmeared field operatorϕ̂()and considering the Fourier mode decomposition

    ϕ̂() = ∫^n â_ u_() + â_^† u_^*() ,

whereu_()andu_^*()are positive- and negative-frequency modes normalized to Dirac delta functions via the KG inner product:

    ⟨u_,u_'|_⟩   = δ^n(-') ,
    ⟨u^*_,u^*_'|_⟩   = -δ^n(-') ,
    ⟨u_,u^*_'|_⟩   = 0 .

These modes are not proper elements of the one-particle Hilbert space but they are convenient to work with. In particular, we see thatâ^*_≡â(u_^*)andâ_^†≡â^†(u_). If we were to pick any other quasifree state, the resulting one-particle structure will differ and this can give rise to Hilbert space representations that are unitarily inequivalent, e.g., by considering Kubo-Martin-Schwinger (KMS) thermal states <cit.>. 

Using these modes, we can decompose the solutionEfin the “eigenmode basis”{u_,u_^*}Ef ≡∫^n ⟨u_,Ef|_⟩u_ + ⟨u^*_,Ef|_⟩u_^* .

From the CCR and the definition of the Wightman function, we see that

    iE(,')    = ∫^n u_()u_^*(') - u_^*()u_(') ,

the positive-frequency partKEfofEfreads

    KEf    = ∫^n ⟨u_,Ef|_⟩u_≡∫^n f_ u_

where

    f_-i∫ V f()u_^*() .
 
Furthermore, sinceKEf lies inℋ, it is convenient to use the notation|KEf⟩∈ℋand|u_⟩for the (improper) basisu_. In this notation, we write

    |KEf⟩ = ∫^n f_|u_⟩ ,

This suggests that we can label the ladder operators using the “momentum space smearing function”f_rather than the “real space positive-frequency solution”KEf, i.e., we can write

    ϕ̂(f)    = â(f_^*) + â^†(f_) ≡∫^n â_^†f^*_ + â^†_f_
       ≡ϕ̂(f_) .

Essentially, we are using the elements of the one-particle Hilbert spaceℋto label the field observables rather than using the spacetime smearing functions. This notation is particularly convenient if the focus is more about which modes of the field that the detectors are coupled to <cit.>.



§ SETUP


In this section, we introduce the UDW detector model that we will use to couple Alice's and Bob's qubits to the field and briefly review the notion of quantum capacity for a quantum channel and coherent information. These are relevant measures of how much quantum information can be transmitted (on average) through the quantum channel between Alice and Bob.



 §.§ UDW detector model


In the UDW framework, Alice and Bob carry their own qubit (`detector') with a free HamiltonianĤ_ν= 1/2Ω_νσ̂_z, whereν∈A, B,Ω_νthe energy gap andσ̂_zthe Pauli-Zoperator. We denote the eigenstates ofσ̂_sby|±_s⟩withs ∈x,y,z, where we do not label the Pauli matrices withνto reduce notational clutter and the context will make it clear.  The excited and ground states of the free HamiltonianĤ_νare given by|±_z⟩with eigenvalues±Ω_ν/2respectively.

The UDW interaction between Alice's and Bob's detectors and the field is assumed to be linear in the field as it represents a simplified model of dipole interaction𝐝̂·𝐄̂in quantum optics, i.e., as a spin-boson type interaction <cit.>. The general form of such an interaction is given by a linear combination of the form[This is a slight modification of the usual spin-boson model since typically spin-boson models are assumed to have time-independent interactions <cit.> and to account for the covariant formulation of the UDW model <cit.>.]
    ĥ_I,ν() ∑_iλ_ν,i f_ν,i() m̂_ν,i(τ()) ⊗ϕ̂() ,

whereλ_ν,iis the coupling strength,m̂_ν,i(τ)is a Hermitian operator acting on the qubit, andf_ν,i()is a spacetime smearing function defining the duration of the interaction and the spatial profile of the detector. The most commonly used UDW model involves only one term inĥ_I,ν() = λ_νf_ν()m̂_ν(τ())⊗ϕ̂()<cit.>, but we will need the more general version in (<ref>) for our purposes. We drop the tensor product symbol if the context is clear. 


The time evolution operator generated by the interaction Hamiltonian (<ref>) reads

    Û = 𝒯_t exp[-i ∫_V∑_νĥ_I,ν]

whereV [n] √(-g)is the invariant volume element and the time-ordering is assumed to be given by some global time function (see <cit.> for discussions about some subtleties of time-ordering for the spatially smeared UDW model). In general, it is not possible to solve the the time evolution analytically due to the time-dependence of the interaction, thus many analyses involving the UDW detectors consider perturbative methods in the weak coupling regimes. However, our goal in this paper is to establish a perfect quantum channel between Alice and Bob and as was explained in <cit.>, this requires our model to be in the non-perturbative regime.

In order to construct the perfect quantum channel that can transmit quantum information perfectly, we need to be able to calculate the interaction unitaryÛnon-perturbatively (in the sense of not truncating the Dyson series in the weak coupling regime). For this purpose, we will need to use the so-called delta-coupling regime <cit.>, where the interaction timescale is assumed to be much faster than all the relevant timescales of the problem so that the interaction can be taken to occur at a single instant in time. Delta-coupled detectors have been used in many contexts such as entanglement extraction, relativistic communication, work extraction, and many others (see, e.g., <cit.>). As each detector has its own rest frame, we need to consider the Fermi normal coordinates (FNC) associated with each observer  <cit.>, whose center-of-mass (COM) trajectory is parametrized by their respective proper times <cit.>.

The physically reasonable assumption we make is that in the FNC, ≡(τ,), the spacetime smearing functions can be decomposed as

    f(()) = χ(τ)F() ,

where(τ,=0)labels the COM trajectory. In other words, Alice (and Bob) can in their rest frames distinguish the spatial profile (which is assumed to satisfy `Born rigidity', i.e., time-independent with respect toτ) from the switching function that controls the duration and strength of the interactions. The Hamiltonian density in this case reads


    ĥ_I,ν() ∑_iλ_ν,iχ_ν,i(τ)F_ν,i() m̂_ν,i(τ ) ϕ̂(())

and the delta coupling interaction corresponds to the case when

    χ_ν,i(τ) = δ(τ-τ_ν,i)

for some constantsτ_ν,i∈. 

For our purposes, we consider the scenario where Alice and Bob each interact with the field via delta coupling twice, so that we need to consider multiple interaction timesτ_ν,iwithτ_ν,1 < τ_ν,2. More generally, if Alice (or Bob) performs a sequence ofNdelta interactions withτ_ν,i<τ_ν,i+1, the corresponding unitary time evolutionÛ_νcan be decomposed into a sequence of simple-generated unitaries<cit.> (see also <cit.>)

    Û_ν   = ∏_j=1^NÛ_ν,j ,  Û_j = exp[-i λ_νm̂_ν(τ_ν,j)  ϕ̂(F_ν,j)]

The time-ordering𝒯is taken care of since the unitariesÛ_iare written in time-increasing order. When two or more observers are involved, we can use the global time coordinates to orderÛ_aandÛ_b: for example, if Alice and Bob are spacelike separated then the ordering does not matter, while Alice unitary acts before BobÛ_bif(f_b)is in the causal future of(f_a). The unitaryÛ_νcan be computed non-perturbatively sinceÛ_jis a controlled unitary: for any two Hermitian operatorsÂ, B̂such thatÂ=∑_i=1^ra_j|a_j⟩⟨a_j|is finite-dimensional operator in its spectral decomposition, we can write

    exp[i Â⊗B̂] = ∑_j=1^r|a_j⟩⟨a_j|⊗exp[ia_j B̂] .

As we will see, for our setup the full unitary time evolutionÛwill consist of products of Alice's two simple-generated unitaries and also Bob's two simple-generated unitaries.

Last but not least, let us recall the unitaries found in <cit.> for Minkowski spacetime. There Alice performs two simple-generated unitaries, but one of them involves coupling to a conjugate momentum.

    Û_a = e^iσ̂_x π̂(F_a)e^iσ̂_z ϕ̂(F_a) ,

where the spatially smeared operators are

    ϕ̂(F_a)    = λ_1∫^3 F_a()ϕ̂(t_a,) ,
    π̂(F_a)    = λ_2∫^3 F_a()∂_tϕ̂(t_a,) .

Note that here they taket_at_A,1andt_A,2 = t_a+ϵand consider the limit asϵ→0^+so that the unitary generated by the conjugate momentum acts later. Bob's unitary is then constructed as a “suitable inverse” of Alice's unitary to faithfully extract Alice's qubit. 

Our construction requires us to make two modifications to the above unitaries. First, we can generalize the conjugate momentum operator in curved spacetime, i.e., 

    π̂() ∂_tϕ̂()≡√(h) t^a∇_a ϕ̂() ,

wheret^ais the forward pointing unit normal, orthogonal toΣ_twith an induced metrich. Second, it is straightforward to see from integration by parts that 

    π̂(f)    = ∫ V f()∂_tϕ̂() = ϕ̂(-∂_tf) .

Thus we can always view the smeared conjugate momentum operator as a smeared field operator with a different smearing function (namely, its time derivative). In Minkowski spacetime, using the spacetime smearing notation we can thus write

    π̂(F_a)≡π̂(δ· F_a) = ϕ̂(-δ'· F_a)  ,

whereδ'(t)is the distributional derivative of the Dirac delta functionδ(t). This second step might appear unconventional sinceπ̂is not usually regarded as a spacetime-smeared operator, but this provides a simple way to recast all basic observables in terms of a single field operatorϕ̂. The connection between the modern AQFT approach based on spacetime-smeared operators (cf. Section <ref>) and the “old” (non-covariant) AQFT formulations based on two spatially smeared operators{ϕ̂|_Σ,π̂|_Σ}at a fixed Cauchy sliceΣwith equal-time CCR[ϕ̂(t,),π̂(t,)]=iδ^(n)(-)can be made more precise and rigorous  (see, e.g., <cit.>, and also <cit.> for details). 

The two modifications above suggest that what we really need for the generalization to curved spacetime is that Alice couples to two smeared field operatorsϕ̂(f_1),ϕ̂(f_2)that do not commute, i.e.,E(f_1,f_2)≠0, and not necessarily to the field and its conjugate momentum. Indeed, in the modern AQFT framework the coupling to momentum is somewhat awkward because the perfect quantum channel in <cit.> requires that Alice couples toϕ̂(F_a)more strongly thanπ̂(F_a), but the coupling strengthsλ_1,λ_2would need to have different units, so one needs an additional length scale to compare their strengths. By considering different smeared field operatorsϕ̂(f_i), we can haveλ_iwith the same units and we could use other length scales (e.g., from the effective size of the spatial smearing) if we insist on interpreting one of them as conjugate momentum. In what follows we assume thatf_i(and henceλ_i) have the same units for simplicity.




 §.§ Quantum capacity and coherent information


Here we briefly review the notion of quantum capacity of a quantum channel and quantum coherent information, following closely the exposition in <cit.> (see <cit.> for more in-depth details). We denote by𝒟(ℋ)the space of density operators associated with some Hilbert spaceℋ. LetΦ:𝒟(ℋ_a)→𝒟(ℋ_b)be a quantum channel — a completely-positive trace-preserving (CPTP) map — between Alice's and Bob's density operators. 

An ideal quantum communication channel is one that is able to send any quantum state from Alice to Bob, or equivalently, one that preserves entanglement reliably since Alice's state can always be viewed as an entangled state with some environment that purifies it.  Schematically the idea is the following. First, Alice prepares a purification of the stateρ_a∈𝒟(ℋ_a), namely,ρ_ea|ψ_ea⟩⟨ψ_ea|∈ℋ_e⊗ℋ_awithℋ_a≅ℋ_e. Then Alice applies an encoding channelℰ:𝒟(ℋ_e⊗ℋ_a)→𝒟(ℋ_e⊗ℋ_a^⊗N)to map her share of pure state intoNquantum systems, i.e.,

    ρ_ea,N (_e⊗ℰ)(ρ_ea) .

After encoding, Alice sends each quantum system throughNindependent uses of the communication channelΦ, i.e.,

    ρ_eb,NΦ^⊗ N(ρ_ea,N)∈𝒟(ℋ_e⊗ℋ_b^⊗ N) .

Bob's task is to use a decoding channel𝒟:𝒟(ℋ_e⊗ℋ_b^⊗N)→𝒟(ℋ_e⊗ℋ_b), such that

    ρ_eb (_e⊗𝒟)(ρ_eb,N) .

This is a good encoding-decoding procedure if for a givenϵwe can have

    ||ρ̃_eb-ρ_eb||_1≤ϵ

whereρ̃_ebis the state obtained ifΦis replaced with a noiseless channel_a→b:(ρ∈𝒟(ℋ_a)) ↦(ρ∈𝒟(ℋ_b)). 

The rate of the communication channel is the number of qubits transmitted per use of the channel, i.e.,

    R 1/nlog_2(ℋ_a)

and this rate is said to be achievable if for anyδ,ϵ>0, there exists an(n,R-δ,ϵ)for sufficiently largen. The quantum capacity𝒬[Φ]of the channelΦis defined by the supremum of all achievable rates (see <cit.> for more details). 

For practical calculations that we need, we consider a useful form of the quantum capacity in terms of an information-theoretic measure called coherent information. Given a quantum channelΦ:𝒟(ℋ_a)→𝒟(ℋ_b), let|ψ_ea⟩be a purification of Alice's input stateρ_aand define

    ρ_eb    (⊗Φ)(|ψ_ea⟩⟨ψ_ea|) .

Usingρ_b _e(ρ_eb) =  Φ(ρ_a), the coherent informationI_c(ρ_a,Φ)is defined to be <cit.>
    I_c(ρ_a,Φ)     S(Φ(ρ_a)) - S(ρ_eb) .

We may sometimes writeI_c(ψ_ea,Φ)to make clear the joint state onEA. The coherent information depends on the input stateρ_aand the channelΦbut is independent of its purification. From this, we can then define the coherent information of a quantum channelI_c(Φ)by maximizing over all input states:

    I_c(Φ)   max_ρ_aI_c(ρ_a,Φ) .

Note that for qubit channels we have that-1≤I_c(ρ,Φ)≤1and0≤I_c(Φ)≤1<cit.>. We can then rewrite the quantum capacity using the Lloyd-Shor-Devetak (LSD) formula <cit.>, given by <cit.>
    𝒬[Φ]lim_n→∞1/nI_c(Φ^⊗ n) ,

where the RHS is also known as regularized coherent information.  

Despite the somewhat physically intuitive expression for𝒬[Φ], the asymptotic nature of Eq. (<ref>) makes quantum capacity very difficult to compute in most cases unless the quantum channel has very specific properties (e.g., when the quantum channel is additive, namely𝒬[Φ^⊗n] = n𝒬[Φ], which is false in general <cit.>). However, for us we do not actually need this: we follow the strategy adopted in <cit.>, which is to construct a quantum channel for qubits that hasI_c(Φ) →1by appropriately adjusting the setup parameters. This works because the quantum coherent information is superadditive<cit.>:

    I_c(Φ^⊗ N) ≥ NI_c(Φ)

which immediately implies that

    𝒬[Φ] ≥ I_c(Φ) ≥ I_c(ρ_a,Φ) .

For qubit channels, we have0≤𝒬[Φ]≤1, thus we can construct an essentially perfect quantum channel if we can show thatI_c[Φ]can be made arbitrarily close to 1. 

The interpretation of the above analysis is that in the limit where𝒬[Φ]→1, the quantum channel can effectively transmit quantum information (qubits) reliably in one use of the channel. It is perhaps not surprising that this is possible if and only if the whole procedure is essentially a “swapping operation”, i.e., Alice swaps her qubit `into the field', and then Bob swaps out the (approximate) embedded qubit in the field state to recover Alice's state. In effect, this suggests that the core of the protocol comprises of two components: (1) Bob being positioned correctly in spacetime in order to recover Alice's qubit, and (2) Alice and Bob being able to implement as high quality `SWAP gates' as possible[ In <cit.>, Alice's unitary is called ENCODING gate, and Bob's unitary is called DECODING gate. While the terminology has physical justification, it does not coincide with the encoding and decoding channels ℰ,𝒟 even in the case of a single use of the communication channel Φ (N=1), as they are not part of Φ whose capacity is being computed. The ENCODING and DECODING gates were part of the definition of the communication channel.].

As a final remark, we would like to point out that the existence of many superadditive quantum channels are in some sense a `curse' to finding a good characterization of quantum capacity — we refer the readers to <cit.> for one of the more recent summary and progress on this front. From Eq. (<ref>) we see that we need0<I_c(Φ) ≤1for the channel to be able to transmit some quantum information and our construction capitalizes on this. However, the case forI_c(Φ) = 0is much more subtle, as many quantum channels exhibit strict superadditivity in both weak and strong sense <cit.>. Weak superadditivity means that there existsNsuch thatI_c(Φ^⊗N) > N I_c(Φ), and strong superadditivity meansI_c(Φ_1⊗Φ_2) > I_c(Φ_1)+I_c(Φ_2)for two channelsΦ_1,Φ_2. In both cases, the `single-letter' formula for coherent informationI_c(Φ)is not sufficient to calculate the quantum capacity,  although some exceptions are known. For instance, all degradable channels (such as some regimes of the dephasing channels) are known to be weakly additive, in which case𝒬[Φ] = I_c(Φ)and hence non-positive channel coherent information would imply zero quantum capacity <cit.>. All anti-degradable channels (such as all entanglement-breaking channels and certain regimes of quantum erasure channels) are known to have zero quantum capacity, though the argument does not rely on additivity and more towards no-cloning type arguments <cit.>. Our construction (similarly for <cit.>) exploits the structure of the detector-field coupling to make the channel coherent information tractable.





§ PROTOCOL FOR THE PERFECT RELATIVISTIC QUANTUM CHANNEL


In this section, we present the protocol for Alice and Bob to implement a perfect quantum channel through the field. While our protocol closely follows <cit.>, our approach naturally extends to arbitrary background spacetimes and choices of vacuum states of the field by recasting the protocol in the AQFT framework. 

A brief visual of the quantum channel is depicted in Fig. <ref>. Mathematically, the quantum channel from Alice to Bob generically takes the form

    Φ(ρ_a) = _a, ϕ[Û_b^†Û_a^†( ρ_a⊗0⊗ρ_b ) Û_a^†Û_b^†]  ,

where|0⟩is the field vacuum[The state |0⟩ can be replaced with any GNS vector state |Ω_ω⟩ associated with any of the quasifree representations, which accounts for the existence of many unitarily inequivalent ground states in QFT <cit.>.],ρ_bis Bob's initial state, andÛ_aandÛ_bare the unitaries implemented by Alice and Bob through the UDW interaction with the field. The task is to construct a perfect quantum channel that can faithfully transmit quantum information to arbitrary accuracy.

Essentially, the basic idea is for Alice to (approximately) perform a `SWAP' gate[Technically speaking, the qubit and the quantum field have different Hilbert space dimensions but we will call them `SWAP' gates anyway since it would be the usual SWAP gate if we were to replace the quantum field with a third qubit. The fact that we are “embedding” the qubit state into the (much higher-dimensional) field can be viewed as a form of encoding operation, which justifies calling Alice's unitary ENCODING gate in <cit.>, and similarly Bob's unitary can be viewed as decoding operation from the field to his qubit system.] on her qubit state to the field, effectively encoding it into some superposition of approximately orthogonal coherent states, so that Bob can extract the qubit state by performing another swap operation. The entire problem thus reduces to analyzing under what circumstances the UDW interaction allows for nearly perfect `SWAP' operations on both parties which will depend on both the coupling strengths and the supports of the interactions. In particular, we will see that Bob can only perform a perfect `SWAP' gate if he is causally connected to Alice and the details of the interactions determine the quality of the swap operation. 

More precisely, Alice starts with an arbitrary state|ψ_a⟩c_1 |+_z⟩ + c_2 |-_z⟩, equivalentlyρ_a=|ψ_a⟩⟨ψ_a|, and encodes that state into the field using a `SWAP' gate. Bob should then be able to decode the state from the quantum field into his own Hilbert space. If the `SWAP' gatesÛ_aandÛ_bare implemented correctly, then we should have a perfect quantum channelΦ(ρ_a) = ρ_ain the idealized limit. We start by discussing how Alice encodes her state into the field:


    Let  |ψ_a⟩ = c_1 |+_z⟩ + c_2 |-_z⟩ be Alice's initial state and let |0⟩ be the initial vacuum state of the field. Consider Alice's unitary operation at time t=t_a given by[The usual convention would be Û_a =  e^-i σ̂_x ϕ̂(f_2) e^-i σ̂_z ϕ̂(f_1), but we opt for the positive sign for the exponent for convenience as many expressions appear more symmetric (this was also implicitly done in <cit.>). This amounts to the replacement ĥ_I→ -ĥ_I, or equivalently f_i→ -f_i. ]
    Û_a = e^i σ̂_x ϕ̂(f_2) e^i σ̂_z ϕ̂(f_1) .

    Then Alice can approximately swap the state |ψ_a⟩ into the field provided the following conditions are satisfied 
    
    E(f_1, f_2)    = π/4 mod  2π ,
    
            E(f_1, f_2)^2    ≫𝖶(f_2, f_2) .

    The resulting state after Alice's operation reads 
    
    Û_a (c_1 |+_z⟩   + c_2 |-_z⟩) |0⟩
       = |+_y⟩ (c_1 |+α⟩ - i c_2 |-α⟩) ,

    where |±α⟩ are coherent states of the quantum field defined by
    
    |±α⟩ e^± i ϕ̂(f_1)|0⟩.

    In order to prove this statement, we will explicitly apply Eq. (<ref>) to Alice's qubit and the field and show that the qubit state is encoded under the conditions in Eqs. (<ref>) and (<ref>).

    The first unitary in Eq. (<ref>) gives 
    
    e^i σ̂_z ϕ̂(f_1)|ψ_a⟩|0⟩   = c_1 |+_z⟩|+α⟩  + c_2 |-_z⟩|-α⟩.

    Next, we need to apply the second unitary e^i σ̂_x ϕ̂(f_2) but we first note that 
    
    ϕ̂(f_2) |±α⟩ = ± E(f_1, f_2) |±α⟩ + e^± i ϕ̂(f_1)ϕ̂(f_2) |0⟩ ,

    which follows directly from the Baker-Campbell-Haussdorff (BCH) formula and the CCR. Now we see that if 
    
    E(f_1, f_2)^2 ≫0ϕ̂(f_2) ϕ̂(f_2)0 = 𝖶(f_2, f_2)

    then |±α⟩ become approximate eigenvectors of ϕ̂(f_2)
    ϕ̂(f_2) |±α⟩≈± E(f_1, f_2) |±α⟩.

    If we pick the eigenvalues to be E(f_1, f_2) = π/4 mod 2π, then the action of Alice's full unitary Û_a gives
    
    Û_a    (c_1 |+_z⟩ + c_2 |-_z⟩) |0⟩
       = e^i σ̂_x ϕ̂(f_2)(c_1 |+_z⟩|+α⟩  + c_2 |-_z⟩|-α⟩)
       ≈ c_1 e^i π/4σ̂_x |+_z⟩|+α⟩  
            + c_2 e^-i π/4σ̂_x |-_z⟩|-α⟩
       = |+_y⟩(c_1 |+α⟩ - i c_2 |-α⟩) ,

    where we used e^i π/4σ̂_x |+_z⟩ = |+_y⟩ and e^-i σ̂_x π/4|-_z⟩ = -i|+_y⟩ in the last equality.



The second of Alice's conditions in Eq. (<ref>) can be interpreted as a “strong coupling condition” that requires Alice's coupling through the field via the interaction profilef_1be much stronger than throughf_2. In <cit.>, this corresponds to choosingf_2such thatϕ̂_2corresponds to conjugate momentumπ̂(F_a)and the condition translates to Alice coupling to the fieldϕ̂(F_a)much more strongly than toπ̂(F_a)(cf. Eq. (<ref>)). We can interpret Eq. (<ref>) as a “fine-tuning condition” that performs near exactπ/4rotations in the Bloch sphere so that Alice's second unitary decouples her detector from the quantum field.

Now we move on to discussing how at a later time, Bob can decode Alice's state from the quantum field into his own qubit detector. We notice that the protocol in Proposition <ref> leaves Alice's system in the state|+_y⟩, therefore if Bob's qubit starts in that initial state, it seems that Bob should simply apply the inverse ofÛ_ato decode the state from the field, i.e.,

    Û_b^†Û_a^† =  e^-i σ̂_z ϕ̂(f_1) e^-i σ̂_x ϕ̂(f_2) ,

where the qubit operatorsσ̂_sare now understood to act on Bob's Hilbert space. If Bob were able to implement this unitary then we have (almost trivially) achieved our goal of creating a perfect quantum channel from Alice to Bob. The issue is that Alice's operations defined throughϕ̂(f_i)are defined at timet=t_a, while we want Bob to decode from the field using only field operatorsϕ̂(g_i)localized at timet=t_b. We need a way to write the operatorsϕ̂(f_i)using different spacetime smearing functionsg_idefined at a different Cauchy slice. To achieve this we use the following theorem:

    Suppose we have two smeared field operators ϕ̂(f) and ϕ̂(g), then the two operators are equal ϕ̂(f) = ϕ̂(g) if the smearing functions satisfy
    
    Ef = Eg.

    The proof is almost immediate from the symplectic definition of smeared operators in Eq. (<ref>). If we assume Ef = Eg[The converse statement is false. From the axioms of AQFT, we also have that ϕ̂(P̂f)=0 which is the KG equation. Thus ϕ̂(g) is only unique up to homogeneous solution of the KG equation, i.e., g∼ g+P̂f.], then 
    
    ϕ̂(f) = σ(Ef, ϕ̂) = σ(Eg, ϕ̂) = ϕ̂(g) .


Some remarks are in order in view of the above Proposition <ref>. First, intuitively Alice's state is encoded into the field amplitude that propagates into the causal future of Alice's interaction region 𝒥^+(f_i), where𝒥^+((f_i))is the causal future of(f_i); for Bob to decode that state, his qubit has to be coupled everywhere where the initial data has propagated at a given timet=t_b, which in turn forces Bob's detector to interact with the field with the correct support and coupling strengths. In particular,(g_i)should essentially beΣ_t_b ∩(Ef_i), where(Ef_i)⊆𝒥^+(f_i). For simplicity, we assume that(f_1)=(f_2)and hence(g_1)=(g_2), which in the delta-coupling scenario is easily enforced as we are fixing the spatial profile of the interaction on Alice's side — this, in turn, fixes the needed spatial profile for Bob's coupling. 

Second, despite its simplicity and simple interpretation, Eq. (<ref>) is largely impractical to use for actually calculating Bob's spatial smearing functionsG_i(x)for a given timet = t_b. Using the eigenmode basis provided in Eq. (<ref>) and the momentum space smearing functions in Eq. (<ref>), the condition in Eq. (<ref>) can be equivalently written in `momentum space' as

    f_ k = g_ k.

These equations in momentum space are often easier to work with since in many simple cases (such as all FRW spacetimes) the spatial part of the mode functionu_k() ∝e^i k ·xand we can use Fourier transforms, which is similar to what was originally done in <cit.> where Bob's spatial smearing functions were related to Alice's in momentum space. More generally, in generic curved spacetimes, we do not have a simple plane-wave basis, so it is essential that all the conditions we demand for the perfect quantum channel are given only in terms of the correlation functions of the field without reference to Fourier transforms. 

Last but not least, note that there is an asymmetry in how we quantify the quality of Alice's and Bob's respective `SWAP' gates, i.e., how close their unitaries are in achieving the ideal `SWAP' operation. On the one hand, Alice needs two smearing functionsf_1andf_2that satisfy the conditions in Proposition <ref> to approximate the ideal `SWAP'. On the other hand, Bob needs two smearing functionsg_1andg_2that are related to Alice's via Proposition <ref> and hence automatically satisfy the conditions in Proposition <ref>. However, Proposition <ref> alone is not enough for the perfect quantum channel: ifEf_i ≠Eg_i, Bob will not be able to decode Alice's state even though by construction Bob's smearing functionsg_1andg_2satisfy Eqs. (<ref>) and (<ref>). In a sense, to decode the state using the `SWAP' operation with the field Bob needs an extra condition, namely that Bob needs to interact “at the correct spacetime regions”. The reason for this asymmetry is that the Hilbert space dimension of the fieldϕ̂is (much) larger than that of the detectors: it is much simpler for Alice to encode her state in a larger system but Bob will need additional information to decode it. 

In summary, we have constructed a perfect relativistic quantum channel that allows Alice to send her qubit state to Bob through the quantum field. The quantum channel consists of two steps:

  *  At time τ= τ_a, Alice encodes her qubit into the field using the unitary Û_a such that the conditions in Proposition <ref> are satisfied.
    
    
  *  At a later time τ = τ_b, Bob decodes the state from the field using the unitary Û_b defined in Eq. (<ref>). However, to implement this protocol, Bob's detector must interact across a spacetime region that satisfies the conditions specified in Proposition <ref>.

As the proper times are not the same in both trajectories, in practice one can use the ambient global coordinate systems and foliation to help with the time-ordering of these operations. Once timesτ_aandτ_bare specified, we demand that Alice's spatial smearing functionsF_i(x)fulfill conditions (<ref>) and (<ref>), and Bob's functionsG_i(x)fulfill the condition (<ref>)[In principle, Alice or Bob could each need up to four spatial smearing functions since f_i or g_i could include both δ(t) and δ'(t) terms. For example, in flat spacetime, we can have f_1 = δ(t) F_11( x) + δ'(t)F_12( x) (similarly for f_2), but for simplicity these can be chosen to be equal. Both g_1 and g_2 are fixed by f_1 and f_2 in the ideal case.]. 

 In order to simplify our subsequent discussions, we make a physically reasonable assumption that Alice's interaction profilesf_iare completely fixed except their coupling constants, i.e., in FNC we have

    f_i = λ_i χ_i(τ)F()

whereχ_iare either the Dirac delta functionδor its distributional derivativeδ'and the spatial profile is the same for bothf_1,f_2(hence they have the same supports(f_i) = (F)⊆). In other words, Alice only varies her interaction profilef_iby tuning the coupling strengthsλ_iwithout changing the `shape' of the interaction profile, and this is sufficient to satisfy the perfect `SWAP' conditions (<ref>)-(<ref>) in Proposition <ref>.




§ ANALYTIC TESTS OF THE QUANTUM CHANNEL


In this section, we will prove that the protocol given in Section <ref> produces a perfect quantum channel in the correct limit without any approximations like Eq. (<ref>). This proves that entanglement can always be perfectly transmitted between two localized qubits through a quantum field regardless of the background spacetime geometry. We also discuss the case when Alice and Bob can in principle perfectly implement the SWAP gates with the field but are spacelike separated from each other. For this channel, the quantum and classical channel capacity are both zero, as we expect with relativistic causality. 




 §.§ Ideal case: perfect quantum channel


Recall that our task is to show that the channel coherent informationI_c(Φ)can be made arbitrarily close to 1, thus proving that the (relativistic) quantum channel can transmit quantum information perfectly. This means computing the coherent informationI_c(ρ_a,0,Φ)and maximizing over all possible input states on Alice's side. Using the definition of coherent information (<ref>), it turns out that it is sufficient to consider the case when Alice's initial state is the maximally mixed stateρ_A, 0 = 1/2 and Bob initial state is|+_y⟩state in order to show that we can attain maximum channel coherent informationI_c(Φ)→1and hence maximal quantum capacity. We stress that this does not mean Alice should send a maximally mixed state to Bob, which is “garbage” as a message and is even classical, since in practice Alice wants to be able to send pure states properly. Also, in general, it is not true thatI_c(Φ)is maximized by the maximally mixed state: for example, the completely depolarizing channelΦ(ρ) = /2hasI_c(/2,Φ) = -1but the channel coherent informationI_c(Φ)=0is attained by any pure state.


In order to show thatI_c(Φ)→1is attainable, we need to introduce a qubit of the environmentEthat purifies Alice's maximally mixed state/2, such that the joint state of Alice and the environment is the maximally entangled state|ψ_ea⟩ = 1/√(2) (|-_z⟩ |+_z⟩ + |+_z⟩ |-_z⟩ )_ea. As discussed in Section <ref>, a perfect quantum channel will preserve all the entanglement between Alice and the environmentEAto Bob and the same environmentEB. To computeI_c (/2, Φ), we must evaluate the stateρ_ebin Eq. (<ref>). 

Writing out the expression forρ_ebin terms of the unitariesÛ_aandÛ_b, we obtain:

    ρ_eb=_Aϕ [Û_bÛ_a( ψ_ea⊗0⊗+_y ) Û_a^†Û_b^†].

The simplest way of continuing is to decompose the unitaries using projections onto the eigenvectors of the qubit operators. That is, forÛ_awe write

    Û_a   = ∑_x,z ∈{±}P̂_x P̂_z ⊗ e^i x ϕ̂(f_2) e^i z ϕ̂(f_1)
       = ∑_x,z ∈{±}P̂_x P̂_z ⊗ W(xEf_2) W(zEf_1)

whereP̂_iare the projectors onto the eigenstates ofσ̂_iand in the last line we used the Weyl algebra notation introduced in Eq. (<ref>). For Bob's unitary, we write

    Û_b = ∑_x,z ∈{±}P̂_z P̂_x ⊗ W(-zEf_1) W(-xEf_2)

where the projection operators are understood to act on Bob's system. We have written Bob's unitary as the inverse of Alice's, the imperfect case whereÛ_b ≠Û_a^-1is discussed in the following sections. We also expand the Bell state as|ψ_ea⟩ = 1/√(2) ∑_j |-j_z⟩ |j_z⟩withj ∈{ ±}to get the expression

    ρ_eb   = 1/2∑_j,k,x_i, z_iω(Ô) k_zP̂_z_1P̂_x_1P̂_x_4P̂_z_4j_z
       ×|-j_z⟩_e⟨-k_z|⊗P̂_z_3P̂_x_3|+_y⟩_b⟨+_y|P̂_x_2P̂_z_2

where we write ω(Ô) ≡⟨0|Ô|0|$⟩ and

    Ô   = W(-z_1Ef_1) W(-x_1Ef_2) W(x_2Ef_2) W(z_2Ef_1) 
       × W(-z_3Ef_1) W(-x_3Ef_2) W(x_4Ef_2) W(z_4Ef_1),

x_i stands for x_1, x_2, x_3, x_4, similarly for z_i, and all sums are taken over the set {±}. Provided that we can obtain an expression for ω(Ô), a computer algebra system can straightforwardly sum over the 2^10 terms in the sum. We start by redefining summation indices x_1 ↦ -x_1, z_1 ↦ -z_1, x_3 ↦ -x_3, and z_3 ↦ -z_3 such that  

    ρ_eb   = 1/2∑_j,k,x_i, z_iω(Ô) k_zP̂_-z_1P̂_-x_1P̂_x_4P̂_z_4j_z
       ×|-j_z⟩_e⟨-k_z|⊗P̂_-z_3P̂_-x_3|+_y⟩_b⟨+_y|P̂_x_2P̂_z_2

and Ô only contains terms without minus signs

    Ô   = W(z_1Ef_1) W(x_1Ef_2) W(x_2Ef_2) W(z_2Ef_1) 
       × W(z_3Ef_1) W(x_3Ef_2) W(x_4Ef_2) W(z_4Ef_1).

In <cit.>, 0Ô0 is evaluated using the Baker-Campbell-Hausdorff (BCH) formula and Wick's theorem. In this paper, we adopt a much simpler approach using the AQFT framework. It is straightforward (albeit somewhat tedious) to apply the third Weyl condition W(Ef) W(Eg) = e^- i/2E(f,g) W(E(f+g)) seven times to obtain

    Ô   = W(E((z_1+z_2+z_3+z_4)f_1+(x_1+x_2+x_3+x_4)f_2)) 
       × e^-i/2 (x_1+x_2)(z_1-z_2-z_3-z_4) E(f_1, f_2)
       × e^-i/2 (x_3+x_4)(z_1+z_2+z_3-z_4) E(f_1, f_2)

where only one Weyl operator remains multiplied by a phase factor. 

Before we evaluate the expectation value of the remaining Weyl operator, it will be convenient to consider the decomposition of a Wightman two-point function into its real (symmetric) and imaginary (anti-symmetric) parts, i.e.,

    𝖶(f,g)    = 1/2ω({ϕ̂(f),ϕ̂(g)})+1/2ω([ϕ̂(f),ϕ̂(g)])
       = 1/2𝖧(f,g) + i/2E(f,g) ,

where we used the CCR to obtain the second term and 𝖧(f,g) ω({ϕ̂(f),ϕ̂(g)}) ≡1/2μ(Ef,Eg) is the smeared Hadamard two-point distribution of the field (cf. Eq. (<ref>)). This decomposition is important because only the symmetric part 𝖧(f,g) is state-dependent, while E(f, g) is independent of the state. This is useful as it tells us which parts of our analysis depend on the field state. Furthermore, while we have assumed that the field starts in some vacuum state for simplicity, but the protocol in Section <ref> automatically extends to all quasi-free states such as thermal states <cit.>, squeezed vacuum state <cit.>, and with slight modifications to more general Gaussian states such as coherent states (see, e.g., <cit.>). 

We can now use the definition of quasifree state ω in Eq. (<ref>) for the vacuum state to evaluate the expectation value ω(Ô) in terms of Wightman and Hadamard functions giving us

    ω(Ô)    = e^-1/2 (z_1+z_2+z_3+z_4)^2 𝖶(f_1, f_1)-1/2 (x_1+x_2+x_3+x_4)^2 𝖶(f_2, f_2)
       × e^-1/2 (x_1+x_2+x_3+x_4)(z_1+z_2+z_3+z_4)𝖧(f_1, f_2)
       × e^-i/2 (x_1+x_2)(z_1-z_2-z_3-z_4) E(f_1, f_2)
       × e^-i/2 (x_3+x_4)(z_1+z_2+z_3-z_4) E(f_1, f_2) .

Note that this expression is valid automatically for any quasifree states such as squeezed vacuum state and (squeezed) thermal state <cit.>.  The density matrix in the ordered basis ρ_eb in the |+_z+_z⟩_be, |+_z-_z⟩_be, |-_z+_z⟩_be, |-_z-_z⟩_be now reads

    ρ_eb = (P_-     0     A     C
     
        0     P_+     X     B
    
        A^*     X^*     P_+     0
    
        C^*     B^*     0     P_-)

where

    P_±   = 1/4(1 ± e^-2 𝖶(f_1, f_1) sin(2E(f_1, f_2)) )
    
        X    = 1/4sin(2E(f_1, f_2)) ( e^- 2 𝖶(f_2, f_2) +  sin(2E(f_1, f_2)) )
    
        A    = - i/4 e^ -2 𝖶(f_1, f_1) cos(2E(f_1, f_2))
       ×( sin(2E(f_1, f_2)) -e^ -2W(f_2, f_2)sinh(4𝖶(f_2, f_1)) )
    
        B    = -i/4 e^ - 2 𝖶(f_1, f_1) cos(2E(f_1, f_2)) 
       ×(sin(2E(f_1, f_2)) + e^-2𝖶(f_2, f_2)cosh(4 𝖶(f_1, f_2)))
    
        C    = 1/4 e^ - 8 𝖶(f_1, f_1)sin(2E(f_1, f_2)) 
       ×( sin(2E(f_1, f_2)) - e^-2 𝖶(f_2, f_2)cosh(4𝖧(f_1, f_2)) ) .



Now that we have an analytic expression for ρ_eb, we can use our two conditions in Proposition <ref> and verify that the coherent information can be made arbitrarily close to 1. First we use the fine-tuning condition E(f_1, f_2) = π/4 which simplifies ρ_eb  considerably to

    ρ_eb = (P̃_-     0     0    C̃
     
        0    P̃_+    P̃_+     0
     
        0    P̃_+    P̃_+     0
    C̃    0    0    P̃_- )

where

    P̃_±   = 1/4(1 ± e^-2𝖶(f_2, f_2))
    C̃   = 1/4 e^-8 𝖶(f_1, f_1)(e^-2𝖶(f_2, f_2)cosh(4𝖧(f_1, f_2)) +1).


Next, we use the strong coupling condition E(f_1, f_2)^2 ≫𝖶(f_2, f_2). Recall that we are assuming for simplicity that on Alice's side, her interaction profile will be kept fixed except for the coupling strengths, thus only λ_1,λ_2 are adjustable parameters. In this case, the strong-coupling condition essentially means that λ_1≫λ_2 and hence we have the hierarchy of scales

        
    𝖶(f_1,f_1) ∼𝒪(λ^2_1)    ≫ E(f_1,f_2)^2∼𝒪(λ^2_1λ^2_2) 
    
            E(f_1,f_2)^2∼𝒪(λ^2_1λ^2_2)    ≫𝖶(f_2,f_2)∼𝒪(λ^2_2)
    𝖶(f_1,f_1)∼𝒪(λ^2_1)    ≫𝖧(f_1,f_2) ∼𝒪(λ_1λ_2)


With this hierarchy, we can express λ_1 in units of λ_2 so that  λ_1 cλ_2 for some c≥ 0, which is also convenient since in general λ_i has physical units[The unit of λ_i is [Length]^n-3/2, and we have assumed for simplicity that λ_i have the same units since f_i are chosen to be (cf. the end of Section <ref>)]. We can thus write


    𝖶(f_1,f_1)    = c^2λ_2^2 𝗐_1 ,  𝖶(f_2,f_2) = λ_2^2 𝗐_2 ,
    
        E(f_1,f_2)    = cλ^2_2𝖾 ,    𝖧(f_1,f_2) = cλ_2^2𝗁 ,


where 𝗐_1,𝗐_2,𝖾,𝗁 are fixed by the spacetime smearings. Proposition <ref> then translates to

    λ_2^2(c^2𝖾^2)≫𝗐_2 ,  λ_2^2(c𝖾) = π/4 2π .

Note that the fine-tuning condition can still be satisfied: for example, for a fixed coupling λ_2 we find c such that 

    λ_2^2(c𝖾) = π/4 2π ,

and we can always ensure that c is sufficiently large so that

    λ^2_2(c^2𝖾^2) = c·𝖾π/4+2π n≫𝗐_2

in order to satisfy both conditions in (<ref>). 

In terms of the rescaled coupling constants, the strong-coupling condition implies that for fixed λ_2 and c≫ 1, we get

    ρ_eb(P̃_-     0     0     0
     
        0    P̃_+    P̃_+     0
     
        0    P̃_+    P̃_+     0
    
        0     0    0    P̃_- ) .

Using negativity 𝒩 as an entanglement measure <cit.>, 

    𝒩[ρ_eb] = ||ρ_eb^Γ_e||_1-1/2 = 1/2e^-2λ_2^2𝗐_2

where ρ_eb^Γ_e is the partial transpose of ρ_eb and ||·||_1 is the trace norm. This means that ρ_eb is always an entangled state for any finite coupling λ_2≪λ_1 and the channel is able to preserve some quantum entanglement. 

Finally, we obtain the perfect quantum channel in the idealized limit where c→∞ and λ_2→ 0 such that λ_1 is finite and still satisfies (<ref>), i.e., 

    ρ_eb = 1/2[ 0 0 0 0; 0 1 1 0; 0 1 1 0; 0 0 0 0 ] ,

i.e., a Bell state with maximum negativity 𝒩[ρ_eb] = 1/2. Therefore, in this limit the coherent information I_c(Φ)→ 1, which corresponds to a perfect quantum channel that can transmit a maximally entangled state perfectly. 




 §.§ Alice and Bob are spacelike separated



The preceding analysis shows that the ability to implement the approximate `SWAP' gate depends on how closely one is able to satisfy the fine-tuning and strong-coupling conditions in Proposition <ref>. While this is straightforward for Alice, Proposition <ref> tells us that for Bob to emulate ϕ̂(f_i) defined at some (global coordinate) time t(τ_a),  Bob's spacetime smearing functions g_i at time t(τ_b) has to be somewhat delocalized over some region of spacetime to catch all the signals from Alice. Since the quantum channel Φ is built out of a relativistic quantum field, we ought to be able to show that there can be no transmission of both classical and quantum information through Φ.

Formally, we start from the CCR which tells us that 

    E(f_i,g_j) = 0    ∀ i,j=1,2

when Alice and Bob are spacelike-separated, i.e., (g_i)∩𝒥^±((f_i)) = ∅. Note that the Hadamard function 𝖧(f_i,g_j) is generically nonzero for spacelike-separated regions due to entanglement contained in the field state <cit.>. The calculations proceeds similarly as in the previous section, except that Bob will have his own arbitrary smearing functions and their respective UDW interaction unitaries are



    Û_A   = ∑_x,z ∈{±}P̂_x P̂_z ⊗ W(xEf_2) W(zEf_1)
    Û_b   = ∑_x,z ∈{±}P̂_z P̂_x ⊗ W(-zEg_1) W(-xEg_2).


The condition for spacelike separation in Eq. (<ref>) will be enforced later in the calculation once everything is written in terms of two-point functions. 

We can analyse this scenario very generally by allowing Alice and the environment to start in any arbitrary state ρ_ea instead of a maximally entangled state or purifications of any initial state ρ_a,0. The expression for the density matrix ρ_eb now reads


    ρ_eb   = ∑_x_i, z_iω(Ô)  _a[ P̂_x_4P̂_z_4ρ_aeP̂_-z_1P̂_-x_1 ]
       ⊗P̂_-z_3P̂_-x_3|+_y⟩⟨+_y|_bP̂_x_2P̂_z_2

where the projection operators act on Alice or Bob and Ô is now defined to be

    Ô   = W(z_1Ef_1) W(x_1Ef_2) W(x_2Eg_2) W(z_2Eg_1)  
       × W(z_3Eg_1) W(x_3Eg_2) W(x_4Ef_2) W(z_4Ef_1).

Applying the third Weyl relation and using Eq. (<ref>), this evaluates to 

    
    ω(Ô) 
           = 
        e^-i/2 (z_1 - z_4)(x_1 + x_4) E(f_1, f_2)  
        e^-i/2 (z_2 + z_3)(x_3 - x_2) E(g_1, g_2)
        e^-i/2 (z_2 + z_3)(z_1 - z_4) E(f_1, g_1)
       × 
        e^-i/2 (z_1 - z_4)(x_2 + x_3) E(f_1, g_2)
        e^-i/2 (z_2 + z_3)(x_1 - x_4) E(f_2, g_1)
        e^-i/2 (x_2 + x_3)(x_1 - x_4) E(f_2, g_2)
       ×  e^-1/2(
          (z_1+z_4) (z_1+z_4)  𝖶(f_1, f_1)
        + (x_1+x_4) (x_1+x_4)  𝖶(f_2, f_2)
        + (z_2+z_3) (z_2+z_3)  𝖶(g_1, g_1)
        + (x_2+x_3) (x_2+x_3)  𝖶(g_2, g_2) )
       × e^-1/2(
          (x_1+x_4) (z_1+z_4)  𝖧(f_1, f_2)
        + (z_2+z_3) (z_1+z_4)  𝖧(f_1, g_1)
        + (x_2+x_3) (z_1+z_4)  𝖧(f_1, g_2) )
       × e^-1/2(
          (z_2+z_3) (x_1+x_4)  𝖧(f_2, g_1)
        + (x_2+x_3) (x_1+x_4)  𝖧(f_2, g_2)
        + (x_2+x_3) (z_2+z_3)  𝖧(g_1, g_2) ) .

This is the most general form that ω(Ô) can take in terms of Alice and Bob's two-point functions. We now enforce spacelike-separation using Eq. (<ref>), giving us

    ω(Ô) 
           = 
        e^-i/2 (z_1 - z_4)(x_1 + x_4) E(f_1, f_2)  
        e^-i/2 (z_2 + z_3)(x_3 - x_2) E(g_1, g_2)
       ×  e^-1/2(
          (z_1+z_4) (z_1+z_4)  𝖶(f_1, f_1)
        + (x_1+x_4) (x_1+x_4)  𝖶(f_2, f_2)
        + (z_2+z_3) (z_2+z_3)  𝖶(g_1, g_1)
        + (x_2+x_3) (x_2+x_3)  𝖶(g_2, g_2) )
       × e^-1/2(
          (x_1+x_4) (z_1+z_4)  𝖧(f_1, f_2)
        + (z_2+z_3) (z_1+z_4)  𝖧(f_1, g_1)
        + (x_2+x_3) (z_1+z_4)  𝖧(f_1, g_2) )
       × e^-1/2(
          (z_2+z_3) (x_1+x_4)  𝖧(f_2, g_1)
        + (x_2+x_3) (x_1+x_4)  𝖧(f_2, g_2)
        + (x_2+x_3) (z_2+z_3)  𝖧(g_1, g_2) ) .


The density matrix ρ_eb can now be calculated 

    ρ_eb = ρ_e⊗(1/2    X
     X^*    1/2)

with ρ_e = _a [ρ_ea] and 

    X    = -i/2 e^-2 𝖶(g_1, g_1)
       [e^-2 𝖶(g_2, g_2)cosh(2𝖧(g_1, g_2)) + sin(2E(g_1, g_2))].


Observe that the density matrix ρ_eb depends only on the initial condition of the environment and Bob's local two point functions, i.e. not involving Alice's smearing functions at all, and is clearly separable across B and E. 
Consequently, the quantum channel preserves no entanglement at all from the joint system EA (viewing it as ⊗Φ). In fact, since we start from any arbitrary state ρ_ea, we have effectively shown that Φ is a PPT channel <cit.>, and hence also an entanglement-breaking channel (because PPT is equivalent to separability for two qubits), and these are known to fall under the class of anti-degradable channels with zero quantum capacity 𝒬[Φ] = 0  <cit.>. 

From an information-theoretic viewpoint, zero quantum capacity says nothing about the ability of the quantum channel to transmit classical information, since the classical channel capacity 𝒞[Φ] is an upper bound for quantum capacity, i.e., 𝒞[Φ] ≥𝒬[Φ]. Quantum channels can also transmit classical bits by, for instance, considering only diagonal density matrix

    ρ_a,0 = (α    0
     0     1-α) ,  α∈ [0,1] .

and indeed it has been shown that for certain delta-coupled UDW models,  the classical capacity 𝒞[Φ] can be computed exactly and is zero if Alice and Bob are spacelike <cit.>. Our quantum channel is more complicated as Alice and Bob each perform a sequence of two simple-generated unitaries, but it is not necessary for us to find the explicit formula for 𝒞[Φ] to show that it is zero. This follows directly from the fact that the density matrix in Eq. (<ref>) does not depend on ρ_a,0 at all, therefore the quantum channel carries no information whatsoever about Alice's initial state, hence 𝒞[Φ] = 0. 

One of the main features of a relativistic quantum channel is precisely that one should be able to show that it respects the causal structure of spacetime by construction, and this requires one to be able to show, for instance, that no information transmission is allowed in spacelike separation. In fact we can say more: our formalism does not require us to specify anything about the (violation of) strong Huygens' principle <cit.>, since everything is encoded in the causal propagator. For example, in (3+1)-dimensional Minkowski spacetime, Bob can be timelike-separated from Alice but for a massless scalar field, the causal propagator has zero support in the interior of the light cone (see, e.g., <cit.>). In this case, if Bob also coupling functions to the field  g_i are strictly in the timelike interior and  (g_i)∩𝒥^+( f_j) = ∅ for any i,j=1,2, then 𝒬[Φ]=𝒞[Φ]=0.




 §.§ Intermediate scenarios


In the two preceding subsections, we have analyzed two opposite regimes: the ideal case with Ef_i = Eg_i and the worst case with Alice and Bob spacelike-separated. Let us make some comments on intermediate regimes. 

When the coherent information of a quantum channel is not close to 1, e.g., when 0 ≤ I_c(Φ) ≪ 1, it is in general very difficult to make quantitative claims about the quantum channel capacity for the reasons like superadditivity described in Section <ref>. In fact, even for quantum channels with I_c(Φ) = 0, only small subsets of these are known to have zero quantum capacity, namely the anti-degradable channels <cit.> and the PPT channels <cit.> (see also <cit.>). For intermediate scenarios, such as when Bob is only partially covering  𝒥^±((f_i)) or implementing faulty SWAP gates we are not able to give generic quantitative statements about 𝒬[Φ], as is the case for generic quantum communication channels. 

However, in some situations we may try to use no-cloning type argument to conclude that 𝒬[Φ]=0 in some scenarios as was done in <cit.> (see also <cit.>). Suppose Bob is allowed to couple only to at most half of Alice's signal at some future time t that defines the Cauchy slice for Bob's interaction. Consider another observer Charlie who now chooses to couple to the other half of the region at the same time slice so that jointly the interaction support is the full region needed for the perfect quantum channel. Since Charlie's interaction is by construction spacelike separated from Bob, Charlie's interaction is invisible to Bob (and vice versa). If Bob and Charlie were able to individually decode Alice's qubit, this would violate the no-cloning theorem, so we can conclude that the quantum capacity must be zero if the UDW interactions are set up this way[It is worth emphasizing that the protocol we have (and also in <cit.>) are generically not ideal, since for any finite interaction strengths the `SWAP' gates are never perfect and the `logical qubit' in the field is only approximate — consequently one has to be careful about using exact cloning argument. In practice, it would be better to think of the scenario in terms of approximate cloning. Indeed, in Figure 8 of <cit.> the coherent information vanishes smoothly as one varies Bob's interaction support when analyzing the cloning-type argument.]. 

The above discussion is not necessarily the only (or natural) way to view this. One way to think about this from the perspective of the quantum field is this: if Bob manages to decode Alice's state at time t, the `logical qubit' state c_1|+α⟩-ic_2|-α⟩ in Eq. (<ref>) would have been taken out by Bob, and the field would now change into a new state that has no logical qubit in it (essentially because Bob swapped out the state into his physical detector). Charlie, who couples to the field at any time t+ϵ for any arbitrary ϵ>0, would no longer have any logical qubit to extract from the field, so Charlie cannot extract another qubit for free. Yet another way to see this is that Bob's interaction is designed to implement the inverse unitary Û_a^-1, and this requires one to find g_i in 𝒥^+((f_j)) that can undo Alice's operation. Therefore, if we choose to pick g_i that does not cover the full region needed for the decoding, then Bob fails to implement Û_a^-1 and instead performs a different unitary operation. In this interpretation, we would say that Bob is unable to perform the required `SWAP' gate to decode Alice's state. Showing that Bob has zero quantum capacity would mean showing that Bob's operations somehow completely depolarize the qubit state and lose all the quantum information in it — we will not pursue this further in this paper and leave this for future work.




§ DISCUSSION AND OUTLOOK



We motivated our work through the question: What is the information-carrying capacity of quantum fields? The UDW framework allows us to formulate a precise statement of this question, in terms of quantum capacity of the relativistic quantum channel between Alice and Bob who try to transmit quantum information between their qubit detectors through the quantum field.  In this work, we have successfully generalized the results of <cit.>, in particular <cit.>, where we construct a quantum channel that can transmit quantum information arbitrarily well through a relativistic quantum scalar field in arbitrary curved spacetimes. 

   

The features of our construction is that the resulting quantum channel, and hence the quantum capacity, are manifestly diffeomorphism-invariant, respect the causal structure of spacetime, and are independent of the details of the background geometry, topology, and the choice of Hilbert space (quasifree) representations of the CCR algebra. Furthermore, in this generalization we no longer need to deal with (violation of) the strong Huygens' principle that may occur in curved spacetimes for non-conformally invariant fields (see, e.g.,<cit.>), since this is fully encoded in the causal propagator. We also showed explicitly that the quantum and classical capacity is zero for spacelike separated observers, as we expect from quantum communication channel built from a relativistic quantum field.

Crucially, in this generalized formulation it becomes clear that there are two essential features of a near-perfect quantum channel that can transmit quantum information arbitrarily well: (1) Alice and Bob's interactions need to satisfy the so-called fine-tuning and strong coupling conditions in Proposition <ref> in order to approximate an ideal `SWAP' gates; (2) Bob needs to couple to the field at appropriate spacetime regions, fixed by Proposition <ref>, so that he could execute the approximately ideal `SWAP' gates. Indeed, where Bob should be is fixed by the causal propagator that determines where Alice's information propagates in spacetime. The asymmetry between Alice and Bob can be viewed as the consequence of the asymmetry between encoding the qubit state into the field, with the logical basis being the coherent states |±α⟩, and decoding the state while taking into account relativistic causality. In this sense, our construction constitutes the most general relativistic qubit channel within the UDW framework that can transmit quantum information arbitrarily well in appropriate regimes and we also provided analyses when the quantum channel is sub-optimal.


There are several future directions that merit investigation, at least from a more fundamental standpoint:

    
  * Is it possible to construct a perfect relativistic quantum channel using higher-dimensional detector models (using qudit detector models <cit.> rather than qubits)? A priori it is not obvious how to construct logical qudits in the field using only controlled displacement operators and what kind of interaction Hamiltonians are needed to do this, if it exists at all. It is also natural to consider several emitters and receivers, exploiting the fact that one can interpret displacement operations can be viewed as Weyl generators with respect to some spacetime smearing functions <cit.>.

    
  * As mentioned in <cit.>, one may be interested in more directional coupling, which can be envisaged in (1+1)D models (see, e.g., <cit.>). Our protocol works automatically simply because only the two-point functions change to reflect that the left- and right- movers of the field modes are independent. A more interesting setup would be to study how realistic settings involving waveguides and non-trivial dispersion relations can be adapted for this purpose. 

    
  * Is the information-carrying capacity different between different fields with different spins and statistics? For example, how does one compare the scalar field with a fermionic field? The issue one has to deal with is that we have to separate the effects coming from the model dependence of the coupling (e.g., we cannot couple σ̂_x⊗ψ̂ where ψ is the fermionic field) from truly field-theoretic physics that contribute to the quantum capacity, if we wish to do it in the same way as we have done in this paper (see, e.g., some proposed fermionic couplings in RQI literature <cit.>).

    
  * Our construction relies on the fact that the quantum field is non-interacting (Gaussian). There are very few studies involving the UDW model with interacting QFTs such as the ϕ^4 theory, even perturbatively and for a single qubit (see, e.g., <cit.> for one example). The effects of interactions are unclear for (long-distance) quantum communication, both perturbatively and non-perturbatively. A possibly relevant testbed would be to consider known rigorous constructions in (1+1)D interacting QFTs (see, e.g., <cit.>). 

    
  * Is it possible to systematically study quantum capacity in the non-relativistic quantum many-body settings? Somewhat related studies to our setup are, for example, <cit.>. It would be interesting to see the similarities and differences when they are compared with setups involving relativistic quantum fields.

We leave these problems open for the future.




§ ACKNOWLEDGMENT



M. K. acknowledges support from the Princeton International Internship Program (IIP). E. T. acknowledges funding from the Munich Center for Quantum Science and Technology (MCQST), funded by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy (EXC2111 - 390814868). 


apsrev4-2


 Paper Title: Estimating quantum Markov chains using coherent absorber post-processing and pattern counting estimator.










School of Mathematical Sciences, University of Nottingham, United Kingdom
Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems,
University of Nottingham, Nottingham, NG7 2RD, UK
Department of Mathematics, Polytechnic University of Milan, Milan, Piazza L. da Vinci 32, 20133, Italy



School of Mathematical Sciences, University of Nottingham, United Kingdom
Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems,
University of Nottingham, Nottingham, NG7 2RD, UK





School of Mathematical Sciences, University of Nottingham, United Kingdom
Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems,
University of Nottingham, Nottingham, NG7 2RD, UK

———————————————————————————————————–
§ ABSTRACT 
We propose a two step strategy for 
estimating one-dimensional dynamical parameters of a quantum Markov chain, which involves quantum post-processing the output using a coherent quantum absorber and a “pattern counting” estimator computed as a simple additive functional of the  outcomes trajectory produced by sequential, identical measurements on the output units. We provide strong theoretical and numerical evidence that the estimator achieves the quantum Cramé-Rao bound in the limit of large output size.




Our estimation method is underpinned by an asymptotic theory of translationally invariant modes (TIMs) built as averages of shifted tensor products of output operators, labelled by binary patterns. For large times, the TIMs form a bosonic algebra and the output state approaches a joint coherent state of the TIMs whose amplitude depends linearly on the mismatch between system and absorber parameters. Moreover, in the asymptotic regime the TIMs capture the full quantum Fisher information of the output state. While directly probing the TIMs' quadratures seems impractical, we show that the standard sequential measurement is an effective joint measurement of all the TIMs number operators; 
indeed, we show that counts of different binary patterns extracted from the measurement trajectory have the expected joint Poisson distribution. Together with the displaced-null methodology of <cit.> this provides a computationally efficient estimator which only depends on the total number of patterns. This opens the way for similar estimation strategies in continuous-time dynamics, expanding the results of <cit.>.  




 


   





Estimating quantum Markov chains using coherent absorber post-processing and pattern counting estimators
    Mădălin Guţă
    
=========================================================================================================




§ INTRODUCTION



Quantum statistical inference <cit.> provides the mathematical framework for enhanced metrology 
<cit.>, imaging <cit.>,  waveform 

and noise 
estimation <cit.>, 
and quantum sensing applications <cit.> including  time keeping <cit.>, magnetometry <cit.>, biomedical sensing <cit.>, 
thermometry <cit.>, gravitational wave detection <cit.>.







The cornerstone of quantum estimation is the quantum Cramér-Rao bound (QCRB)  <cit.> 
which places a fundamental restriction on the precision in estimating unknown  parameters of a quantum state. 

For one-dimensional parameters the bound is attainable in the limit of many copies, by measuring a specific observable called the symmetric logarithmic derivative, which is the quantum analogue of the classical score function. However, when dealing with complex models involving correlated states of many-body systems,

the optimal measurement may be hard to compute and implement in practice, and in the case of pure states (and more generally, rank deficient states) such a measurement is highly non-unique. Therefore, it is particularly important 
to devise realistic measurement schemes which allow the estimation of unknown parameters with close to optimal precision, by means of computationally efficient estimators. 


In this paper we provide a general measurement and data processing protocol for estimating an arbitrary  dynamical parameter of a quantum Markov chain (QMC), which satisfies the above criteria. A QMC is a discrete time model of an open quantum system, in which the system interacts successively with a sequence of identically prepared “noise units” representing the environment, cf. Figure <ref> a). The setup is similar to Haroche's photon-box one-atom maser <cit.> and to that used in quantum collision models <cit.>, and provides a physical mechanism for generating versatile many-body states such  as matrix product states <cit.> and finitely correlated states <cit.>. By discretising time, QMCs can  be used to model continuous-time dynamics of a Markovian open system coupled with Bosonic input-output channels <cit.>.
   




For clarity, it is useful to distinguish between two mainstream approaches to parameter estimation in quantum open systems. In the setting of <cit.>, the quantum system undergoes a noisy evolution depending on an unknown parameter, and the experimenter tries to extract information about the parameter by repeatedly applying instantaneous direct measurements and control operations while the system is evolving. In contrast, the setting adopted in this paper is commonly used in quantum optics and input-output theory 
<cit.> where the experimenter does not have direct access to the system but can measure the output field of an environment channel coupled to the system. This allows the experimenter to track the conditional state of the system by means of stochastic filtering equations <cit.> and control it using feedback.
As these techniques require full knowledge of the system's dynamical parameters, it is important to devise tools for estimating such parameters from the stochastic trajectory of the measurement record. Since the early works <cit.>, many aspects of continuous-time estimation have been investigated, including adaptive estimation <cit.> filtering methods <cit.>, Heisenberg scaling <cit.>, sensing with error correction <cit.>, Bayesian estimation <cit.>, quantum smoothing <cit.>, 
estimation of linear systems <cit.>, central limit and large deviations theory for trajectories <cit.>, concentration bounds for time averaged observables <cit.>, estimation with feedback control <cit.>.


However, a major problem in this area has been that standard  measurement protocols such as counting and homodyne do not achieve the ultimate precision limit prescribed by the QCRB in terms of the quantum Fisher information (QFI) of the output state <cit.>.


Two recent papers <cit.> have addressed this problem by introducing the idea of quantum post-processing of the output state using a quantum coherent absorber <cit.>. For a given QMC reference dynamics, the absorber takes the system's output as its own input and is characterised by the property that it “reverts” the action of the system, so that its output is a trivial product state (vacuum in continuous-time dynamics), cf. Figure <ref> b). In the statistical estimation framework, the absorber is set to a particular reference parameter of the QMC dynamics, so that a small deviation of the true parameter from this value will lead to non-trivial output statistics which can be used to estimate the deviation as illustrated in Figure <ref> c). In <cit.>, two of the present authors proved that the QCRB can be achieved by performing sequential, adaptive measurements on the output units, after the interaction with the coherent absorber. In addition, the adaptive measurement can be implemented efficiently in a Markovian fashion. The work <cit.> deals with the same problem but in the continuous-time setting, and proposes to perform a standard counting measurement (instead of an adaptive one) in conjunction with post-processing using the coherent absorber. In both papers the final estimator was computed from the measurement trajectory using the maximum likelihood method, which becomes inefficient for long trajectories. 



The strategy proposed here is similar to that of <cit.>, albeit in discrete rather than continuous time, but strengthens it in several important aspects. Firstly, we employ the technique of displaced-null measurements <cit.> to 
provide a precise recipe for choosing the coherent absorber parameter. As we explain below, this is an important technical detail, as the intuitive choice of parameters fails to achieve the QCRB in the limit of long trajectories. Secondly, our strategy employs a two step adaptive procedure which allows us to compute the final estimator as a simple linear transformation of the total number of “pattern counts” which can be easily extracted from the measurement trajectory, cf. Figure <ref> d). This circumvents the computational issues associated with the maximum likelihood estimator. Thirdly, we provide strong theoretical evidence that the final estimator achieves the QCRB in the limit of large times.

This is based on a novel representation of the output in terms of translationally invariant modes which are shown to satisfy the quantum local asymptotic normality property <cit.>. 



We now give a brief summary of the two steps estimation strategy proposed here and the related mathematical results. We consider a QMS whose dynamics depends on a one-dimensional parameter 
θ which we aim to estimate by measuring the output state produced after n time steps. In the first stage we run the QMC dynamics with the unknown parameter θ for 
1≪ñ≪ n time steps and measure the noise units in a fixed basis, cf. Figure <ref> a). From the total counts statistics we construct a rough estimator θ̃_n by matching the empirical frequency to its expected value. In general this estimator is not optimal but its mean square error has the standard 1/ñ scaling (<cit.>). In the second stage we run the system and absorber QMC for the remaining n-ñ time steps and measure the output in the standard basis. If the absorber parameter matched the true system parameter θ, this measurement would produce a string of 0s (corresponding to no counts in continuous-time) cf. Figure <ref> b). Therefore, it would seem natural to choose the absorber parameter to be θ̃_n, our best guess at the unknown parameter θ. However, this choice is unsuitable since for small deviations Δ_n = θ-θ̃_n, the counting statistics depends quadratically on Δ_n, which prevents the estimation of θ at standard 1/n rate. This non-identifiability issue is explained in detail in <cit.>, which also provides the solution to this problem. We deliberately set the absorber parameter at θ_ abs= θ̃_n -δ_n, which is away from the best guess by a small “displacement” δ_n↓ 0 chosen to be larger than the uncertainty |Δ_n|. This allows us to unambiguously identify θ from counts statistics. Stage two of the estimation procedure is illustrated in Figure <ref> c).


We turn now to the question of estimating θ from the counts trajectory ω= (ω_1,ω_2,…, ω_n) of the second stage measurement (setting n-ñ to n for simplicity). Since θ-θ_ abs is vanishingly small (even with the extra displacement), ω will typically contain a small number of 1s while most of the outcomes will be 0, cf. Figure <ref> d). This allows us split the trajectory into long sequences of 0s and in between them, binary “excitation patterns” starting and ending with a 1. For each pattern 
α (e.g. 1, 11, 101 etc.) we count the number of occurrences N_α,n. In Theorem <ref> we show that in the limit of large n the counts N_α,n become independent Poisson variables whose intensities are λ_α u^2 where u= √(n)(θ-θ̃_ abs) is the “local parameter” and λ_α is a model dependent coefficient. Moreover the total Fisher information of the Poisson variables is equal to the output QFI, which shows that the pattern counts statistics capture the full information of the output state. Using this asymptotic behaviour, we construct a simple estimator θ̂_n (cf. equations (<ref>) and (<ref>)) which is linear in the total pattern count, and we argue why it should achieve the QCRB in the limit of large n. To summarise, the two step procedure provides a computationally and statistically efficient estimation method which involves only standard basis measurements and a minimal amount of “quantum post-processing” implemented by the coherent absorber.

For a more in-depth understanding of why the  excitation pattern counts have asymptotically Poisson distributions, we refer to sections <ref> and <ref> where we develop a theory of translationally invariant modes (TIMs) of the output. These modes turn out to capture all statistical information about the unknown parameter, and can be measured simultaneously and optimally by performing the sequential standard output measurement. For each excitation pattern α=(α_1,… ,α_k)∈{0,1}^k we define the creation operator A^*_α(n) on the output chain of length n. This consists of a running average 

    A^*_α(n)=1/√(n)∑_i=1^n-k+1σ^α_i

where σ^α_i is the tensor product of the 
the type 
σ^α = σ^α_1⊗…⊗σ^α_k where σ^0= 1 and σ^1 = σ^+ = |1⟩⟨ 0|, with first tensor acting on position i of the output chain. In Proposition <ref> and Corollary  <ref> we show that asymptotically with n, by applying the creation operators to the reference (vacuum) state |0⟩^⊗ n, we obtain Fock-type states with different excitation pattern numbers. The creation and annihilation operators satisfy the Bosonic commutation relations with each excitation pattern being an independent mode. For large n, a Fock state is a superposition of basis states consisting of long sequences of 0s interspersed with the corresponding patterns appearing in any possible order. One of our key results, Theorem <ref> shows that when the gap between system and absorber parameters scales as θ - θ_ abs=u/√(n), the quadratures of the excitation pattern modes satisfy the  Central Limit Theorem and the corresponding joint state is a product of coherent states whose amplitudes are linear in the local parameter u. The total QFI of this multimode coherent state is equal to the output QFI, showing that the TIMs contain all statistical information about the dynamics. We also prove separately, that the number operators of the TIMs have asymptotic Poisson distributions, as expected for a coherent state. Together with the result of Theorem <ref>, this completes a circle of ideas, which played a crucial role in formulating our estimation strategy. In a nutshell, when looking at the output from the perspective of the TIMs, one deals with a simple Gaussian estimation problem. Using the displaced-null method, we can achieve the QCRB by measuring the number operators of the TIMs and such a measurement can be implemented by simple sequential counting measurements followed by the extraction of pattern counts from the measurement trajectory ω.



 

The paper is organised as follows. In section <ref> we give a brief review of quantum estimation theory and the displaced-null measurement technique developed in <cit.>. In section <ref> we introduce the notion of QMC and the estimation problem, together with the idea of quantum post-processing using a coherent absorber. In section <ref> we define the translationally invariant modes of the output and establish their Fock space properties. In section <ref> we show that the restriction of the output state to the TIMs is a coherent state whose amplitude is linear in the local parameter and whose QFI is equal to the output QFI (cf. Theorem <ref> and Corollary <ref>). 
In section <ref> we establish that the excitation pattern counts obtained from the sequential output measurement have asymptotically Poisson distribution (cf. Theorem <ref>). In section <ref> we formulate our measurement and 
estimation strategy and define the "pattern counts" estimator. Finally in section <ref> we present results of a simulation study confirming the earlier theoretical results. 










§ QUANTUM ESTIMATION AND THE DISPLACED NULL MEASUREMENT TECHNIQUE




In this section we give a brief overview  of the quantum parameter estimation theory <cit.> used in this paper, with an emphasis on asymptotic theory and the displaced-null measurement technique developed in our previous work <cit.>. In particular, we explain  why this method is asymptotically optimal, by employing the Gaussian approximation technique called local asymptotic normality 
<cit.>. Later on, this picture will guide our intuition when dealing with the Markov estimation problem. For our purposes it suffices to discuss the case of one-dimensional parameters, and we refer to <cit.> for the multi-dimensional setting.

Let ρ_θ∈ M(ℂ^d) be a family of quantum states depending smoothly on a one-dimensional parameter 
θ. Consider a measurement described by a positive operator valued measure 
{M_1,…, M_k} and let X be the measurement outcome with probability distribution p_θ(X=i) =  Tr(ρ_θ M_i). The quantum Cramér-Rao bound (QCRB) <cit.> states that the variance of any unbiased estimator θ̂ =θ̂(X) is lower bounded as

    Var(θ̂) =𝔼_θ(θ̂-θ)^2 ≥ 
    F_θ^-1

where F_θ is the quantum Fisher information (QFI) defined as 
F_θ =  Tr (ρ_θℒ_θ^2) with ℒ_θ the symmetric logarithmic derivative (SLD) operator which satisfies 

    dρ_θ/dθ =1/2(ℒ_θρ_θ + ρ_θℒ_θ).

In general, the QCRB is not achievable when only a single copy of ρ_θ is available. However, the bound is attainable in the asymptotic limit of large number of samples by the following two step adaptive procedure <cit.>. Given n copies of ρ_θ one can use a small proportion of the samples (e.g. ñ = n^1-ϵ for a small ϵ >0) to compute a preliminary (non-optimal) estimator 
θ̃_n of θ; reasonable estimators will concentrate around θ such that 
|θ̃_n-θ|=O (n^-1/2+ϵ) with high probability, which will be assumed throughout. In the second step, one measures the SLD operator 
ℒ_θ̃_n on each of the remaining copies. If X_1,… , X_n^' are the outcomes of these measurements (with n^' = n-ñ) then the estimator

    θ̂_n:= θ̃_n+ 1/F_θ̃_nn^'(∑_i=1^n^' X_i)

is asymptotically optimal in the sense that 

    n𝔼_θ(θ̂_n -θ)^2 → F_θ^-1
 
in the limit of  large n and in addition θ̂_n is asymptotically normal, i.e. 
√(n)(θ̂_n -θ) converges in distribution to the normal N(0, F_θ^-1). 

However, for certain models including that considered in this paper,  measuring the SLD may not be feasible experimentally. Instead, we will  use a different method called displaced-null measurement <cit.>, which aims to estimate the parameter of a pure state models 
ρ_θ = |ψ_θ⟩⟨ψ_θ| 
by measuring each copy in a basis that contains the vector |ψ_θ̃⟩ with θ̃ close to the true parameter 
θ. The Fisher information of such a measurement is known to converge to the QFI as θ̃ approaches θ <cit.>. This suggests that the QCRB can be achieved asymptotically by using a two-step strategy similar to the SLD case: one first obtains a preliminary estimator θ̃_n and then measures each copy in a basis containing the vector |ψ_θ̃_n⟩. However, it turns out that this “null measurement" strategy fails due to the fact that for small deviations from θ̃_n, the outcome probabilities depend on (θ-θ̃_n)^2 and one cannot distinguish between left and right deviations from θ̃_n, cf. <cit.> for the precise mathematical statement. This non-identifiability issue can be sidestepped by deliberately changing the reference parameter from θ̃_n to 
θ_0:=θ̃_n - δ_n where δ_n =n^-1/2+3ϵ, so that θ= θ_0 +(u+τ_n)/√(n) with 
τ_n=n^3ϵ. The choice of τ_n is not unique but we refer to <cit.> for the general requirements. Since 
|θ -θ̃_n| =O(n^-1/2+ϵ) ≪δ_n, it means that 
θ lies on the right side of θ_0 and can be unambiguously identified from the outcomes of a measurement in a basis {|e_0⟩, … , |e_d-1⟩} such that 
|e_0⟩≡ 
|ψ_θ_0⟩. In addition, as θ_0 approaches θ in the limit of large n, the displaced-null measurement exhibits the optimality properties of the “null measurements" without sharing their non-identifiability issues. 


Let X_1,… , X_n'∈{0, 1,… , d-1} 
be the independent outcomes of basis {|e_0⟩, … , |e_d-1⟩} measurements performed on the remaining n^' = n-ñ systems, and let N_j,n denote the counts of the outcome j=0,… d-1. The displaced-null estimator based on the two-stage measurement strategy is defined as follows

    θ̂_n := θ̃_n +û_n/√(n)

with local parameter estimator

    û_n= 
    
    2/τ_n f∑_j=1^d-1N_j,n - τ_n/2

where f= 4 ψ̇_θ̃_n^2 is the QFI at θ̃_n. The estimator θ̂_n is asymptotically optimal in the sense of equation (<ref>) and asymptotically normal.



In appendix <ref> we give more insight into this method by analysing its  properties using the theory of local asymptotic normality <cit.>. In a nutshell, for large n the original model becomes equivalent to a Gaussian one consisting of a multi-mode coherent state whose amplitude depends linearly on the parameter, while the SLD and displaced-null strategies translated into measuring a quadrature and respectively the modes number operators of the shifted state. While this i.i.d. setup is different from the Markovian one studied in this paper, the overall asymptotic picture is similar and reader may find the i.i.d. case useful in guiding the intuition.    








§ QUANTUM MARKOV CHAINS AND POST-PROCESSING  USING COHERENT ABSORBERS







We start this section by reviewing the problem of  estimating dynamical parameters of quantum Markov chains (QMC). We then introduce the notion of quantum coherent absorber, which will play a key role in designing an optimal sequential measurement strategy.


A quantum Markov chain consists of a system interacting successively with a chain of independent "noise units" (the input) modelling the environment. In this paper the system's space is taken to be ℂ^d while the "noise units" are two dimensional systems prepared in the state |0⟩ where {|0⟩, |1⟩} is the standard basis in ℂ^2. We expect that the theory developed here works for general finite dimensional inputs, but we restrict here to this minimal setup which can be used to represent a discretised version of a continuous-time Markovian model with a single Bosonic field <cit.>.

At each time step  the system interacts with the input unit via a unitary
U on ℂ^d⊗ℂ^2. If the system is initially prepared in a state |φ⟩, the joint state of system and noise units (output) after n times steps is 

    |Ψ_n⟩   =    U_n |φ⊗ 0^⊗ n⟩
       =    
    U^(n)·…· U^(2)· U^(1)|φ⊗ 0^⊗ n⟩∈ℂ^d⊗(ℂ^2)^⊗ n

where U^(i) is the unitary acting on the system and the i-th noise unit. By expanding the state (<ref>) with respect to the standard product basis in the output we have

    |Ψ_n⟩ = 
    ∑_i_1,… , i_n ∈{0,1} 
    K_i_n… K_i_1 |φ⟩⊗ |i_1⟩⊗…⊗ |i_n⟩

where K_i= ⟨ i|U|0⟩ are Kraus operators acting on ℂ^d.



From equation (<ref>) it follows that the reduced system state of the system at time n is given by 

    ρ^ sys_n := 
    
     Tr_ out
    (|Ψ_n⟩⟨Ψ_n|) = 
    𝒯_*^n(ρ^ sys_ in),   ρ^ sys_ in = |φ⟩⟨φ|,

where the partial trace is taken over the output noise units, and 
𝒯_*:M(ℂ^d) → M(ℂ^d) 
is the Markov transition operator 
(Schrödinger picture)

    𝒯_*: ρ↦∑_i∈{0,1} 
    K_i ρ K_i^*

whose dual (Heisenberg picture) will be denoted by 𝒯.  

On the other hand, the reduced state of the output is 

    ρ^ out_n    :=    Tr_ sys (|Ψ_n⟩⟨Ψ_n|)
    
       =   ∑_ i, j∈{0,1}^n⟨φ |
    K^*_ jK_ i 
    |φ⟩·
    | i⟩⟨ j|

where K_ i:= K_i_n… K_i_1 for i= (i_1,… i_n).

 
Throughout the paper we will assume that the dynamics is primitive in the sense that 𝒯_* has a unique stationary state ρ_ ss>0 so that 𝒯_*(ρ_ ss)= ρ_ ss and it is aperiodic, i.e. the only eigenvalue of 𝒯_* with unit absolute value is 1.



Estimation of dynamical parameters


We investigate the following quantum estimation problem: assuming that the dynamics depends smoothly on an unknown parameter 
θ∈ℝ, we would like to estimate θ by performing measurements on the output state 
ρ_n^ out generated after a number n of interaction steps. In particular, we are interested in designing measurement strategies which achieve the highest possible precision, at least in the limit of large times. 

Let θ↦ U_θ be a smooth map describing how the dynamics depends on an unknown parameter θ, which is assumed to belong to an open bounded interval Θ of ℝ. We use similar notations |Ψ_θ,n⟩, K_θ,i, 𝒯_θ to denote the dependence on θ of the system-output state, Kraus operators, transition operator, etc. Two sequences of quantum statistical models indexed by time are of interest here: the system-output state 
𝒮𝒪_n:= {|Ψ_θ,n⟩ :θ∈Θ} defined in equation (<ref>) and the output state 
𝒪_n:= {ρ^ out_θ,n : θ∈Θ} defined in equation (<ref>). While the former is more informative than the latter and easier to analyse, we are particularly interested in estimation strategies which involve only measurements on the output, hence the importance of the model 𝒪_n. The following Theorem <cit.> shows that for primitive dynamics the QFI of both models scale linearly with n with the same rate, so having access to the system does not change the asymptotic theory. To simplify the expression of the QFI rate (<ref>) we assume the following "gauge condition"

    ∑_j  Tr 
    (ρ_θ^ ssK̇_θ,j^*
    K_θ,j) =0.

The condition (<ref>) means 
that ∑_j 
K̇_θ,j^*
K_θ,j belongs to the subspace {X:  Tr(ρ_θ^ ssX) =0}⊆ M(ℂ^d) on which 
the resolvent ℛ_θ:= ( Id - 𝒯_θ)^-1 is well defined as the Moore-Penrose inverse. The condition can be satisfied by choosing the complex phase of the Kraus operators appropriately, or equivalently the phase of the standard basis in the noise unit space ℂ^2.



Consider a primitive discrete time Markov chain whose unitary U_θ depends smoothly on θ∈Θ⊂ℝ, and assume that condition (<ref>) holds true. The QFI F^ s+o_n(θ) of the system and output  state |Ψ_θ,n⟩ and the QFI F^ out_n(θ) of the output state ρ^ out_θ, n scale linearly with n with the same rate: 

    lim_n→∞1/n F^ s+o_θ(n) =lim_n→∞1/n F^ out_θ(n) = f_θ
          =4∑_i=1^k   Tr[ρ^ ss_θK̇_θ, i^*K̇_θ,i] 
    
          
    +8∑_i=1^k  Tr[ Im ( K_θ, iρ^ ss_θK̇_θ,i^* ) ·ℛ_θ(  Im∑_jK̇_θ,j^*
    K_θ,j)]

where 


ℛ_θ is the Moore-Penrose inverse of Id- 𝒯_θ.


In the following, we will always assume that f_θ>0 for every θ∈Θ. In general, the classical Fisher information associated to simple repeated measurements (measuring the same observable on each output unit) does not achieve the QFI rate f_θ. However, the class of available measurements can be enlarged by unitarily "post-processing" the output before performing a standard measurement, so that  effectively one measures the original output in a rotated basis. While this shifts the difficulty from measurement to "quantum computation", it turns out that the post-processing can be implemented with minimal computational cost by employing  the concept of a coherent absorber introduced in <cit.>. Indeed <cit.> demonstrated that the QFI rate is achievable by combining post-processing by a coherent absorber with a simple adaptive sequential measurement scheme. Furthermore, <cit.>, argued that one can also achieve the QFI by performing  simple counting measurements in the output, without the need for adaptive measurements. Our goal is to revisit this scheme and to provide a new, computationally and statistically effective estimation strategy.




 §.§ Quantum postprocessing with a coherent absorber




The working of the coherent absorber 
is illustrated in Figure <ref> b).  Consider a QMC with a fixed and known  unitary U. After interacting with the system, each output noise unit interacts with a separate d dimensional system (the coherent absorber), via a unitary V. The system and absorber can now be regarded as a single doubled-up system which interacts with the input via the unitary W:= V U on ℂ^d⊗ℂ^d ⊗ℂ^2, where U acts on first and third tensors and V on second and third. The defining feature is that the system plus absorber have a pure stationary state. One can arrange this by requiring 

    VU: 
    |χ^ ss⟩⊗ |0⟩↦ 
    |χ^ ss⟩⊗ |0⟩

where |χ^ ss⟩∈ℂ^d⊗ℂ^d is a purification of the system stationary state, i.e. ρ^ ss =   Tr_ abs ( |χ^ ss⟩⟨χ^ ss|). This 
implies that in the stationary regime the output is decoupled from system and absorber, and is in the "vacuum" state |0⟩^⊗ n. 
We briefly recall a few expressions related to the construction of V that will be useful later on, and refer to Lemma 4.1 in <cit.> for more details; for clarity, in the following we will use the labels S,A,N to indicate system, absorber and noise unit.

Let us consider a spectral representation of ρ^ ss and the corresponding purification:

    ρ^ ss=∑_i=1^dλ_i |i_S⟩⟨i_S|,   |χ^ ss⟩=∑_i=1^d√(λ_i)|i_S⟩⊗|i_A⟩.

For simplicity we assume that the eigenvalues λ_i are strictly positive and are ordered in decreasing order;  

one can check that the following vectors are orthonormal:

    |v_i⟩=∑_k=0^1∑_j=1^d√(λ_j/λ_i)⟨i_S|K_k |j_S⟩|j_A⟩⊗|k_N⟩,    i=1,…, d.


For any choice of v_d+1,…, v_2d such that {v_1,…, v_2d} is an orthonormal basis for ^d ⊗^2 (the Hilbert space corresponding to the absorber and the ancilla), a suitable choice for V is given by

    V=
    1_S ⊗ (
    ∑_i=1^d |i_A ⊗ 0_N⟩⊗⟨v_i|+ 
    ∑_i=d+1^2d 
    |i_A ⊗ 1_N⟩⊗⟨v_i| ) .


Note that V is not uniquely defined: there is freedom in the spectral resolution of ρ^ ss if there are degenerate eigenvalues and in picking v_d+1,… , v_2d. 

The Kraus operators corresponding to the reduced dynamics W= VU of the system and the absorber together are given by the following expression:

    K̃_k:    =   ⟨ k| W|0⟩ =
    ∑_l=0^1⟨k_N| V  |l_N⟩⟨l_N| U |0_N⟩
       =   ∑_l=0^1  K_l ⊗ V_kl


where V_kl and K_l are the Kraus operators of V and 
U considered without ampliation. 
The following Lemma prescribes the structure of the blocks V_kl. 
We first define the “recovery” channel <cit.> with Kraus operators K^'_i= √(ρ) K_i^*√(ρ^-1) and note that they satisfy the normalisation condition and the recovery channel has ρ as invariant state.


The absorber operators V_kl are of the following form. The blocks V_0l are determined as 
V_0l = K_l^' T
where the transpose is taken with respect to the eigenbasis of ρ. Assuming that 
1 - |V_00|^2 and 1 - |V_01|^2 are strictly positive, then the V_1l blocks are determined up to an overall arbitrary unitary u

    V_10 = u |V_10| , V_11= uw |V_11|

where   
|V_10| = √(1 - 
|V_00|^2), 
|V_11| = √(1 - |V_01|^2 ) are fixed, as well as the unitary 
w= -|V_10|^-1 V_00^* V_01 
|V|_11^-1.




Proof. 
From the definition of |v_i⟩ we have


    V_0k   =
     
     ∑_i,j=1^d√(λ_j/λ_i)⟨j_S| K^*_k |i_S⟩|i_A⟩⟨j_A| = K_k^' T.

which proves the first statement.
From the fact that V is unitary we obtain 

    V_00^*V_00 +V_10^*V_10   =   1
    
    V_01^*V_01 +V_11^*V_11   =   1

from which we get 

    |V_10| = √(1 - |V_00|^2),    
    |V_11| = √(1 - |V_01|^2)

which means that the absolute values of V_10, V_11 are fixed. 
Let V_10= U_0 |V_10| and V_11= U_1 |V_11| be their polar decompositions. Then from

    V_00^*V_01 + V_10^*V_11 =0

we get V_10^*V_11= -V_00^*V_01 and 

    U_0^* U_1= |V_10|^-1 V_10^*V_11 |V_11|^-1 = 
    -|V_10|^-1 V_00^*V_01 |V_11|^-1

which is a fixed unitary w. This proves the claim.



Later on we will require that the system and absorber transition operator 𝒯̃(·) = ∑_k K̃^*_k ·K̃_k is primitive, in addition to the 
system's transition operator 𝒯 satisfying the same property. At the moment we are not able to establish what is the connection between the two properties. However we performed extensive numerical simulations with randomly chosen QMC dynamics and corresponding absorbers which indicate that “generically” with respect to the original dynamics, for every primitive 𝒯 there exists a corresponding absorber such that 
𝒯̃ is primitive. In fact, a stronger statement seems to hold, which is that the spectral gap of 𝒯̃ is smaller or equal to that of 𝒯 and one can always choose the absorber such that the two are equal. For more details on the absorber theory we refer to the recent paper <cit.>.


Returning to the parameter estimation setting where U= U_θ depends on the unknown parameter θ, we note that one cannot implement a coherent absorber which precisely matches 
the system dynamics. Instead, one can implement the absorber for an approximate value θ_0 of θ and try to estimate the offset 
θ-θ_0 by measuring the output. This setting is closely related to that of displaced null measurements discussed in section <ref>. Indeed, the joint state of system, absorber and output is pure for all θ and at θ=θ_0 it is of the product form 
|χ^ ss_θ_0⟩⊗|0⟩^⊗ n, assuming that system and absorber are initially in the stationary state. Therefore, repeated standard basis measurements on the output units constitute a null measurement (in conjunction with a final appropriate measurement on system and absorber). 


The exact procedure for determining θ_0 will be described in section <ref> and follows the important displacement prescription outlined in section <ref>. For the moment it suffices to say that θ_0 will be informed by the outcome of a preliminary estimation stage involving simple (non-optimal) measurements on the output (without post-processing), and it will converge to θ in the limit of large n.  
While for θ_0=θ the output state is the "vacuum", for θ≠θ_0 the output could be seen as carrying a certain amount of "excitations" which increases with the parameter mismatch |θ-θ_0|. 

In section <ref> we show how these "excitations" can be given a precise meaning by fashioning the output Hilbert space into a Fock space carrying modes labelled by certain "excitation patterns". In section 
<ref> we show that from this perspective, the output state converges to a joint coherent state of the excitation pattern modes whose displacement depends linearly on θ-θ_0. This will allow us to devise a simple "pattern counting" algorithm for estimating θ in section <ref>.  









 







§ TRANSLATIONALLY INVARIANT MODES IN THE OUTPUT







In this section we introduce the concept of translationally invariant modes (TIMs) of a spin chain. We show that in the limit of large chain size, certain translationally invariant states acquire the characteristic Fock space structure, and that the corresponding creation and annihilation operators satisfy the bosonic commutation relations. This construction will then be used in analysing the stationary Markov output state in section <ref>.

Let (ℂ^2)^⊗ n be a spin chain of length n and let |Ω_n⟩: = |0⟩^⊗ n be the reference “vacuum” state. For every pair of integers (k,l) with 1≤ l≤ k we define an excitation pattern of length 
k and number of excitations l to be an ordered sequence
α := (α_1, …α_k) ∈{0,1}^k such that α_1=α_k =1 and ∑_i=1^k α_i =l. 
For instance, for k=1,2 the only patterns are 1 and respectively 11 while for k=3 the possible patterns are 
111 and 101.

For each pattern α of length k we define `creation and annihilation' operators

    A^*_α(n) = 1/√(n)∑_i=1^n-k+1σ^α_i, 
       A_α(n)=1/√(n)∑_i=1^n-k+1σ_i^α*

where σ_i^α=∏_j=0^k-1σ_i+j^α_j+1, with σ^0 := 1, 
σ^1:= σ^+ := |1⟩⟨ 0| and the index i denotes the position in the chain.

In particular for α=1 we have

    A^*_1(n)=1/√(n)∑_i=1^nσ_i^+,    A_1(n)=1/√(n)∑_i=1^nσ_i^-.



We further define the “canonical coordinates” and “number operator” of the “mode” α as

    Q_α(n)   =   A_α(n)+A^*_α(n)/√(2),
    
    P_α(n)    =   A_α(n)-A^*_α(n)/√(2)i,
     
    N_α(n)   =   A^*_α(n)A_α(n).

We now introduce “Fock states” obtained by applying 
creation operators to the vacuum. 

Let 𝒫 denote the ordered set of all patterns, where the order is the usual one when regarding patterns as integer numbers in binary representation. Let n:𝒫→ℕ be pattern counts 
n = (n_α)_α∈𝒫 such 
that all but a finite number of counts are zero, and let 
n!:= ∏_α∈𝒫n_α! and | n|:=∑_αn_α the total number of patterns.



We define the approximate Fock state associated to the set of 
counts n as

    | n
    ; n⟩:= 
    1/√( n)!∏_α∈𝒫 
    A^*_α(n)^n_α
    
    |Ω_n⟩ ,    n ≥ 1,

where the product is ordered according to the order on 𝒫. For any fixed n these vectors are not normalised or orthogonal to each other, and indeed they are not linearly independent since the Hilbert space is finite dimensional; however, Proposition <ref> shows 
that the expected Fock structure emerges in the limit of large 
n. Let us first illustrate this with a simple example. 
The state containing 2 patterns α =1 is given by 

    |n_1= 2;n⟩   :=   1/√(2)(A_1^*(n) )^2 |Ω_n ⟩
       =   1/√(2)n∑_i≠ j=1^n 
    |0… 0
    10 … 01 0… 0⟩

where i≠ j indicate the positions of the excitations, while the state containing a single 11 pattern is 

    |n_11=1;n⟩   :=    
    A_11^*(n) |Ω_n ⟩
       =   1/√(n)∑_i=1^n-1 
    |0… 0
    11 0… 0⟩

Now it is easy to check that as n→∞

    ⟨ n_1= 2;n| n_1= 2;n ⟩   =    1/2n^24n(n-1)/2→ 1,
    ⟨ n_11=1;n| n_11=1;n ⟩   =    n-1/n→ 1,
    ⟨ n_1= 2;n| n_11=1;n ⟩   =   2(n-1)/2√(n)n= O(1/√(n)).

This is generalised in the following Proposition which establishes the familiar structure of the bosonic Fock space in the limit of large n.



Let | n;n ⟩ be the "Fock states" defined in equation (<ref>). In the limit of large n the "Fock states" become normalised and are orthogonal to each other

    lim_n→∞⟨ n;n|  m ;n⟩ =
    δ_ n,  m .

Moreover, the order of the creation operators in (<ref>) becomes irrelevant in the limit of large n.


The proof of Proposition <ref> can be found in Appendix <ref>. From Proposition <ref> we obtain the following Corollary which shows that the action of the creation operators on the "Fock states" converges to that of a bosonic creation operator in the limit of large n.



Let β be a pattern and let δ^(β) be the counts set with δ^(β)_α = δ_α, β. Let | n; n⟩ be a "Fock state" as defined in equation (<ref>).
In the limit of large n the action of creation and annihilation operators A^*_β(n) and A_β(n) satisfy

    A^*_β(n)| n ;n⟩   =   √(n_β +1) | n + δ^β; n⟩+o(1)
    
    A_β(n)| n ;n⟩   =   √(n_β) | n - δ^β; n⟩+o(1)


The proof of Corollary <ref> can be found in Appendix <ref>.




§ LIMIT DISTRIBUTION OF QUADRATURES AND NUMBER OPERATORS.





 
In this section we analyse the structure of the output state obtained by post-processing the output of a QMC with a coherent absorber, as described in section <ref>. Motivated by the fact that the statistical uncertainty scales as 
1/√(n)

we choose the absorber parameter 
θ_0 to be fixed and known, and write the system parameter as
θ= θ_0+u/√(n),
where u is an unknown local parameter. For a more in depth motivation we refer to local asymptotic normality theory in appendix <ref>.

Since the output state becomes stationary for long times, it is natural to focus on the state of the translationally invariant modes introduced in section <ref>. In Theorem <ref>
we show that asymptotically with n, the restricted state of these modes is a joint coherent state whose amplitude depends linearly on u, i.e. a Gaussian shift model. Moreover, Corollary <ref> shows that the QFI of this model is equal to the QFI rate f_θ_0 of the output state characterised in Theorem <ref>. This means that the TIMs capture the entire QFI of the output. Together with Theorem <ref> of section <ref>, these  results will be the theoretical underpinning the  estimator proposed in section <ref>.



We consider the system and absorber together as an open system with space ℂ^D= ℂ^d⊗ℂ^d with D= d^2, interacting with the noise units. The corresponding unitary is W_θ= V_θ_0U_θ where the absorber parameter is a fixed value θ_0 (which later will be determined based on a preliminary estimation procedure) and θ is the unknown parameter to be estimated. We distinguish between the system Kraus operators K_θ,i =
⟨ i|U_θ |0⟩∈ M(ℂ^d) and the system and absorber Kraus operators 
K̃_θ,i = ⟨ i| W_θ|0⟩∈ M(ℂ^D), and similarly between the system transition operator 𝒯_θ and the system and absorber transition operator 𝒯̃_θ (X) =∑_i=0^1 K̃_θ,i^*XK̃_θ,i. 
 
We will be interested in the probabilistic and statistical properties of the output state ρ̃^ out_θ of the system and absorber dynamics, for parameters θ in the neighbourhood of a given θ_0. For clarity we state the precise mathematical properties we assume throughout.


The following properties of the system-absorber QMC are assumed to be true:


  * 𝒯̃_θ has a unique faithful invariant state ρ̃^ ss_θ and is aperiodic for θ in a neighborhood of θ_0;

  * The Kraus operators K̃_θ,i and the stationary state ρ̃^ ss_θ are analytic functions of θ around θ_0;

  * At θ_0 the stationary state is pure 
ρ̃^ ss_θ_0=
|χ^ ss_θ_0⟩⟨χ^ ss_θ_0| and 
K̃_θ_0,i |χ^ ss_θ_0⟩=(1-i)|χ^ ss_θ_0⟩ for i=0,1.



To formulate the result we use the 
local parametrisation θ= θ_0 + u/√(n)
where u is to be seen as a local parameter to be estimated from the output of length n. To simplify the notation, we denote the derivatives at θ_0 as 

    K̇_i := . dK_θ,i/dθ|_θ_0,  K̇̃̇_i := . dK̃_θ,i/dθ|_θ_0,
 and drop the subscript θ_0 in K_θ_0,i =: K_i, K̃_θ_0,i =: K̃_i, etc.; we also use the local parameter instead of θ e.g. 
K_θ_0+u/√(n),i =: K_u,i. 

Note that properties 1. and 2. in Hyothesis <ref> imply that ρ̇̃̇^ ss=ℛ̃_*𝒯̇̃̇_*(ρ̃^ ss) where ℛ̃ is the Moore-Penrose inverse of Id - 𝒯̃. In addition, K̃_0|χ^ ss⟩= K̃^*_0|χ^ ss⟩ = |χ^ ss⟩.
Indeed by construction K̃_0|χ^ ss⟩ =|χ^ ss⟩ and 
K̃_1|χ^ ss⟩ =0, and by applying  
K̃_0^*K̃_0+ K̃_1^*K̃_1=1 to |χ^ ss⟩ we get K̃_0^*|χ^ ss⟩ = |χ^ ss⟩. 


Since for large n the dynamics reaches stationarity, we consider the output state corresponding to the system starting in the stationary state ρ̃^ ss_u

    ρ̃^ out_u,n=
    ∑_ i,  j∈{0,1}^n Tr[ 
    ρ̃^ ss_u 
    K̃^*_u, jK̃_u, i]| i⟩⟨ j|


To formulate our results below we need to introduce several superoperators acting on the system and absorber space. For  x∈ M(ℂ^D), we define 

    A_i(x)=𝒯̃(x)     i=0 
    K̃^*_1xK̃_0     i=1 
    K̃^*_0xK̃_1     i=-1,

and

    Ȧ_1(x)=K̇̃̇_1^*xK̃_0 + K̃_1^*xK̇̃̇_0


Furthermore, for every pattern  α of length l, we denote

    A^α= A_α_1⋯ A_α_l and Ã^α=Ȧ_1 A_α_2⋯ A_α_l.


With a slight abuse of notation we will denote the expectation with respect to a density matrix ρ as 
ρ(X) =  Tr(ρ X).


 

Let θ_0 be a fixed parameter and assume that the dynamics satisfies the assumptions in Hypothesis (<ref>). Let θ=θ_0 +u/√(n) be the system parameter with fixed local parameter u. Let α be a fixed pattern and let z=β+iγ∈ℂ with |z|=1. Then the following convergence in distribution hold in the limit of large n with respect to the output state ρ̃^ out_u,n as defined in equation (<ref>).

    
  i)
    The quadratures (<ref>) and (<ref>) of the TIM mode α satisfy the joint Central Limit Theorem

    β Q_α(n)+γ P_α(n)  N(u μ_α,z, 1/2),

where N(μ, V) denotes the normal distribution with mean μ and variance V and  μ_α,z= √(2) (z̅μ_α) with

    μ_α= ρ̃^ ss((𝒯̇̃̇ℛ̃ A^α+Ã^α)(1)).


  ii) The number operator (<ref>) of the TIM mode α satisfies the Poisson limit:

    N_α(n)  Poisson(u^2λ_α),

where
λ_α:=|μ_α|^2.





The proof of Theorem <ref> can be found in Appendix <ref>. 


We now provide a simpler expression for the parameters μ_α in terms of Kraus operators and their first derivatives. As a by-product we show that the sum of all (limiting) QFIs of the Gaussian modes 
(Q_α(n), P_α(n)) is the QFI rate of the output state. This means that the TIMs capture all the QFI of the output state.

 
    Let α be any excitation pattern and let μ_α 
    
    be the constant defined in (<ref>).
    Then μ_α can also be expressed as

    μ_α   =   ⟨K̃_α_|α|⋯K̃_α_1(1-K̃_0)^-1K̇̃̇_0 χ^ ss | χ^ ss⟩
       +   ⟨K̃_α_|α|⋯K̃_α_2K̇̃̇_1 χ^ ss|χ^ ss⟩.
    
    Moreover with
    λ_α=|μ_α|^2, one has
    
    ∑_αλ_α=(K̃_1(1-K̃_0)^-1K̇̃̇_0+K̇̃̇_1)χ^ ss^2.



The proof of Lemma <ref> can be found in Appendix <ref>.

 
Asymptotically with n, the total QFI of the TIMs is equal to the QFI rate of the output state (<ref>), that is

    4∑_αλ_α=4 (K̃_1(1-K̃_0)^-1K̇̃̇_0+K̇̃̇_1)χ^ ss^2 =f_θ_0 .


The proof of Corollary <ref> can be found in Appendix <ref>. 

The upshot of this section is that (asymptotically in n) the statistical information of the output state is concentrated in the TIMs and the state's restriction to the TIM Bosonic algebra is a coherent state. Formally, in order to optimally estimate the parameter, one would only need to measure the appropriate quadrature of the Gaussian shift model, as explained in section <ref>. However, it is not obvious how to perform such a measurement, and the theoretical insight does not seem to help on the practical side. Surprisingly, it turns out that the standard sequential counting measurement is an effective joint measurement of all the TIMs' number operators! This will be the main result of the next section, which in conjunction with the displaced null strategy discussed in section <ref>, provides the ingredients of a counting-based estimation strategy.




§ LIMIT THEOREM FOR COUNTING TRAJECTORIES




In this section we continue to investigate the probabilistic properties of the output state and consider the distribution of the stochastic process obtained measuring the output units sequentially in the canonical basis {|0⟩, |1⟩}. We consider the system and absorber dynamics with 
fixed absorber parameter θ_0 and system parameter 
θ= θ_0+u/√(n) for a fixed local parameter u. The  output state is given by equation (<ref>). The probability of observing a sequence ω =(ω_1,… , ω_n)∈{0,1}^n as the outcome of the first n measurements is given by

    ν_u,n(ω):=ρ^ ss_u ( B_u,ω_1⋯ B_u,ω_n(1) )

where 

    B_u,j(x)=K̃^*_u,1 xK̃_u,1    j=1 
    K̃^*_u,0xK̃_u,0    j=0 
    .

In order to state the main result of this section, we need to introduce a collection of events: first of all we define

    B_ 0(n)={(0,…,0)}⊂{0,1}^n.

Let us fix any real number 0<γ<1. Let m = {m_α}_α∈𝒫 be a set of pattern excitation counts where all  occupation numbers are zero except (m_α^(1), …,m_α^(k)), and define B_ m(n) as the empty set for every n strictly smaller than 
∑_i=1^km_α^(i) |α^(i)|+(k-1)n^γ (|α| is the length of the pattern α), otherwise B_ m(n) is the set of all binary sequences obtained concatenating m_α^(1) copies of α^(1)s, up to m_α^(k) copies of α^(k)s in any order, and inserting 0s before, between and after them in order to reach a total length of n and making sure that between two consecutive patterns there are at least n^γ 0s.


The following Theorem shows that the distribution of the 
pattern counts m converges to a product of Poisson distributions with the same intensities as those of the number operators of the TIM in Theorem <ref>. This means that performing a standard output measurement and extracting the pattern counts provides an effective joint measurement of the number operators of the TIMs. This finding is essential in constructing an optimal estimator in section <ref>.

 
For every positive constant C>0 and finite collection of excitation patterns counts m=(m_α^(1),… ,m_α^(k)),

the following limit holds true:

    lim_n → +∞sup_|u|<C |ν_u,n (B_ m(n) )-e^-λ_totu^2∏_i=1^k(λ_α^(i)u^2)^m_α^(i)
    /m_α^(i)! |=0.

where 
    
    λ_tot    :=   -1/2⟨χ^ ss|(2Ḃ_0* R_0*Ḃ_0*+ B_0*)(|χ^ ss⟩⟨χ^ ss|)|χ^ ss⟩
       =   -(⟨χ^ ss, (2K̇̃̇_0(1-K̃_0)^-1K̇̃̇_0+K̃_0) χ^ ss⟩ ).

Moreover,

    ∑_αλ_α = λ_tot.



The Proof of Theorem <ref> can be found in Appendix <ref>.

The previous result has some relevant consequences. Let us define the “pattern extraction” function 



which associates to each trajectory ω∈{0,1}^n a set of pattern counts { N_α,n(ω): α∈𝒫}∈ℕ^𝒫, which is uniquely determined by the condition that ω is a maximal union of contiguous patterns separated by sequences of 0s of length at least n^γ with a fixed 0<γ<1; moreover, let us consider the stochastic process given by the infinite collection of independent random variables {N_α: α∈𝒫} where N_α is a Poisson random variable with parameter λ_α u^2.
 
  For every u ∈ℝ the law of the stochastic process { N_α,n: α∈𝒫} under the measure ν_u,n converges to the one of {N_α:α∈𝒫}. Moreover, for every α∈𝒫, p≥ 1 one has
  
    lim_n→ +∞𝔼[N_α,n^p]=𝔼[N_α^p].


The Proof of Corollary <ref> can be found in Appendix <ref>.

However, in the following we will be interested in local parameters with growing size, i.e. |u| ≤ n^ϵ^' for some 0 <ϵ^' <1/2. In this case we can show the following result.
 
 For 0 <ϵ^' <1/6 and for every finite collection of excitation patterns counts m the following holds true:

    lim_n → +∞sup_|u| ≤ n^ϵ^' | ν_u,n (B_𝐦(n) )/ e^-λ_totu^2∏_i=1^kλ_α^(i)^m_iu^2m_i/m_i!-1  |=0.
  

The Proof of Proposition <ref> can be found in Appendix <ref> as well. Upgrading this result to a  weak convergence one similar to Theorem <ref> remains the subject of future research.



§ PATTERN COUNTING ESTIMATOR



In this section we describe our adaptive estimation scheme which exploits the asymptotic results presented in sections <ref> and <ref>. The scheme involves four key ingredients: 

    
  i) 
    perform a simple output measurement (no absorber) to compute a preliminary estimator;
    
    
    
  ii) set the absorber parameter by using the displaced-null measurement technique developed in <cit.> and run the system-absorber dynamics for the remainder of the time;
    
  iii) perform a sequential counting measurement in the output and extract counts for the TIMs from the outcomes trajectory;
    
  iv)
    construct a simple estimator expressed in terms of total counts of patterns for different TIM modes.
    


The first step of the adaptive protocol is to use ñ := n^1-ϵ≪ n output units to produce a rough preliminary estimator 
θ̃_n. This can be done by performing a repeated standard basis measurement on the output (without using an absorber). 
Typically, the estimator
θ̃_n will have variance scaling with the standard rate ñ^-1= n^-1+2ϵ, and ñ^1/2(θ̃_n-θ) will satisfy the central limit theorem and a concentration bound ensuring that |θ̃_n-θ| =O(n^-1/2+ϵ) with high probability. 

For instance, one can define 
θ̃_n by equating the empirical counting rate with the theoretical rate n_τ:=  Tr(ρ^ ss_τ K_τ, 1^*K_τ,1), see <cit.>. 

In the second step we set the absorber at a parameter value θ_0 and run the system-absorber quantum Markov chain for the reminder of the time n^' = n-ñ. 
The naive choice for θ_0 is our best guess θ̃_n about θ, based on the first stage measurement. However, with this choice, the counting measurement suffers from the non-identifiability issue described in section <ref>. This can be resolved by further displacing the absorber parameter by an amount 
δ_n:= τ_n/√(n) where τ_n= n^3ϵ so that 
θ_ abs:= 

θ̃_n - δ_n. As usual we write 
θ = θ̃_n+u_n/√(n) where u_n is a local parameter satisfying |u_n|≤ n^ϵ, so that 
θ= 

θ_ abs
+(u_n+τ_n)/√(n).

In the third step we perform standard basis measurements in the output of the modified dynamics which includes the absorber. For simplicity, in the discussion below we ignore the fact that we have n^' = n-ñ rather than n output units, which does not affect the error scaling of the estimator. Let ω= (ω_1,… , ω_n) be the measurement outcome with distribution (<ref>), where the local parameter u_n is replaced by 
u_n+τ_n to take into account the displacement. 


We now describe the construction of the estimator from the outcomes of this last stage measurement. The estimator will be built using the “pattern extraction” function { N_α,n(ω): α∈𝒫}∈ℕ^𝒫 that we defined in the previous section; we recall that it
associates to each trajectory ω∈{0,1}^n a set of pattern counts, which is uniquely determined by the condition that ω is a maximal union of contiguous patterns separated by sequences of 0s of length at least n^γ with a fixed 0<γ<1, cf. Figure <ref> d) for illustration. This means that the algorithm will not detect any pattern which contains a sequence of zeros of length larger than n^γ, since this would be seen as being made up of several identified patterns.
We now introduce the final estimator using an intuitive argument based on extrapolating the results of Theorem <ref> from fixed to slowly growing local parameters u_n+τ_n. This means that N_α,n(ω) is approximately distributed as Poisson((u_n+τ_n)^2 |μ_α|^2), for large n; since τ_n= n^3ϵ is larger than u_n= O(n^ϵ), the intensity of the Poisson distribution diverges with n, and the distribution can be approximated further by the normal  
N((u_n+τ_n)^2|μ_α|^2, (u_n+τ_n)^2|μ_α|^2 ) with the same mean and variance. Using 

    1/τ_n(u_n+τ_n)^2    =   
    2u_n + τ_n + o(1) 
    1/τ^2_n(u_n+τ_n)^2    =    1+ o(1)

we obtain that 

    Y_α,n := 1/|μ_α|(N_α,n/τ_n -τ_n |μ_α|^2 )

has approximate distribution N(2u_n|μ_α| , 1). A simple computation shows that the optimal estimator of u_n based on the (approximately) normal variables Y_α,n is the linear combination

    û_n:= Y_n: = 2/ f_θ_ abs∑_α |μ_α| Y_α,n =2/f_θ_ absτ_n∑_α N_α,n - τ_n/2

where f_θ_ abs= 4∑_α |μ_α|^2 is the quantum Fisher information rate of the output by Corollary <ref>. Note that Y_n depends only on the total number of patterns of the trajectory ω, not to be confused with the total number of 1s. 

Since Y_α,n is approximately normal with distribution N(2u_n|μ_α| , 1), we obtain that û_n has approximate distribution N(u_n, f_θ_ abs^-1). 
The final estimator of θ is 

    θ̂_n:=θ̃_n+û_n/√(n).

and it attains the QCRB in the sense that 
√(n)(θ̂_n-θ) converges in distribution to N(0,f_  θ^-1).

At the moment we only have a rigorous proof of this statement assuming a stronger version of Proposition <ref>, which we were not able to obtain; however, we point out that Theorem 2 in <cit.> establishes a similar optimality result in the case of multi-parameter estimation with independent, identical copies.

Before proceeding, we need to introduce some more notations. Let us consider the following collection of random variables:

    N_n,θ,θ̃∼Poisson(λ_ tot(θ̃)(√(n)(θ-θ̃)+τ_n)^2),

where n ∈ℕ, and θ̃, θ∈Θ. We recall that τ_n=n^3ϵ is the displacement size. Let us define

    Y_n,θ,θ̃:=1/2τ_n λ_ tot(θ̃)
    N_n,θ,θ̃
    - τ_n/2.

 
Let fix θ∈Θ at which f_θ is continuous and let θ̃_n be a preliminary estimator which uses ñ:=n^1-ϵ samples with ϵ small enough, such that it satisfies the concentration bound

    ℙ_θ(| θ̃_n-θ |>n^-1/2 +ϵ)
    ≤ C e^-n^ϵ r

for some constants C,r>0. Let 

    θ̂_n:=θ̃_n+Y_n/√(n)  ,

be the final estimator as defined in (<ref>) and (<ref>).

If for every a ∈ℝ one has

    lim_n → +∞sup_θ̃:|θ̃-θ|<n^-1/2+ϵ |𝔼_θ[e^iaY_n|θ̃_n=θ̃]-𝔼[e^iaY_n,θ, θ̃]|=0,

then θ̂ is asymptotically optimal and asymptotically normal, i.e. the following convergence in law holds for large n

    √(n)(θ̂_n-θ)  N (0, 1/f_θ ).



The proof of the following theorem and the relationship between the extra hypothesis in Eq. (<ref>) and Proposition <ref> can be found in Appendix <ref>.




























In the following section we illustrate our method with results of numerical simulations on a qubit model.











§ NUMERICAL EXPERIMENTS




In this section we illustrate the estimation protocol through numerical simulations, using a qubit model inspired by the previous work <cit.>.
The simulations are implemented in Python using the QuTiP package <cit.>. 


The quantum Markov chain model consists of a two-dimensional system coupled to two-dimensional noise units by a unitary U_θ with unknown parameter θ∈ℝ, where the noise units are all prepared in the same initial state | 0⟩. Since the system interacts with a fresh noise unit at each step, it suffices to specify the action of  U_θ on the states |00⟩ and |10⟩, and we define

    U_θ : |00⟩   →   cos(θ)  √(1-θ^2)|00⟩
          + i sin(θ) √(1-θ^2)|10⟩ + θ|11⟩, 
    
        U_θ : |10⟩   →    i sin(θ)  √(1-λ)|00⟩
          + cos(θ) √(1-λ)|10⟩ + √(λ) e^i ϕ|01⟩,

where λ, ϕ are known parameters. In  simulations we  used ϕ=π/4, λ=0.8 and θ=0.2 for the true values of the parameter. 


In a first simulation study we verify the predictions made in Corollary <ref>. We run the dynamics for a total of n=6× 10^5 time steps, with absorber parameter 
θ_ abs= θ_0 (cf. section <ref>) and system parameter θ= θ_0+ u/√(n) such that θ=0.2 and the local parameter is u =2.

For each run we perform repeated measurements in the standard basis of the output units, to produce a measurement record 
ω =(ω_1, …, ω_n). For each such trajectory, the pattern counts N_α, n(ω) are obtained by identifying patterns (sequences starting and ending with a 1) which are separated by at least n^γ 0s and no pattern contains more than n^γ 0s, where γ>0 is a small parameter. This can be done by combing through the sequence and identifying occurrences of a given such pattern padded by n^γ 0s to the left and right (taking care of the special case of the first and last patterns). Note that for any given n this procedure will not count patterns with more than  n^γ successive 0s. Since the mean counts for each pattern α is |μ_α|^2 u^2 and |μ_α|^2 decays exponentially with |α|, we find that patterns of such length are unlikely to occur for large n. The results of N=2000 independent repetitions of the experiment are illustrated in Figure <ref> which shows a good match between the counts histograms corresponding to several patterns (in blue) and the theoretical Poisson distributions 
Poisson(|μ_α|^2 u^2) (in orange) , as predicted by Corollary <ref>.  








In a second simulation study we use system parameter
θ=0.2 and set the absorber at θ_ abs = θ-δ_n, with displacement δ_n= n^-1/2τ_n for n=6× 10^5 and 
τ_n = 7. We perform the same measurement as above and extract the pattern counts N_α,n(ω) for each trajectory ω. We then use the pattern counts estimator (<ref>) to estimate θ, taking 
θ̃_n=θ. This amounts to assuming that the first stage of the general estimation procedure outlined in section <ref> gives a perfect estimator, which is then used in setting the absorber parameter in the second step. While this procedure cannot be used in a practical situation, the study has theoretical value in that it allows us to study the performance of the pattern counts estimator in its own right, rather than in conjunction with the first step estimator. Figure <ref> shows that the final estimator 
θ̂_n has Gaussian distribution with variance closely matching 1/(n f_θ), thus achieving the QCRB in this idealised setup. This can also be seen by comparing the “effective” Fisher information 
F_ eff:= (n (θ̂_n-θ)^2)^-1 where the mean square error is estimated from the data, with the QFI rate f_θ; the former is equal to F_ eff = 13.8 while the latter is f_θ = 13.5.





In the third simulations study we implement the full estimation procedure described in section <ref> including the first stage estimator. In the first step we fix θ =0.2, and run the Markov chain (without the absorber) for ñ= 4× 10^5. We then perform sequential measurements in the standard basis on the output noise units to obtain a measurement trajectory 
from which we compute θ̃_n by equating the empirical mean (the average number of 1 counts)

with the stationary expected value 
 c_τ: =  Tr (ρ^ ss_τ K_τ,1^* K_τ,1), where K_τ,i are the system's Kraus operators. The effective Fisher information of this estimator is 
 F_ eff = 4.06, significantly lower than the QFI f_θ =13.5.
 

We then set the absorber to 
θ_ abs = θ̃_n - n^-1/2τ_n where τ_n = 25.5 and n=6.6× 10^6 and we run the system and absorber chain for n^' = n-ñ steps. The system and absorber  is initialised in the pure stationary state corresponding to θ̃_n, but since the system and absorber dynamics is assumed to be primitive, any other initial state will 
result in equivalent asymptotic results. We then perform the counting measurement as in the previous simulation studies to obtain a trajectory ω∈{0,1}^n' and extract the pattern counts N_α,n^'(ω). 


From these average counts we compute the estimator (<ref>), where the Fisher information f is computed at θ̃_n and multipled by n^'/n to account for the smaller number of samples used in the last step. The results are illustrated in Figure <ref> which compares the histogram of θ̂_n (in blue) with the density of a normal distribution with mean θ and variance 1/(n f_θ). We find that there is a good fit with the normal distribution but less accurate that that of the second simulation study, cf. Figure <ref>. This is expected, since the final estimator is based on a two stage estimation process and does not use any prior information about θ. A more accurate measure of the protocol's performance is given by the  effective Fisher information which works out as F_ eff = 10.8 compared to the QFI rate f_θ=13.5, while the effective Fisher information of the first stage was only 4.06. These simulation results are in agreement with the theoretical arguments put forward in section <ref> which indicate that the two stage estimator attains the QCRB asymptotically.















 












































































    

      





 §.§ Construction of the estimator





To construct a suitable estimator, we start by considering the local parameters of our model. We assume we've performed some generally sub-optimal preliminary estimation on n^1 - ϵ (ϵ>0) of our total samples. With some reasonable assumptions on the initial estimator θ̃, we can assume that θ∈ I = {θ̃ - n^-1/2n^ϵ, θ̃ + n^-1/2n^ϵ} as n→∞ [previous paper]. Therefore, we write θ = θ̃ + u.n^-1/2, where u is a local parameter satisfying  |u|<n^ϵ. We also introduce a local parameter for the absorber offset τ_n=n^aϵ such that the parameter used by the absorber θ_0 is

    θ_0 = θ̃ + τ_n/√(n),

where a>2. This requirement on a ensures that the absorber is offset sufficiently (justify need for 2?) outside this localized region I. The difference between these parameters is then

    θ - θ_0 = 1/√(n)(u - τ_n)

producing a local parameter u-τ_n. 

Need to add in new factor involving proportion used in final estimation


We now want to relate this local parameter to the rate of occurrence of a specific pattern 
α. From section <ref> we known that the counts N_α(n) of a pattern α are distributed as 

    N_α(n) ∼ Poisson(|μ_α|^2 (u - τ_n)^2 )

in the limit of large n.

Furthermore, by Theorem <ref> we also known that in this limit the quadratures are normally distributed with mean μ_α (u - τ_n) and variance 1/2.

Needs the most work. Need to justify link to this view of measuring the number operator wrt a coherent state


To construct an estimator, we consider the number operator in the Fock space setup introduced in sections <ref> and <ref>. which corresponds to measuring a displaced number operator N_μ_ατ_n = (a - μ_ατ_n)^* (a - μ_ατ_n) with respect to the coherent state |√(2)μ_α u⟩:

    N_α(n)    = N_μ_ατ_n (|√(2)μ_α u⟩) 
       = (a - μ_ατ_n)^* (a - μ_ατ_n) 
       = a^* a - μ_α^* τ_n a - μ_ατ_n a^* + |μ_α|^2τ_n^2,

where the creation/annihilation operators are defined such that a|√(2) u⟩ = u |√(2) u⟩. We then calculate

    ⟨ a^* a ⟩_|√(2)μ_α u⟩   = |μ_α|^2 u^2 = O(n^2ϵ), 
        
    ⟨μ_α^* τ_n a + μ_ατ_n a^* ⟩_|√(2)μ_α u⟩   = 
        2 |μ_α|^2 τ_n u = O(n^(1+a)ϵ),
        
    ⟨ |μ_α|^2 τ_n^2 ⟩_|√(2)μ_α u⟩   = 
        |μ_α|^2 τ_n^2 = O(n^2aϵ)
        .


Note: we can safely neglect the first term in the limit of large n as a>2 ensures (<ref>) ≪ (<ref>) ≪ (<ref>). We then have

    N_α(n) - |μ_α|^2 τ_n^2 ∼ -√(2)|μ_α|τ_n 𝒩( √(2)|μ_α|u, 1/2)


To construct an estimator from the random variable N_α(n) we want to find a function of the random variable with the correct mean √(2) u and minimum variance 1/2. To this end, we consider 

    Y_α = 
        
        -N_α(n) + |μ_α|^2 τ_n^2
        /√(2) |μ_α| τ_n
        ∼𝒩(√(2)|μ_α|u, 1/2).



To include multiple patterns in the final estimator û, we then take a linear combination of these Y_α terms: û = ∑_α c_α Y_α. We then want to find coefficients c_α such that û has the correct mean and minimal variance. We find

    ⟨û⟩   = ∑_α c_α√(2) |μ_α| u 
         c_α = 1/√(2)
        |μ_α|
        /
        ||μ||^2

where ||μ||^2 = ∑_α |μ_α|^2. Checking the variance, we find

    Var(û)    = ∑_α c_α^2 . 1/2
       = 1/4 ||μ||^2 = 1/F

where F=4||μ||^2 is the QFI. Therefore, combining these together we have constructed an estimator with the correct mean and minimal variance in the limit of large n:

    û   = ∑_α
        |μ_α|
        /√(2) ||μ||^2
        
        .
        
        -N_α(n) + |μ_α|^2 τ_n^2
        /√(2) |μ_α| τ_n
        
       =
        2/F τ_n∑_α (-N_α(n)) + τ_n/2



Finally, we restrict the patterns α to the patterns counted in our simulations α^*, which results in the final estimator 

    û = 2/F_rτ_n∑_α^* (-N_α(n)^*) 
        + τ_n/2

where F_r = ∑_α^* |μ_α^*|^2 is the restricted Fisher information.












§ CONCLUSIONS AND OUTLOOK



In this paper we developed a computationally efficient strategy to estimate  dynamical parameters of a quantum Markov chain and provided strong theoretical evidence that the estimator achieves the quantum Cramér-Rao bound in the large time limit. In addition, we established asymptotic results pertaining to the mathematical structure of the output state which are of more general interest.


The estimation strategy consisted of two estimation stages. In the first stage a rough  estimator is computed from outcomes of simple output measurements by using a fast but non-optimal procedure, e.g. equating the empirical counts with the expected value at the estimated parameter. In the second stage, we used a coherent quantum absorber <cit.> to post-process the output <cit.>. When tuned to the true value of the system parameter, the absorber “reverts” the evolution such that, in the stationary regime, the post-processed output is identical to the input “vacuum” product state. On the other hand, small  mismatches between system and absorber parameters lead to slight rotations away from this product state, which can be detected by simple sequential measurements on the output “noise units”.
To achieve the perfect “null” setup it would seem natural to use the first stage estimator as absorber parameter, but as shown in <cit.>, this leads to non-identifiability issues and sub-optimal final estimators. Instead, we applied the displaced-null technique <cit.> which prescribes an extra parameter shift, calibrated to 
remove the non-identifiability issue while preserving the optimality of the sequential measurement.

The key theoretical contributions of this work are related to the understanding of the output state and the stochastic measurement process. We introduced the concept of translationally invariant modes (TIMs) of the output and showed how they generate a Bosonic algebra in the asymptotic limit. Each mode is labelled by a binary sequence called a “pattern”
and its creation operator is an average of shifted blocks consisting of tensor products operators. 
We then showed that when the mismatch between system and absorber parameter scales at the estimation rate n^-1/2, the restriction of the output state to the TIMs becomes a multi-mode coherent state whose displacement depends linearly on the mismatch (quantum Gaussian shift model). Moreover, we showed that the TIMs carry all the quantum Fisher information of the output state and are therefore the relevant quantum statistics of the problem. While homodyne is the standard optimal measurement for such models, 
in the presence of the additional parameter displacement the 
modes amplitudes become large and counting measurements become effectively equivalent to homodyne. Surprisingly, we discovered that the sequential counting measurement acts as a joint measurement of all TIMs number operators. Due to the proximity to the vacuum state, typical trajectories consist of a relatively small number of patterns separated by long sequences of 0s. We showed that for large times the patterns counts distribution converges to the Poisson distribution of the TIMs coherent state. These insights allowed us to devise a simple “pattern counting” estimator for estimating the unknown parameter.


Our discrete-time results open the way for fast and optimal continuous-time estimation strategies based on coherent absorber post-processing <cit.> and simple pattern counting estimation, as opposed to expensive maximum likelihood methods. Interesting and important topics for future investigations concerns the robustness of the pattern counting method with respect to various types of noises, improving the estimation accuracy for short times, extensions to multi-parameter models and the relationship between the general theory of quantum absorbers <cit.> and quantum estimation.  




Acknowledgements. This work was supported by the EPSRC grant EP/T022140/1. MG thanks Juan Garrahan, Mankei Tsang, Dayou Yang, Martin Plenio and Susana Huelga for fruitful discussions. F.G. has been partially supported by the MUR grant Dipartimento di Eccellenza 2023–2027 of Dipartimento di Matematica, Politecnico di Milano and by the INDAM GNAMPA project 2024 “Probabilità quantistica e applicazioni”.







§ DISPLACED-NULL MEASUREMENTS AND ASYMPTOTIC NORMALITY




In this section we review the concept of quantum local asymptotic normality (QLAN) in its simplest form involving a one-parameter family of pure states, and use it to better understand the asymptotic theory of SLD and displaced-null measurements. For the general QLAN theory we refer to <cit.> and <cit.>. For our purposes, QLAN expresses the fact that the statistical model |ψ_θ⟩^⊗ n describing an ensemble of n identically prepared systems can be approximated by a simpler quantum Gaussian model consisting of a coherent state of a continuous variables (CV) system, whose mean is related to θ by a linear transformation. Below, we will show that the SLD measurement and the null measurement in the multi-copy model correspond to measuring canonical variables and respectively displaced number operators in the Gaussian picture.


More precisely, let  
θ_0 be a fixed parameter value and write θ= θ_0 + u/√(n) where u is a local parameter describing the deviation from θ_0. The latter can be thought of as being the first stage estimator θ̃_n, and will be used as such later on, but for the moment is considered to be fixed and known. 
To simplify the presentation we assume that ⟨ψ_θ_0 |ψ̇_θ_0⟩= 0, which can always be satisfied by appropriately choosing the phase of |ψ_θ⟩ around θ_0. Let {|0⟩, … , |d-1⟩} be an orthonormal basis such that 
|0⟩ = |ψ_θ_0⟩. 
Then in the first order of approximation we can write the state as 

    |ψ_θ⟩   =   
    |0⟩  + u/√(n) |ψ̇_θ_0⟩
    +O(n^-1)
       =   
    |0⟩  + u/√(n)∑_j=1^d-1 c_j |j⟩ + O(n^-1),   
    c_j = ⟨ j |
    ψ̇_θ_0⟩ .

The SLD is given by 

    ℒ_θ_0  = 
    2(|ψ_θ_0⟩⟨ψ̇_θ_0 | + 
    |ψ̇_θ_0⟩⟨ψ_θ_0 |)

and the QFI is 
F_θ_0= 4ψ̇_θ_0^2 = 4 c^2, where c= 
(c_1,…, c_d-1)∈ℂ^d-1.

We now construct a Gaussian model which approximates the multiple copies model in the neighbourhood of θ_0. Let 
ℱ_d-1=ℱ^⊗(d-1) be the Fock space of d-1 CV modes with annihilation operators 
(a_1,… ,a_d-1), and let | n⟩ = ⊗_j=1^d-1 |n_j⟩ be the Fock basis, where n= (n_1, … , n_d-1)∈ℕ^d-1. We further denote by 
| z⟩ = ⊗_j=1^d-1 |z_i⟩ a coherent state with amplitude z= (z_1,… , z_d-1)∈ℂ^d-1 such that 
⟨ z |a_j| z⟩ =z_j. On the Fock space we consider the quantum statistical model with parameter 
u∈ℝ consisting of coherent states

    | cu⟩ = ⊗_j=1^d-1 |c_j u⟩.

We now show (in three different ways) that the multiple copies model  
|ψ^n_u⟩ := |ψ_θ_0+u/√(n)^⊗ n⟩, which capture the local properties of the original model around θ_0 in terms of the rescaled parameter u, is approximated by the Gaussian model | cu⟩. Firstly, one can show that the collective variables 

    Q^n_j   : =   1/√(2n)∑_i=1^n(|0⟩⟨ j| + |j⟩⟨ 0|)_j
    
    P^n_j   : =   1/√(2n)∑_i=1^n( -i|0⟩⟨ j| +i |j⟩⟨ 0|)_j

 converge in distribution to the canonical variables Q_j, P_j of the CV Gaussian model, with respect to the state | cu⟩. Secondly, the convergence holds on the geometric level, i.e. it can be expressed in terms of the overlaps 

    lim_n→∞⟨ψ^n_u| ψ^n_v⟩ =
    ⟨ cu |  cv⟩.

where u,v are fixed local parameters. The reader will recognise that for d=2 these statements are statistical reformulations of results from coherent spin states <cit.>. 




The third type of QLAN convergence is akin to the Le Cam strong convergence of statistical models in classical statistics <cit.>; it states that the multi-copy and Gaussian models can be mapped into each other operationally by means of quantum channels. Let 
𝒮_n be the symmetric subspace of 
(ℂ^d)^⊗ n and let V_n: 𝒮_n→ℱ_d-1 be the isometry 

    | n; n⟩↦ | n⟩

where | n; n⟩ is the normalised projection onto S_n of the basis vector (⊗_j=1^d-1 |j⟩^⊗ n_j )⊗|0⟩^n-| n|∈(ℂ^d )^⊗ n. Then the following holds for any 0<γ<1 <cit.>

    lim_n→∞sup_|u|≤ n^1-γ V_n |ψ^n_u⟩  - | cu⟩ =0

which means that the original model can be mapped isometrically (with no loss of information) into the Gaussian model with vanishing asymptotic error, uniformly over all the relevant local parameters. 


From the estimation perspective, QLAN can be used to devise an asymptotically optimal, two steps measurement strategy for the multi-copy model. 
We first compute a preliminary estimator θ̃_n by using a small sub-sample and set θ_0 =θ̃_n. Subsequently, we map the ensemble state into the (approximately) Gaussian one by applying the isometry V_n, and then measure the SLD a c^† +  c a^† of the Gaussian shift model, which is the limit of 1/√(n)∑_i=1^nℒ^(i)_θ_0 where ℒ_θ_0 is the SLD (<ref>). Joining the two estimation stages, we compute the final estimator as
θ̂_n = θ̃_n +û_n/√(n) where 
û_n = X/F_θ̃_n and X is the result of the second stage measurement. Similarly to  (<ref>), the estimator
θ̂_n defined above is optimal in the sense that it achieves the QCRB asymptotically, cf. equation (<ref>).




We now move away from SLD measurements and consider the QLAN perspective on displaced-null measurements. An important point here is that the parameter u of the Gaussian shift model |  cu⟩ is not completely arbitrary but can be considered to be bounded by n^ϵ, where n is the sample size of the qudit ensemble. Indeed, recall that 
the reference parameter θ_0 should be thought of as a preliminary estimator θ̃_n obtained by measuring a sub-sample of ñ= n^1-ϵ qudits. 
Assuming this has standard concentration properties i.e. |θ -θ̃_n|< n^-1/2 +ϵ with high probability, and writing θ= θ̃_n+u/√(n) we obtain |u|≤ n^ϵ <cit.>. 



While the collective SLD observable in the qudit model maps onto a quadrature of the limit model, a measurement in the null basis 
{ |e_0 ⟩≡|ψ_θ_0⟩, … ,|e_d-1⟩} corresponds to a simultaneous measurement of the number operators N_j = a_j^*a_j, for all the CV modes a_j. More precisely, in the limit of large n the joint distribution of the counts {N_j,n}_j=1^d-1 for outcomes corresponding to vectors {|e_j⟩}_j=1^d-1, converges to the joint distribution of the number operators {N_j = a_j^* a_j}_j=1^d-1 with respect to the coherent state | cu⟩, which is the product of independent Poisson distributions 
Poisson (|c_j|^2 u^2). Since the latter depends on u^2, the local parameter is non-identifiable so the measurement does not distinguish the parameters θ_±:= θ_0± u/√(n). Consequently, any final estimator has an error of the same order as the statistical error of the initial estimator, and is therefore far from optimal. Let us now change the reference point to 
θ_0 = θ̃_n - δ_n, with δ_n = τ_n/√(n) and τ_n = n^3ϵ. From this vantage point the limit model is 
 | c(u+ τ_n)⟩, and measuring in the displaced-null basis { |e_0 ⟩≡|ψ_θ_0⟩, … |e_d-1⟩} is again asymptotically equivalent to measuring the number operators {N_j}_j=1^d-1, with the difference that the limiting distribution is the product of Poisson(|c_j|^2(u+τ_n)^2). Since |u|<n^ϵ (with high probability) due to the preliminary estimation step, and τ_n≫ n^ϵ this means that u is uniquely determined by the measurement distribution.
 Moreover, for large n the Poisson distribution can be approximated by the Gaussian N(λ_i^n, λ_i^n ) where 
 λ_i^n= |c_i|^2(u+τ_n)^2. By expanding (u+τ_n)^2/τ_n and neglecting u^2/τ_n= O(n^-ϵ) we obtain that the rescaled variable
 
    Y_i,n = 1/|c_i|(N_i,n/τ_n -τ_n |c_i|^2)

converges in  distribution to N(2u|c_i|, 1). A simple signal to noise analysis shows that the best estimator of u is the linear combination

    û_n= 1/2 c^2∑_j=1^d-1 |c_j| Y_j,n = 1/2τ_n c^2∑_j=1^d-1N_j,n - τ_n/2

Its limiting distribution is N(u, F_θ^-1) where F_θ= 4 c^2 is the quantum Fisher information of the original model. In particular
the final estimator θ̂_n =θ̃_n+ û/√(n) achieves the QCRB in the sense of (<ref>).









§ PROOFS OF PROPOSITION <REF> 
AND COROLLARY <REF>
 




Proof of Proposition <ref>. Let α = (α^(1),…, α^(p)), β=(β^(1), … ,β^(q)) be the patterns with non-zero counts for n and respectively m; let (n_1, …, n_p) and   (m_1,…, m_q) be their counts, and let M be the maximum length of all patterns in α and β. 

Since pattern creation operators A_α^*(n) involve sums of σ^α_i for different positions i, the Fock state 
| n ;n⟩ is a superposition of vectors obtained by applying the following type of ordered products to the vacuum

    ∏_k=1^n_pσ^α^(p)_i_p,k…∏_k=1^n_1σ^α^(1)_i_1,k

where i_j,k is the index marking the location of the left end of the k-th pattern α^(j), with j∈{1,…, p} and k∈{1,… , n_j}. In general, some of these operators may `overlap' (act on the same spin) and the computation of the superposition becomes cumbersome.
However, since the total length of the patterns is finite, for large n, the main contribution in the superposition comes from arrangements in which there are no overlapping patterns. Even more, we can restrict to arrangements where the patterns are separated by at least M zeros, in which case each pattern  in the sequence of zeros and ones can be identified unambiguously. 








Therefore

    | n;n⟩
          =
    1/√(n^| n|)1/√( n!)∑_ i∈ℐ( n;n) ∏_j=1^p 
     ∏_k=1^n_j σ^α^(j)_i_j,k 
    |Ω_n  ⟩ + o(1) 
    
          =
    1/√(n^| n|)1/√( n!)∑_ i∈ℐ( n;n)  
    |ω( i,  n)⟩
    + o(1)

where n!:=n_1!… n_p! and  ℐ( n;n) is the subset of locations i={ i_j,k} leading to non-overlapping patterns with counts set n, such that all patterns are at distance of at least M from each other, and ω( i,  n)∈{0,1}^n is the basis vector (trajectory) obtained by placing the all the patterns corresponding to the counts set n at locations prescribed by i. More precisely i∈ℐ( n;n) if for any two pairs (j,k) and (j̃, k̃) with 
i_j,k≤ i_j̃,k̃ one has 
    i_j,k+ |α^(j)|+M-1 <i_j̃, k̃. The basis vector |ω( i,  n)⟩ consist of zeros except patterns α^(j) written at positions i_j,k for j∈{1,… ,p} and k∈{1,… , n_j}. 
    
    
    To show that the remainder term is o(1) note that any term in the superposition | n  ,n ⟩ corresponding to a specific product of σs is a vector of the standard basis, and any such vector 
    |ω⟩ has at most a fixed number M· | n| of 1s. Note also that the action of applying a pattern 
    σ^α_i to the vacuum cannot be reversed by applying subsequent σs, since these contain only creation operators. This means that for a given 
    |ω⟩, the locations i ={i_j,k} of the σs  producing this vector are limited to an area of size 2 M around each 1 in the sequence, or in other words, the coefficient 
    of |ω⟩ is bounded by (2 M)^M| n|. On the other hand, the number of basis vectors 
    |ω⟩ obtained by applying patterns such that at least two overlap or are at distance smaller than M from each other is o(n^| n|) since the number of possible locations for two such σs is O(n). Therefore, the remainder term in (<ref>) is o(1).
     



Using equation (<ref>) we obtain

    ⟨ m;n | n;n⟩=
          1/√(n^| n|· n^| m|· n! · m!)∑_ i , ĩ⟨ω( i,  n)|ω(ĩ,  m) ⟩ +o(1)

where the sum runs over 
i∈ℐ( n;n) and 
ĩ∈ℐ( m;n). Since each i and ĩ uniquely determines the patterns it contains, the basis vectors |ω( i,  n)⟩  and 
|ω(ĩ,  m) ⟩ have non-zero overlap (coincide) only if their sets of patterns coincide, 
i.e. n=  m. Therefore, if n≠ m  hold then

    lim_n→∞⟨ m;n | n;n⟩ =0.


On the other hand,
if m =  n then

    lim_n→∞⟨ n;n | n;n⟩ = 
    lim_n→∞1/n^| n| n! ( n!)^2
    |ℐ( n,n)|/ n! =1

where we took into account that each basis 
vector |ω ( i,  n)⟩ appears n! times in the sum over i∈ℐ( n;n), and that

    lim_n→∞|ℐ( n , n)|/n^| n| =1 .





Proof of Corollary <ref>. As shown in the proof of Proposition <ref> the "Fock state" | n; n⟩ can be approximated by a superposition of basis states in which the patterns in n are non-overlapping and are situated at least at a certain distance from each other. Moreover the remainder term contains o(n^| n|) basis vectors with bounded coefficients. After applying A^*_β(n), the multiplicity of each basis vector |ω⟩ is finite and the number of possible basis vectors is 
o(n^| n|+1). Taking account of the factor n^-(| n|+1)/2 we find that the action of A^*_β(n) on the 
o(1) remainder in (<ref>) is still o(1). On the other hand, the action on the main term in (<ref>) is to add a pattern β separated by the other patterns by max(M, |β|), with a negligible term coming from locations in which β overlaps with or is too close to one of the existing patterns. The factor √(n_β +1) comes from the definition of the "Fock state" | n+ δ^(β);n⟩.

The action of A_β(n) on one of the vectors |ω( i,  n)⟩ is to produce a superposition of basis vectors in which the pattern 
β has been removed from the set 
n of patters, at all possible locations. This may include removing part of a existing pattern α^(j) which coincides with β. These two case will produce orthogonal vectors and can be evaluated separately. In the first case, the pattern β is removed from one of the locations where such pattern existed. 

In the Fourier decomposition of 
√(n_β)| n-δ^β; n⟩, 
a basis vector 
|ω⟩ given by non-overlapping patterns has coefficient 

    c(ω) = 
    √(n_β) n!/n_β√(n· n_β)/√(n^| n| n!) =
     n! √(n)/√(n^| n| n!)

Such a vector can be obtained in approximately n ways by removing a pattern β from a basis vector which appears in the decomposition of 
| n; n⟩. Therefore its coefficient in A_β | n, n⟩ is approximately

    n!/√(n)· n·1/√(n^| n| n!)

where we took into account a factor 1/√(n) from the definition of A_β. We therefore obtain that the coefficients of the non-overlapping terms in A_β | n; n⟩ and 
√(n_β) | n -δ^β; n⟩ agree asymptotically. The fact that the o(1) terms remain small after applying A_β can be shown similarly to the above.












§ PROOFS OF THEOREM <REF> 
AND LEMMA <REF>
 



In order to prove the theorem, we will make use of the method of moments; this can be done since the moment problem corresponding to the moments of Poisson and Gaussian random variables admits a unique solution (this can be seen for instance using Cramér condition, see <cit.>).

Number operators. We will show the convergence of the moments of N_α(n) in the state ρ^ out_u,n to those of a Poisson random variable of intensity λ(u):=u^2 λ_α. We recall the the r-th moment of a Poisson random variable of intensity λ(u) is equal to ∑_m=1^r S(r,m) λ(u)^m where S(r,m) are the Stirling numbers of the second type: a combinatorial interpretation of S(r,m) is the number of partitions in m non-empty subsets of a set of cardinality r. 





Let us focus on the expression of the r-th moment of N_α(n). For simplicity we 
denote ⟨ X⟩_u := ρ̃^ out_u,n(X). From (<ref>) we have

    ⟨ N_α(n)^r ⟩_u=1/n^r∑_i_1,…,i_r=1 
     j_1,…,j_r=1^n-|α|+1⟨σ^α_i_1σ^α*_j_1⋯σ^α_i_rσ^α*_j_r⟩_u.


Splitting of the sum based on non-overlapping groups of σs. Let us consider a term σ^α_i_1σ^α*_j_1⋯σ^α_i_rσ^α*_j_r and represent each σ^α_i or σ^α*_i  as a block of length |α| covering positions {i, i+1, … i+|α|-1} of the string {1,2,… n}.

Depending on the overlapping pattern of the blocks, the indices can be split (uniquely) in a number s= s(i_1, j_1,… ,i_r, j_r) of groups (1 ≤ s ≤ 2r) such that blocks in different groups do not overlap, and each group cannot be split into further non-overlapping sub-groups. Among these groups we identify  
g =g(i_1, j_1,… ,i_r, j_r) special groups characterised by the fact that they are made up of one or more pairs of blocks of the type (σ_z^α*σ^α_z) for some z. 

We call P_0 the set of such groups. Note that not all groups associated to a product of σs can be in 𝒫_0 because the order of σ_z^α* and σ_z^α in the latter is opposite to that in which such terms appear in N_α(n).

For example, consider the pattern α="11" for n=10 and the term

    σ^α_8σ_1^α *σ^α_6 σ^α*_8 σ^α_1 σ_2^α^*σ_2^ασ_5^α*

This has 3 non-overlapping groups of operators (that commute with each others): the first one

    σ^α*_1σ_1^ασ^α*_2  σ_2^α

belongs to P_0, while the second and  third ones 

    σ^α_6σ^α*_5
    
        and    σ^α_8 σ_8^α*

are not in P_0. Therefore s=3 and g=1.


Let us now look at the expected value of a given product of σs. We have

    ⟨σ^α_i_1σ^α*_j_1⋯σ^α_i_rσ^α*_j_r⟩_u=
    ρ^ ss_u ( C_u,1𝒯̃^x_1_u⋯𝒯̃^x_s-1_u C_u,s (1) )

where x_i's are the distances between the non-overlapping groups and C_u,i is the map corresponding to the i-th group, which is computed according to the following rule. In  every group we multiply the operators on a given position (we may have several σ^+, σ^- or σ^0 on one position) and the result will be an element of the set 𝒪:= {σ^+, σ^-, σ^0, |0⟩⟨0|, |1⟩⟨1|,  0}. For example the first group above gives

    σ^α*_1σ_1^ασ^α*_2  σ_2^α 
    =(|0⟩⟨0|)_1(|0⟩⟨0|)_2(|0⟩⟨0|)_3

while

    σ^α_6σ^α*_5
    =σ^-_5(|1⟩⟨1|)_6 σ^+_7

and 

    σ^α_8 σ_8^α*
    =(|1⟩⟨1|)_8(|1⟩⟨1|)_9.

Suppose the result of this computation is 
O^1_i⋯ O^k_i+k-1, with O^k∈𝒪; then in the expectation, this translates into a  superoperator 
C_u =  C_u[O^1]∘…∘ C_u[O^k] obtained by composing in the same order basic maps C_u[O] defined as follows

    C_u [σ^+]:=  A_u,1,    C_u [σ^0]:=  A_u,0,    C_u [σ^-]:=  A_u,-1,
        C_u [0]:= 0,
     C_u [|j⟩⟨j|]:=  B_u,j,    j∈{0,1}.

where A_u,i are defined as in equation (<ref>) by replacing K̃_i with K̃_u,i
and

    B_u,j(x)=K̃^*_u,1 xK̃_u,1    j=1 
    K̃^*_u,0xK̃_u,0    j=0 
    .

In what follows we will drop the label u when u=0.










We can now compute the expectation of our example product of σs as

    ⟨σ^α_8σ_1^α *σ^α_6 σ^α*_8 σ^α_1 σ_2^α^*σ_2^ασ_5^α*⟩_u=
       ρ^ ss_u (  B_u,0^3𝒯̃_u A_u,-1 B_u,1 A_u,1 B^2_u,1(1) ).



Let us define the space

    1^⊥:={x ∈ M(ℂ^D): ρ̃^ ss(x)=0},

and note that the following properties hold true:

    
  * A_0 and B_0 leave 1^⊥ invariant;
    
  * the image of A_i and B_j  for i,j ≠ 0 is contained in 1^⊥;
    
  * for every m ≥ 1, the norm of A^m_0=𝒯̃^m restricted to 1^⊥ is less or equal than cλ^m for some c>0 and 1>λ>0. 
    


The reason why we singled out groups in P_0 is given by the following key observation: such groups are the only ones for which  the corresponding map C_u,i is only composed by A_u,0s and B_u,0s. Any other group will contain at least one 𝒜_u,± 1 or ℬ_u,1 factor.  From this and the properties above it follows that if 𝒞 is not in 𝒫_0 then 

    𝒯̃^m ∘𝒞≤ cλ^m.



We will often identify the maps with the corresponding patterns saying, for instance, that the map is in P_0 when the corresponding group is. 




We will prove the convergence of moments by expanding in Taylor series in u and showing the convergence at each order. 
The moment of order r of the Poisson distribution with intensity 
u^2λ_α is 

    m_r = ∑_k=1^r S(r, k) u^2kλ_α^k

where S(r, k) are the Stirling numbers of second kind; the mth derivative with respect to u at u=0 is 
S(r, m/2)m!λ_α^m/2 for m even, and zero otherwise. 

Recall that each term in (<ref>) gives rise to 
a a certain set of groups of indices made up of overlapping blocks, and each group corresponds to a map  C_u,i such that the expectation is expressed as in equation in (<ref>). We will show that in limit of large n, the only terms which contribute to the m-th derivative are those coming from certain configurations with s-g=m where s is the number of groups and g is the number of groups in 𝒫_0. 

 
Taylor approximation for a given set ( C_u,1, … C_u,s) up to order m=s-g.
 Let us consider the sum of all the terms in Eq. (<ref>) coming from all the correlations corresponding to a given sequence of maps 𝒞:=
 ( C_u,1, … , C_u,s). This contribution is given by the product between a combinatorial factor (independent of n) counting how many products of σs in (<ref>) lead to the same set of maps 
 𝒞 and the following sum

    1/n^r∑_x_0+… +x_s = n-Kρ̃^ ss_u
     ( C_u,1𝒯̃^x_1_u⋯𝒯̃^x_s-1_u C_u,s (1) ),

where K is the total length of all the s blocks in  𝒞, which is smaller than 2|α| r. 
Note that the factors 𝒯̃^x_0_u and 
𝒯̃^x_s_u have been suppressed due to stationarity but the indices x_0, x_s are still present in the sum. We denote by g the number of maps in ( C_u,1, …,  C_u,s) that belong to P_0. 

Let us now consider the Taylor expansion of the correlations in (<ref>). The 0-th order term is 0 because at least one C_i is not in P_0 and hence ρ̃^ ss annihilates the result. This is because on one hand, C_i*(ρ̃^ ss)=ρ̃^ ss for all C_i∈𝒫_0, but on the other hand at least one  C_i is not in 𝒫_0 and hence it contains a term 𝒜_±1 or ℬ_1 for which 𝒜_± 1 *(ρ̃^ ss)=ℬ_1* (ρ̃^ ss) =0 since K̃_1|χ^ ss⟩ =0. 

Before addressing in detail the derivatives of (<ref>) we make some remarks concerning the magnitude of the sum. Note that this contains O(n^s) terms which are uniformly bounded and the largest possible value of s is 2r (all patterns are non-overlapping). On the other hand the sum is preceded by the factor n^-r and any derivative further multiplies it by n^-1/2 since all operators depend on u via u/√(n). These arguments alone are not sufficient to deduce the convergence of (low order) derivatives, at least for configurations 𝒞 with large s. 

The key additional ingredient is the fact that for 𝒞_i∉𝒫_0, the factors 𝒯̃^x_i𝒞_i are exponentially decreasing, cf. equation (<ref>). This will provide more conservative upper bounds for the derivatives of the sum (<ref>), as detailed below. Before considering that, note that the exponential bound can also be used to get an alternative proof of the convergence to 0 of

    1/n^r∑_x_0+… +x_s = n-K C_u,1𝒯̃^x_1_u⋯𝒯̃^x_s-1_u C_u,s (1).

Indeed, by summing over x_is for groups not in 𝒫_0,  we get that the sum (<ref>) is O(n^g/n^r) rather than O(n^s/n^r). Since in any configuration 𝒞 , at least one group is not in 𝒫_0, and each group in 𝒫_0 has at least two of the original 2r blocks, we see that g<r so that the whole sum is O(n^-1). The upshot for derivatives will be that in order to create contributions that do not decay, the derivatives have to be applied in an "efficient" way, namely to factors of the type 𝒯^x_i∘𝒞_i with 
𝒞_i not in 𝒫_0. 
This will break the exponential decay and allow for the balancing of the terms in the derivative with the pre-factor n^-r.


 



Consider now the case of the first order derivative of the sum (<ref>). This will split into a sum of sub-sums, one for each of the terms 𝒞_u,i or 
𝒯^x_i_u that are differentiated. Each sub-sum will be shown to have a decaying contribution to the derivative.



  i) 
If a term 𝒞_u,i in 𝒫_0 is differentiated, then the sub-sum is similar to the original sum with the difference that 𝒞_u,i is replaced by another bounded term 𝒞̇_i). The overall contribution is then of the order n^-r× n^g× n^-1/2 which decays since g<r.


  ii) 
If a term 𝒯̃^x_i-1_u is differentiated, for which 𝒞_u,i is in 𝒫_0 then the product 𝒯̃^x_i-1_u 𝒞_u,i becomes the sum 

    S_i = ∑_l=1^x_i-1𝒯̃
    ^x_i-1-l𝒯̇̃̇𝒯̃^l-1 C_i(·)

By ergodicity 𝒯̃^l-1(𝒞_i(·)) converges exponentially fast to ρ̃^ ss(𝒞_i(·))1 for large l so 

    S_i = 
    ∑_l^'=1^x_i-1𝒯̃^l^'(𝒯̇̃̇(1)) ·ρ̃^ ss(𝒞_i(·)) + O(1)

By the same argument 𝒯̃^l^'𝒯̇̃̇(1) converges to ρ̃^ ss(𝒯̇̃̇(1)) = 0 for large values of l^'. The latter follows from differentiating ρ̃^ ss_u (𝒯̃_u(·)) = ρ̃^ ss_u(·). Therefore S_i is O(1) which does not change the magnitude of the overall sum before differentiation, so this contribution decays as well, as argued in point i).


  iii) 
If a term 𝒯̃^x_i-1_u is differentiated, for which 𝒞_u,i is not in 𝒫_0, one obtains a sum S_i as in equation (<ref>). Since 𝒞_i leaves 1^⊥ invariant we have 
𝒯̃^l ∘𝒞_i≤ c λ^l so the terms with large l become negligible while for small l (large x_i-1-l) 
𝒯̃^x_i-1-l(·) converges to 
ρ̃^ ss(·)1. Therefore, using 
∑_l^' =0^∞𝒯̃^l^'𝒞_i = ℛ̃𝒞_i where ℛ is the Moore-Penrose inverse of Id- 𝒯̃, we obtain

    S_i=ρ^ ss(𝒯̇̃̇ℛ̃ C_i(·)) 1+o(1).

Hence differentiating 𝒯̃^x_i-1_u for which 𝒞_u,i is not in 𝒫_0 removes the exponential decay associated to un-differentiated  term 𝒯̃^x_i-1_u𝒞_u,i so that the upper bound for this contribution to the derivative is 
n^g+1n^-1/2n^-r. Since g<r we have 
g+1 -1/2-r<0 so the contribution to the derivative 
decays (even though it decays slower than the previous sub-sums, which will be important when considering higher derivatives).






  iv)
If a term 𝒞_u,i not in 𝒫_0 is differentiated then for large x_i-1

    𝒯̃^x_i-1Ċ_i(·)=ρ̃^ ss(Ċ_i(·))1 + o(1)

This means that such a derivative breaks the exponential decay of the product 𝒯̃^x_i-1 C_i but the contribution to the sum is still bounded by  
n^g+1n^-1/2n^-r.




The upshot of the argument for the first order derivative is that differentiating a 𝒞_u,i in 𝒫_0 or the factor 𝒯^x_i-1 in front of it, does not increase the magnitude of the sum and the contribution to the overall sum decays. On the other hand, differentiating a 𝒞_u,i which is not in 𝒫_0 or the factor 𝒯^x_i-1 in front of it, breaks the overall exponential decay of the product and contributes with an additional factor n^1/2 compared to the un-differentiated term.



We now consider higher order derivatives. 
One can again consider the possible positions where derivatives are applied and evaluate their separate contributions to the derivative. Following the same argument as in point i) above, one can see that differentiating a term 𝒞_u,i in 𝒫_0 once or multiple times does not bring any change compared to the term before differentiation. Similarly, as in point ii) one finds that differentiating 𝒯^x_i-1_u in front of 𝒞_u,i which is in 𝒫_0, once or multiple times does not bring any change (this is due to the fact that d^k/du^k𝒯_u^x(1)=0 for every k ≥ 1). Now we move our attention to terms 𝒯^x_i-1_u𝒞_u,i for which 𝒞_u,i is not in 𝒫_0. As in cases iii) and iv) above, differentiating either one of the two factors will break the exponential decay and bring an extra overall multiplicative factor n^1/2 compared to the un-differentiated terms. However if any of the two factors is differentiated further (more than one derivative in the product 𝒯^x_i-1_u𝒞_u,i) then the additional derivative does not change the bound except for the multiplicative factor n^-1/2 due to differentiation. This shows that in order to obtain contributions that do not vanish asymptotically, the derivatives have to be placed 
on the products 𝒯^x_i-1_u𝒞_u,i for which 𝒞_u,i is not in 𝒫_0, with at most one derivative for each product.


Recall that we are considering derivatives up to order 
m=s - g which is equal to the number of 𝒞_u,i not in 𝒫_0. According to the argument above, the most favourable positions for the derivatives is on different terms 𝒯^x_i-1_u𝒞_u,i. Therefore, for a derivative of order k≤ m the entire derivative can be upper bounded as 

    n^-r× n^-k/2× n^g× n^k =
    n^-r+g+k/2

Since k≤ m = s-g the exponent is smaller than 
t:= -r+g+ (s-g)/2. We will show that t≤ 0 with equality if and only if m= s-g and the configuration 𝒞 is such that all groups in 𝒫_0 consists of 2 perfectly overlapping σs and the groups not in 𝒫_0 consists of single blocks. Indeed for given total number 2r of σ 
blocks, s+g is maximum if all groups in 𝒫_0 have just two σ blocks and all others have a single block, so 2g+ s-g = s+g ≤ 2r, which implies t ≤ 0. In conclusion, for any configuration 𝒞, the derivatives below order m=s-g decay asymptotically, and that of order m does as well unless 𝒞 is of the special type described above. Derivatives of order above m will be treated below. Note that for the special configurations, m is even so all odd derivatives decay, as is expected from the fact that the intensity of the limit Poisson distribution is proportional to u^2. We will now compute the 
limit of the mth derivative for the special configuration. 

If 𝒞_i∈𝒫_0 consists of two blocks of σs then the corresponding product is

    σ^α *σ^α= O_α_1… O_α_|α|

where 
α=(α_1, …, 
α_|α|) and O_0 := 1,  
O_1= |0⟩⟨ 0|. Therefore 

    𝒞_i = C^(α_1)… 
    C^(α_|α|)

with C^(0) = 𝒜_0 = 𝒯 and C^(1)= ℬ_0.
Therefore when x_1,…, x_s are large the following holds

    𝒯^x_i-1 C_i(·) = 
    ρ^ ss( C_i (·)) 1
     + o(1) =  ρ^ ss(· ) 1 + o(1)

since 
ρ^ ss∘𝒜_0 =ρ^ ss and ρ^ ss∘ℬ_0 =ρ^ ss
This means that for large x_1,…, x_s the derivative of order 
m=2(m/2) factorises as 

    |ρ^ ss(𝒯̇ℛ A^α(1)+ρ^ ss(Ã^α(1))|^2(m/2)

where Ã^α is obtained from A^α differentiating the first A_1. This formula follows from the contributions obtained at points iii) and iv) above and the fact that the groups that are differentiated contain a single σ block.

We remark that the term for which we computed the limit appears in the m-th derivative with a factor m! in front.

Finally, we need to compute the combinatorial factor counting the number of terms in the expectation which produce the desired set 𝒞 consisting of g= r-m/2 groups of two σs in 𝒫_0 plus 
m groups of single σs (with s=r+m/2 total number of groups). 

We will show that this is exactly S(r,m/2) (Stirling number of second type) i.e. the number of partitions of a set of r elements into m/2 non-empty subsets. For this it is enough to show that the numbers of ways in which we can pair the σs is in a bijection with the partitions in m/2 classes of r elements. Consider the collection of σs

    σ_i_1^ασ_j_1^α*⋯σ_i_r^ασ_j_r^α*,

which, for our purposes, we can identify with the set of "dipoles"

    (α, α*)_1, ⋯, (α, α*)_r

In order to create a C in P_0, we need to pair a σ_j_z^α* with a σ_i_w^α such that z<w. This induces an equivalence relation on the collection of dipoles where we identify two dipoles (α, α*)_z, (α, α*)_w if j_z=i_w and we impose transitivity of this relation. By construction,  in each equivalence class 
{ (α, α*)_z_1,(α, α*)_z_k} we have j_z_l = i_z_l+1 so all positions of α*s are equal to the position of α in the next dipole. In other words, an equivalence class uniquely determines a set of pairs of equal indices, and altogether the set of equivalence classes uniquely determine the splitting into 𝒫_0 groups. Since i_z_1 and j_z_k are the only indices that are not paired in the chosen equivalence class, the number of classes is m/2, i.e. half of the number of groups not in 𝒫_0. The number of ways to group the r dipoles in m/2 equivalence classes is the Stirling number of second type S(r,m/2) which provided the combinatorial factor for our mth order derivative.

With this we conclude that the mth order derivative of (<ref>) converges to the corresponding derivative of (<ref>). 



Reminder of the Taylor approximation (order s-g+1). The last thing we need to check is that the remainder corresponding to the sum of all the terms in Eq. (<ref>) with s non-overlapping blocks is negligible; the reminder is given by

    u^m^'/m^'!n^r+m^'/2∑_x_1+… +x_s = n-K .d^m^'/du^m^'f_x_1,…,x_s(u) |_u=η_x_1,…,x_s-1

for some |η_x_1,…,x_s-1|≤ |u|/√(n), where m^'=s-g+1 and

    f_x_1,…,x_s(u)=ρ^ ss_u ( C_1,u𝒯^x_1_u⋯𝒯^x_s-1_u C_s,u (1) ).

Notice that r+m^'/2>s: indeed,

    r+s-g+1/2>s ⇔ 2r+1>g+s.

Therefore, it is enough to show that

    .d^l+1/du^l+1f_x_1,…,x_s(u) |_u=η_x_1,…,x_s-1,    2 ≤ l ≤ 2r

are uniformly bounded: the only terms that requires some care are the derivatives of 𝒯^x_u, which are of the type

    ∑_0≤ l_1 ≤⋯≤ l_k ≤ x𝒯_η^x-l_k𝒯_η^(m_k)⋯𝒯_η^(m_1)𝒯_η^l_1-1,

where 𝒯^(m)_η stays for the m-th derivative of 𝒯_u evaluated at η. Using

    
  * the spectral decomposition of 𝒯_η(·)=ρ^ ss_η(·)1+ℛ_η, with ρ^ ss_η(ℛ_η(·))=0 and ℛ_η≤λ<1 (for n big enough) and
    
  * the fact that 𝒯^(m)(1)=0 (m ≥ 1),

we have that

    ∑_0≤ l_1 ≤⋯≤ l_k ≤ x𝒯_η^x-l_k𝒯_η^(m_k)⋯𝒯_η^(m_1)𝒯_η^l_1-1= 
       ∑_0≤ l_1 ≤⋯≤ l_k ≤ x𝒯_η^x-l_k𝒯_η^(m_k)ℛ_η^l_k-l_k-1-1⋯𝒯_η^(m_1)ℛ_η^l_1-1≤
       C∑_0≤ l_1 ≤⋯≤ l_k ≤ xλ^l_k,

which is bounded.




















  
  
  



 
    



























Quadratures. For the sake of keeping notation simple, we will show the proof in the case of z=1, but the same reasoning applies to the general case inserting z and z where needed. We recall that k=|α|.

First of all, notice that the mean of Q_α(n) converges to uμ_α,1: indeed, its first moment is given by

    1/√(n)∑_i=1^n-k+1ρ̃^ ss(u/√(n)) (   √(2)( A^α) (1) )  → u μ_α,1.


In order to simplify the proof, we consider the standardised random variable

    √(2) (Q_α(n) - n-k+1/nu μ_α, 1 )=
       A^*_α(n)+A_α(n)-2(μ_α)u=
       1/√(n)∑_i=1^n-k+1σ̃^α_i(u) + σ̃^α*_i(u),

where

    σ̃^α(u)=σ^α-u/√(n)ρ((𝒯̇ℛ A^α+Ã^α)(1))1.
 In order to prove the statement, we need to show that the sequence of standardized quadratures converges in law to a standard Gaussian random variable.


For r ≥ 2, the r-th moment of the standardised random process at time n has the following expression:

    ∑_j_1,…,j_r ∈{α,α*}1/n^r/2∑_i_1,…,i_r =1^n-k+1⟨σ̃^j_1_i_1⋯σ̃^j_r_i_r⟩_u/√(n).


Consider the correlation corresponding to a choice of indices i_1,…,i_r such that there are exactly s non-overlapping groups and let g be the number of groups with overlapping terms of the form σ̃_z^α*(u)σ̃^α_z(u) (as before, we denote the set of such groups as P_0). As before, these are the only overlapping groups that at u=0 will produce an operator which is not in 1^⊥.

First suppose that s-g>0. We can prevent the maps which do not belong to P_0 to be annihilated by ρ and to cause an exponential decay differentiating and repeating the same computations as in the case of number operators. If we consider the m-th term in the Taylor expansion up to m=s-g, we can see that it grows at most as n to the power g+m/2-r/2; if we want the exponent to be bigger or equal than 0, we need that m≥ r-2g; since m≤ s-g, one realises that g+m/2-r/2 can at most be equal to 0 and this is true when m=s-g and s+g=r, which means that every group in P_0 is of the form σ̃_z^α*(u)σ̃^α_z(u) and all the other groups are singletons. However, if a group is composed by a single element, since we centered the random process, the first derivative will not be enough to cancel the exponential decay. Therefore, if s-g>0, the corresponding terms will decrease to 0.

On the other hand, if r is even and s=g=r/2, the 0-th order term is equal to 1, hence the leading term comes from the case when r is even and g=r/2. In this case one can see that the limit quantity is equal to 1 times the way we can pair the σ̃(u)'s in groups of the type σ̃_z^α*(u)σ̃^α_z(u) and this is given by (r-1)!! (which is the number of partition into pairs of a set of r elements).

The reminder can be controlled as in the case of number operators.





Alternative expression for μ_α. 
Using that K̃_0|χ^ ss⟩=|χ^ ss⟩ and K̃_1|χ^ ss⟩=0, one can write

    𝒯̇̃̇_*(ρ̃^ ss)=   ∑_i=0^1K̇̃̇_i|χ^ ss⟩⟨χ^ ss|K̃_i^* + K̃_i|χ^ ss⟩⟨χ^ ss|K̇̃̇^*_i
    
    =   |K̇̃̇_0χ^ ss⟩⟨χ^ ss| + |χ^ ss⟩⟨K̇̃̇_0χ^ ss|.


Under the "gauge condition (<ref>) 

and using the explicit expression of K̃_θ,i's in Eq. (<ref>) we obtain ⟨χ^ ss|K̇̃̇_0 χ^ ss⟩=0.


 



From 
K̃_0^*K̃_0 + K̃_1^*K̃_1 = 1 we obtain K̃_0^*|χ^ ss⟩ = |χ^ ss⟩.
Therefore 

    K̃_0 = |χ^ ss⟩⟨χ^ ss| + P^ ss_⊥K̃_0 P^ ss_⊥
 
where P^ ss_⊥ = 1- |χ^ ss⟩⟨χ^ ss|, which implies

    ∑_k ≥ 0K̃_0^k|K̇̃̇_0 χ^ ss⟩= |(1-K̃_0)^-1K̇̃̇_0 χ^ ss⟩

and

    ℛ̃_*
    𝒯̇̃̇_*
    (ρ̃^ ss)
       =   |(1-K̃_0)^-1K̇̃̇_0 χ^ ss⟩⟨χ^ ss|
       +   |χ^ ss⟩⟨(1-K̃_0)^-1K̇̃̇_0 χ^ ss|.

When we evaluate it against
A^α(1), the first term in the previous equation gets killed, while the second one produces the term

    ⟨K̃_α_|α|⋯K̃_α_1(1-K̃_0)^-1K̇̃̇_0 χ^ ss | χ^ ss⟩.

The rest of the proof is just a trivial check.

Expression for the total intensity. Since λ_α≥ 0, one has that ∑_αλ_α =C ∈ [0,+∞] and the limit is always the same irrespectively of the choice of partial sums. Notice that

    λ_(1)   =   |⟨ (K̃_1(1-K̃_0)^-1K̇̃̇_0 + K̇̃̇_1)χ^ ss|χ^ ss⟩|^2
       =   (|χ^ ss⟩⟨χ^ ss|Y),

where

    Y   :=   |(K̃_1(1-K̃_0)^-1K̇̃̇_0 +K̇̃̇_1)χ^ ss⟩
          ⟨(K̃_1(1-K̃_0)^-1K̇̃̇_0 +K̇̃̇_1)χ^ ss|.


For any α such that |α|≥ 2, λ_α is equal to:

    | ⟨K̃_α_|α|⋯K̃_α_2 (K̃_α_1(1-K̃_0)^-1K̇̃̇_0 +K̇̃̇_1)χ^ ss | χ^ ss⟩|^2=
       | ⟨K̃_1K̃_α_|α|-1⋯K̃_α_2 (K̃_1(1-K̃_0)^-1K̇̃̇_0 +K̇̃̇_1)χ^ ss | χ^ ss⟩|^2=
       (X K̃_α_|α|-1⋯K̃_α_2YK̃^*_α_2⋯K̃^*_α_|α|-1 ),

where X:=|K̃_1^*χ^ ss⟩⟨K̃_1^*χ^ ss| and Y is the same as in Eq. (<ref>).

Therefore

    ∑_ 2 ≤ |α| ≤ Nλ_α= (∑_k=0^N-1𝒯̃^k(X)Y  )

and

    lim_N→ +∞∑_ 2 ≤ |α| ≤ Nλ_α=∑_ 2 ≤ |α|λ_α=(ℛ̃(X)Y),

therefore

    ∑_αλ_α =(( Id+ℛ̃ B_1)(|χ^ ss⟩⟨χ^ ss|)Y).

Let us massage a little the expression we obtained:

    (( Id+ℛ̃ B_1)(|χ^ ss⟩⟨χ^ ss|)Y)=
       ⟨χ^ ss|Yχ^ ss⟩ + ⟨χ^ ss| B_1*ℛ̃_*(Y-(Y)|χ^ ss⟩⟨χ^ ss|) χ^ ss⟩=
       ⟨χ^ ss|Yχ^ ss⟩ + ⟨χ^ ss|𝒯̃_*ℛ̃_*(Y-(Y)|χ^ ss⟩⟨χ^ ss|)χ^ ss⟩-
       ⟨χ^ ss|ℛ_*(Y-(Y)|χ^ ss⟩⟨χ^ ss|)χ^ ss⟩=
       ⟨χ^ ss|Yχ^ ss⟩ -⟨χ^ ss|(Y-(Y)|χ^ ss⟩⟨χ^ ss|) χ^ ss⟩=(Y).

We used the fact that B_0(|χ^ ss⟩⟨χ^ ss|)=|χ^ ss⟩⟨χ^ ss| and that 𝒯̃_*ℛ̃_*=ℛ̃_*- Id. Finally,

    (Y)=(K̃_1(1-K̃_0)^-1K̇̃̇_0+K̇̃̇_1)χ^ ss^2






§ PROOF OF COROLLARY <REF>
 





We will use the expression of ∑_αλ_α given by

    -(⟨χ^ ss, 2K̇̃̇_0(1-K̃_0)^-1K̇̃̇_0+K̃_0 χ^ ss⟩ )

as states in equation (<ref>) in Theorem <ref>. The proof of this identity can be found in Appendix <ref>; the expression we use here has the advantage that it immediately shows that ∑_αλ_α does not change for different choices of the postprocessing. Before proceeding, we recall the expression of all the terms appearing in the previous equation using quantities of the dynamics of the system alone.



  * |χ^ ss⟩=∑_i=1^d√(λ_i)|φ_i⟩_S ⊗|φ_i⟩_A, where ρ^ ss=∑_i=1^dλ_i |φ_i⟩⟨φ_i| is the spectral resolution of the stationary state of the system dynamics;

  * The Kraus operator K̃_0 is uniquely determined as follows (left tensor is the system)

    K̃_0=∑_k=0^1∑_i,j=1^d√(λ_j/λ_i)⟨φ_j,K_k^*φ_i ⟩ K_k ⊗|φ_i⟩⟨φ_j|.


  * The derivative of K̇̃̇_0 is (where we keep in mind than only the system unitary depends on θ)

    K̇̃̇_0=∑_k=0^1∑_i,j=1^d√(λ_j/λ_i)⟨φ_j,K_k^*φ_i ⟩K̇_k ⊗|φ_i⟩⟨φ_j|,


  * The second derivative is

    K̃_0=∑_k=0^1∑_i,j=1^d√(λ_j/λ_i)⟨φ_j,K_k^*φ_i ⟩K_k ⊗|φ_i⟩⟨φ_j|.



Note that

    ⟨χ^ ss, K̃_0 χ^ ss⟩   = ∑_k=0^1∑_i,j=1^dλ_j ⟨φ_i, K_k φ_j ⟩_S ⟨φ_j,K_k^* φ_i ⟩_S
       =∑_k=0^1(K_k ρ^ ssK^*_k).

Hence, using that ∑_k=0^1K_k^*K_k +K_k^*K_k +2K̇^*_kK̇_k=0, one has

    -(⟨χ^ ss, K̃_0 χ^ ss⟩)=∑_k=0(ρ^ ssK̇^*_kK̇_k).

Let us consider the rest of the total intensity: using that (1-K̃_0)^-1K̇̃̇_0|χ^ ss⟩=∑_l=0^+∞K̃_0^lK̇̃̇_0|χ^ ss⟩, one gets


    ⟨χ^ ss, K̇̃̇_0(1-K̃_0)^-1K̇̃̇_0 χ^ ss⟩
          =∑_i,j=1^d∑_a,b=0^1∑_l=0^+∞∑_k_1,… k_l=0λ_j ⟨φ_i,K̇_a K_k_l⋯ K_k_1K̇_b φ_j ⟩·
          ·⟨φ_j,K_b^* K^*_k_1⋯ K_k_l^* K_a^* φ_i ⟩
          
        =∑_a,b=0^1∑_l=0^+∞∑_k_1,… k_l=0^1(K̇_a K_k_l⋯ K_k_1K̇_b ρ^ ssK_b^* K^*_k_1⋯ K_k_l^* K_a^* )
          = ( ∑_a=0^1K^*_a K̇_a ℛ ( ∑_b=0^1K̇_b ρ^ ss K_b^* ) ).

Therefore,

    -2(⟨χ^ ss, K̇̃̇_0(1-K̃_0)^-1K̇̃̇_0 χ^ ss⟩)
          =2  ( (∑_a=0^1K^*_aK̇_a  ) ℛ ( ( ∑_b=0^1K̇_b ρ^ ss K_b^* )  )  ).





§ PROOF OF THEOREM <REF> AND PROPOSITION <REF>
 



First of all, let us show that λ_ tot given by Eq. (<ref>) is equal to ∑_αλ_α, whose expression is given by equation (<ref>). We have

    (K̃_1(1-K̃_0)^-1K̇̃̇_0+K̇̃̇_1)χ^ ss^2=
       ⟨ (1-K̃_0)^-1K̇̃̇_0χ^ ss,K̃_1^*K̃_1 (1-K̃_0)^-1K̇̃̇_0χ^ ss⟩
       +⟨ (1-K̃_0)^-1K̇̃̇_0χ^ ss,K̃_1^*K̇̃̇_1 χ^ ss⟩
       +⟨χ^ ss,K̇̃̇^*_1K̃_1 (1-K̃_0)^-1K̇̃̇_0 χ^ ss⟩
       +⟨χ^ ss,K̇̃̇_1^*K̇̃̇_1 χ^ ss⟩.

Note that

    ⟨χ^ ss,K̇̃̇_1^*K̇̃̇_1 χ^ ss⟩=1/2⟨χ^ ss,  B_1(1) χ^ ss⟩=-1/2⟨χ^ ss,  B_0(1) χ^ ss⟩
       =-(⟨χ^ ss,K̃_0 χ^ ss⟩ )-K̇̃̇_0 χ^ ss^2.

Moreover, since K̃_0^*K̃_0 +K̃_1^*K̃_1 = 1,

    ⟨ (1-K̃_0)^-1K̇̃̇_0χ^ ss,K̃_1^*K̃_1 (1-K̃_0)^-1K̇̃̇_0χ^ ss⟩
       =(1-K̃_0)^-1K̇̃̇_0χ^ ss^2-K̃_0(1-K̃_0)^-1K̇̃̇_0χ^ ss^2.

As for the other terms, one has

    ⟨ (1-K̃_0)^-1K̇̃̇_0χ^ ss,K̃_1^*K̇̃̇_1 χ^ ss⟩
       =⟨ (1-K̃_0)^-1K̇̃̇_0χ^ ss,Ḃ_1(1) χ^ ss⟩
       =-⟨ (1-K̃_0)^-1K̇̃̇_0χ^ ss,Ḃ_0(1) χ^ ss⟩
       =-⟨K̇̃̇_0(1-K̃_0)^-1K̇̃̇_0χ^ ss,χ^ ss⟩
       -⟨K̃_0(1-K̃_0)^-1K̇̃̇_0χ^ ss , K̇̃̇_0 χ^ ss⟩

and analogously

    ⟨χ^ ss,K̇̃̇^*_1K̃_1 (1-K̃_0)^-1K̇̃̇_0χ^ ss⟩
       =-⟨χ^ ss,K̇̃̇_0(1-K̃_0)^-1K̇̃̇_0χ^ ss⟩
       -⟨K̇̃̇_0 χ^ ss,K̃_0(1-K̃_0)^-1K̇̃̇_0χ^ ss⟩.

Putting everything together, one gets

    ∑_αλ_α=(K̃_1(1-K̃_0)^-1K̇̃̇_0+K̇̃̇_1)χ^ ss^2
       =(1-K̃_0)^-1K̇̃̇_0χ^ ss^2-K̃_0(1-K̃_0)^-1K̇̃̇_0χ^ ss^2
       -2(⟨K̇̃̇_0(1-K̃_0)^-1K̇̃̇_0χ^ ss,χ^ ss⟩)
       -2(⟨K̃_0(1-K̃_0)^-1K̇̃̇_0χ^ ss , K̇̃̇_0 χ^ ss⟩)
       -(⟨χ^ ss,K̃_0 χ^ ss⟩ )-K̇̃̇_0 χ^ ss^2
       =λ_ tot+(1-K̃_0)^-1K̇̃̇_0χ^ ss^2
       -(K̃_0(1-K̃_0)^-1+1)K̇̃̇_0χ^ ss^2=λ_ tot.












Let us now prove the first part of the theorem. What we are actually going to show is that for n → +∞ and |u| ≤ n^ϵ^', one has

    e^λ_totu^2ν_u,n (B_𝐦(n) )≍∏_i=1^k(λ_α^(i)u^2)^m_α^(i)/m_α^(i)!.

Let us consider a fixed ordered sequence of excitation patterns α^(1),…,α^(k) (here we do not require them to be distinct). For an observation time n big enough, the probability of observing such a sequence of patterns separated one from the other by more than n^γ consecutive 0s is given by


    ∑_x_1 +… +x_k+1 = n-K
    x_2,…,x_k≥ n^γ      ρ̃^ ss_u  ( B^x_1_u,0 B_u,α^(1) B^x_2_u,0⋯.
    
           .⋯ B^x_k_u,0 B_u,α^(k) B^x_k+1_u,0(1)) ).


where K=∑_i=1^k|α_i| and B_u,α(x):= B_u,α_1⋯ B_u,α_|α|(x).

The rest of the proof follows a similar line as the proof of Theorem <ref>: we study the Taylor expansion of the series and identify the leading terms in the limit n → +∞. We will often use the spectral decomposition of B_0, i.e.

    B_0(x)=ρ̃^ ss(x) |χ^ ss⟩⟨χ^ ss| + 
     E_0(x)

such that

    
  * for any k ≥ 1, for  some constants C>0, 0 <λ<1, one has E_0^k≤ Cλ^k,
    
  * ρ̃^ ss( E_0(·))=0 and 
    E_0(|χ^ ss⟩⟨χ^ ss|)=0.

Notice that, in general, ρ̃^ ss(u) and the eigenvector of B_u,0* corresponding to the spectral radius only coincide for u=0.

We will also use the following identities


  * ρ̃^ ss (𝒯̇̃̇ℛ̃ B_0^x B_α(· )) =0 follows from (<ref>) and the fact that K̃_1|χ^ ss⟩=0.

  * ρ̃^ ss(Ḃ_α(·))=0 follows from K̃_1|χ^ ss⟩=0

  * ρ̃^ ss(ℬ̇_0(·)) =0 follows from the "gauge condition" (<ref>) 

and using the explicit expression (<ref>).




Significant terms in the Taylor approximation (up to the 2k-th term).
We will show that up to derivatives of order 2k the only contribution in the Taylor expansion of (<ref>) which does not decay with n is that coming from the part of the order 2k derivative in which one takes the second order derivative to each of the blocks B^x_i_u,0 B_u,α^(i), for i=1,… k. This follows from the observations below:






1) the first derivative of (<ref>) at u=0 is zero since 
ρ̃^ ss( B_α(· ))=0 and the first derivative of ρ̃^ ss_u( B_u,0^x  B_u,α(·)) at u=0 is equal to

    ρ̃^ ss (𝒯̇̃̇ℛ̃ B_0^x B_α(· ))+∑_l=0^x-1ρ̃^ ss(Ḃ_0 B_0^l  B_α(·))+ρ̃^ ss(Ḃ_α(·))

and the three terms have been shown to be equal to zero above.





















2) Each block of the type 
B^x_u,0 B_u,α needs to be differentiated twice. Indeed if it is not differentiated at all, then it will bring an exponential decaying contribution since 

    ∑_n^γ≤ x ≤ n B^x_0 B_α≤ C ∑_n^γ≤ x ≤ nλ^x=C(λ^n^γ-λ^n+1)/1-λ→ 0 ;

This follows from the fact that since α contains at least one 1, we have ρ̃^ ss( B_α (·)) =0 which means that B_α(·) belongs to the subspace on which 
ℬ_0 acts as the strict contraction ℛ_0. 

Alternatively, if we take a first order derivative we get a vanishing contribution since 

    ∑_n^γ≤ x ≤ n B_0^x Ḃ_α≤ C∑_n^γ≤ x ≤ nλ^x
 
which decays as the term in the previous equation and

    ∑_n^γ≤ k < x ≤ n B_0^x-kḂ_0  B_0^k-1 B_α
       ≤ C∑_n^γ≤ k < x ≤ nλ^x≍ Cn^γλ^n^γ→_n → +∞ 0.


On the other hand, by differentiating a block of the form B^x_u,0 B_u,α twice, one stops the exponential decay and obtains a linear growth in n; however, we need to remember that every time we use a derivative, everything gets multiplied by u/√(n), which is of the order n^-1/2+ϵ^'. Therefore, if we consider the terms in the Taylor expansion up to order 2k, the only one that does not decay to 0 is the one with 2k derivatives where we use two of them on the block ρ̃^ ss( B^x_1_u,0 B_u,α^(1)(·)) at the beginning and other two on each following block of the form B_u,0^x_i B_u,α^(i)(·). Any other term where we have less derivatives involved or where we spend them in a different way is either 0 or decays at least as (for k ≥ 2)

    (n^1/2+γλ^n^γ)n^2kϵ^'

uniformly in 
u for |u|≤ n^ϵ^'; indeed, for any derivative used in a different way (they cannot be more than 2(k-1) because two of them need to be used for the first block), one gains a growth of n^1/2, but suffers a decay of at least n^γλ^n^γ.







We now focus on the leading (order 2k) term of the Taylor expansion. The second derivative of a block of the type 
ℬ^x_u,0ℬ_u,α looks like

    ℬ^x_0ℬ̈_α + 
    ∑_k=1^x 
    ℬ^k-1_0ℬ̈_0ℬ^x-kℬ_α
          +
    2∑_k=1^xℬ^k-1_0ℬ̇_0ℬ^x-k_0ℬ̇_α
          +
    2∑_1≤ k< s ≤ xℬ^k-1_0ℬ̇_0ℬ^s-k-1_0ℬ̇_0ℬ^x-s_0 ℬ_α

For large n this becomes



    ρ̃^ ss[
    (
    ℬ̈_α   + 
    2 ℬ̇_0ℛ_0ℬ̇_α )(·)]
     |χ^ ss⟩⟨χ^ ss|
     
          
    +ρ̃^ ss[(ℬ̈_0 + 2 ℬ̇_0ℛ_0ℬ̇_0 )
     ℛ_0 ℬ_α )(·)]
     |χ^ ss⟩⟨χ^ ss|

where 
ℛ_0 
is the Moore-Penrose inverse of Id - ℬ_0, and we have used the spectral decomposition of ℬ_0. Moreover, the rightmost term 
B_0^x_k+1(1) (which is not differentiated) converges to 
|χ^ ss⟩⟨χ^ ss|. This means that the full 2k derivative of (<ref>) becomes

    u^2k/k!∏_i=1^k1/2ρ̃^ ss[ B^(2)_α^(i) (|χ^ ss⟩⟨χ^ ss|) ]

where 

    B^(2)_α :=
     B_α+2 Ḃ_0  R_0Ḃ_α 
    +( B_0+
        2Ḃ_0  R_0 Ḃ_0) R_0  B_α^.



 
 





The number of terms with two derivatives in each block is equal to (2k)!/2^k (it is the same as the number of terms of the form 2^k= (d^2/dx^2(x^2)  )^k in the 2k-th derivative of (x^2)^k): (2k)! simplifies with the one coming from the Taylor expansion, while the factor 2^-k can be distributed to each factor. The expression of the factors and the upper bound on the error can be obtained differentiating and using the spectral decomposition of
 B_0; the 1/k! in front comes from the sum: indeed one can see
 that
 
    ∑_x_1 +… +x_k+1 = n-K
    x_2,…,x_k≥ n^γ≍∑_x_1 +… +x_k+1 = n≍n^k/k!,

 which is the number of ways one can choose k numbers out of n. 





The first term of the form

    1/2ρ̃^ ss( B_α(|χ^ ss⟩⟨χ^ ss|))= |⟨χ^ ss|K̃_α_|α|⋯K̃_α_2K̇̃̇_1 χ^ ss⟩|^2.

This follows from the fact that K_1|χ^ ss⟩ =0, so the derivatives need to be applied the the first K_1 terms of ℬ_α.
For the second term, one gets

    ρ̃^ ss(Ḃ_0  R_0Ḃ_α(|χ^ ss⟩⟨χ^ ss|))=
       2 (⟨K̃_α_|α|⋯K̃_α_2K̇̃̇_1 χ^ ss| χ^ ss⟩×.
       .
     ⟨χ^ ss|K̃_α_|α|⋯K̃_α_1(1-K̃_0)^-1K̇̃̇_0 χ^ ss⟩).

Finally, for the last term, below we  will show that

    1/2ρ̃^ ss(( B_0+2Ḃ_0  R_0 Ḃ_0) R_0  B_α(|χ^ ss⟩⟨χ^ ss|))=
       |⟨χ^ ss| K̃_α_|α|⋯K̃_α_1(1-K̃_0)^-1K̇̃̇_0 χ^ ss⟩|^2.

Note that the sum of the three terms is equal to |μ_α|^2 
where μ_α is given in equation (<ref>) in Lemma <ref>.

In conclusion, we obtained that for large n, the probability of any sequence of n outcomes showing the ordered sequence of excitation patterns given by α^(1), …, α^(k) is asymptotically equivalent to

    1/k!∏_i=1^k (λ_α^(i)u^2)

plus a reminder coming from neglecting the terms of order bigger than 2k in the Taylor expansion. If we are able to show that the reminder is negligible compared to the term in Eq. (<ref>), then we can prove the statement in Eq. (<ref>). Indeed, suppose that the sequence we are analysing belongs to B_𝐦; then we can partition B_𝐦 into

    k!/∏_i=1^k m_α^(i)!

disjoint subsets containing the excitation patters in (α^(1),m_α^(1)), …, (α^(k),m_α^(k)) in a fixed order and whose probability asymptotically behaves as (as we just showed)

    1/k!∏_i=1^k (u^2λ_α^(i))^m_α^(i).




We now prove (<ref>).
Notice that, since

    |⟨χ^ ss| K̃_α_|α|⋯K̃_α_1(1-K̃_0)^-1K̇̃̇_0 χ^ ss⟩|^2=
       ⟨ (1-K̃_0)^-1K̇̃̇_0 χ^ ss | B_α(|χ^ ss⟩⟨χ^ ss| )|(1-K̃_0)^-1K̇̃̇_0 χ^ ss⟩,

we need to prove that

    R_0*( B_0*+2Ḃ_0* R_0*Ḃ_0*)(ρ̃^ ss))=
       2|(1-K̃_0)^-1K̇̃̇_0 χ^ ss⟩⟨(1-K̃_0)^-1K̇̃̇_0 χ^ ss|+ rem,

where by R_0* we mean the Moore-Penrose inverse and "rem" is a term which is gives 0 when evaluated against B_α(|χ^ ss⟩⟨χ^ ss|). 
Equivalently,

    ( B_0*+2Ḃ_0* R_0*Ḃ_0*)(ρ̃^ ss)=
       2( Id- B_0*)(|(1-K̃_0)^-1K̇̃̇_0 χ^ ss⟩⟨(1-K̃_0)^-1K̇̃̇_0 χ^ ss|)+ rem^',

where R_0*( rem^')= rem. 

By explicit computations, one can see that

    ( Id- B_0*)(|(1-K̃_0)^-1K̇̃̇_0 χ^ ss⟩⟨(1-K̃_0)^-1K̇̃̇_0 χ^ ss|)=
       |(1-K̃_0)^-1K̇̃̇_0 χ^ ss⟩⟨(1-K̃_0)^-1K̇̃̇_0 χ^ ss|-
       |K̃_0(1-K̃_0)^-1K̇̃̇_0 χ^ ss⟩⟨K̃_0(1-K̃_0)^-1K̇̃̇_0 χ^ ss|.

Using that K̃_0(1-K̃_0)^-1=(1-K̃_0)^-1-1, one gets

    ( Id- B_0*)(|(1-K̃_0)^-1K̇̃̇_0 χ^ ss⟩⟨(1-K̃_0)^-1K̇̃̇_0 χ^ ss|)=
       |(1-K̃_0)^-1K̇̃̇_0 χ^ ss⟩⟨K̇̃̇_0 χ^ ss|+
    |K̇̃̇_0 χ^ ss⟩⟨(1-K̃_0)^-1K̇̃̇_0 χ^ ss|-
       |K̇̃̇_0 χ^ ss⟩⟨K̇̃̇_0 χ^ ss|.

On the other hand,

    ( B_0*+2Ḃ_0* R_0*Ḃ_0*)(ρ)
          =
       |K̃_0 χ^ ss⟩⟨χ^ ss|+2|K̇̃̇_0χ^ ss⟩⟨K̇̃̇_0χ^ ss|+|χ^ ss⟩⟨K̃_0 χ^ ss|
          +2|K̇̃̇_0(1-K̃_0)^-1K̇̃̇_0 χ^ ss⟩⟨χ^ ss|
          +2|χ^ ss⟩⟨K̇̃̇_0(1-K̃_0)^-1K̇̃̇_0 χ^ ss|
          +2|K̃_0(1-K̃_0)^-1K̇̃̇_0 χ^ ss⟩⟨K̇̃̇_0χ^ ss|
          +2|K̇̃̇_0χ^ ss⟩⟨K̇̃̇_0(1-K̃_0)^-1K̇̃̇_0 χ^ ss|
          =|(K̃_0 +2K̇̃̇_0(1-K̃_0)^-1K̇̃̇_0)χ^ ss⟩⟨χ^ ss|
          +|χ^ ss⟩⟨(K̃_0 +2K̇̃̇_0(1-K̃_0)^-1K̇̃̇_0)χ^ ss|
          +2|(1-K̃_0)^-1K̇̃̇_0 χ^ ss⟩⟨K̇̃̇_0. χ^ ss|
          +
    2|K̇̃̇_0 χ^ ss⟩⟨(1-K̃_0)^-1K̇̃̇_0 χ^ ss|-2|K̇̃̇_0 χ^ ss⟩⟨K̇̃̇_0 χ^ ss|.

Note that the last two lines are exactly equal to 2( Id- B_0*)(|(1-K̃_0)^-1K̇̃̇_0 χ^ ss⟩⟨(1-K̃_0)^-1K̇̃̇_0 χ^ ss|). Let us look at the remaining part:

    rem^'=   |(K̃_0 +2K̇̃̇_0(1-K̃_0)^-1K̇̃̇_0)χ^ ss⟩⟨χ^ ss|+
       |χ^ ss⟩⟨(K̃_0 +2K̇̃̇_0(1-K̃_0)^-1K̇̃̇_0)χ^ ss|.

one can easily see that

    R_0*( rem^')=
       |(1-K̃_0)^-1(K̃_0 +2K̇̃̇_0(1-K̃_0)^-1K̇̃̇_0)χ^ ss⟩⟨χ^ ss|
       +|χ^ ss⟩⟨(1-K̃_0)^-1(K̃_0 +2K̇̃̇_0(1-K̃_0)^-1K̇̃̇_0)χ^ ss|.

In the previous Eq. we used (1-K_0)^-1 for the Moore-Penrose inverse: in general, |(K̃_0 +2K̇̃̇_0(1-K̃_0)^-1K̇̃̇_0)χ^ ss⟩ is not orthogonal to |χ⟩^ ss.
It is now clear that 
( R_0*( rem^') B_α(|χ^ ss⟩⟨χ^ ss|)=0
which proves (<ref>).


Remainder. Now, we need to take care of the reminder: it is enough to show that the following expression is o(n^2kϵ):

    u^2k+1/n^k+1/2∑_x_1 +… +x_k+1 = n-K
    x_2,…,x_k≥ n^γ   d^2k+1/u^2k+1ρ̃^ ss (u/√(n) ) (B̃^x_1_0,uB̃_α_1,uB̃^x_2_0,u⋯.
        . .⋯B̃^x_k_0,uB̃_α_k,uB̃^x_k+1_0,u(1)) )  |_u=η


for any |η|≤ u/√(n), where

    B̃_α,u=e^λ_tot |α|u^2/n B_α,u

for any string α.










Let us first point out some properties of the maps 
B̃_α,u:


  * for u small enough,

    B̃_0,u(·)=a(u) l(u)(·)r(u) + Ẽ_0,u(·)

where

    
  * a(u)=1+O(u^3/n^3/2), 
    
  * l(u)(r(u))≡ 1, 
    
  * l(u)(Ẽ_0,u(·))=0,
    
  * Ẽ_0,u(r(u))=0 and
    
  * Ẽ^k_0,u≤ Cλ^k for some C ≥ 0 and 0<λ <1.

The order of the reminder in the expression a(u) is due to the fact that λ̇ is equal to the first derivative at 0 of the spectral radius of B_0,u which is equal to 0 because it attains a maximum there; the fact that λ=0 as well is due to the multiplicative factor in front of B_0,u in the definition of B̃_0,u.


  * ρ̃^ ss(u/√(n))(Ẽ_0,u(·))=O(u/√(n)).

  * For any excitation pattern α one has

    l(u)(B̃_α,u(·))=O(u^2/n),
       l(u)(Ḃ̃̇_α,u(·))=O(u/√(n)).

Indeed, the first derivative of l(u)(B̃_α,u(·)) at 0 is given by

    ρ̃^ ss(Ḃ_α(·))+ l̇( B_α(·)).

That the first addend is 0 has been shown earlier, while for the second one, it is clear using l̇=ρ̃^ ss(Ḃ_α ( Id- B_0)^-1(·)).

  * Moreover,

    l(u)(Ḃ̃̇_0,u(r(u)))=O(u^2/n)  and
       l(u)((B̃_0,u+2Ḃ̃̇_0,uẼ_0,uḂ̃̇_0,u(r(u)))=O(u/√(n)).

This can be seen differentiating

    l(u)B̃_0,u(x(u))=a(u)

and evaluating at 0.

  * Finally,

    l(u)(Ḃ̃̇_0,uẼ_0,uB̃_α,u(·))=O(u/√(n)),

since 
ρ̃^ ss(Ḃ_0)(·)=0.






Let us now study the growth of the derivatives of B̃_0,u^xB̃_α,u(·) for 0 ≤ x  ≤ n→ +∞:


  * 0^ th order:

    B̃_0,u^xB̃_α,u(·)   =a(u)l(u)(B̃_α,u(·))x(u)+O(λ^x) 
       =O  (u^2/n+λ^x  );


  * 1^ st order:

    ∑_1 ≤ l ≤ xB̃_0,u^x-lḂ̃̇_0,uB̃_0,u^l-1B̃_α,u(·)+B̃_0,u^xḂ̃̇_α,u(·)=
       xa(u)^2l(u)(Ḃ̃̇_0,u(x(u)))l(u)(B̃_α,u(·))+
       a(u)R̃_0,uḂ̃̇_0,u(x(u)))l(u)(B̃_α,u(·))+
       a(u)l(u)(Ḃ̃̇_0,uR̃_0,uB̃_α,u(·))x(u) + O (u/√(n) +xλ^x )
        
       =O ( u/√(n)+xλ^x ),

    where we used that ϵ <1/6;
    
    We remark that in the case where the block B̃_0,u^xB̃_α,u(·) is the first one, due to the action of ρ̃^ ss(u/√(n)), the 0^ th-order term becomes O(u^2/n+uλ^x/√(n)) and the 1^ st-order one becomes O(u/√(n)), while if the block is not the first one, λ^x decays exponentially fast in n since x ≥ n^γ. 
    
    
    
  * 2^ nd order:
    
    ∑_1 ≤ l ≤ xB̃_0,u^x-lB̃_0,uB̃_0,u^l-1B̃_α,u(·)+
       2∑_1 ≤ l<k  ≤ xB̃_0,u^x-kḂ̃̇_0,uB̃_0,u^k-l-1Ḃ̃̇_0,uB̃_0,u^l-1B̃_α,u(·)+
       2∑_1 ≤ l ≤ xB̃_0,u^x-lḂ̃̇_0,uB̃_0,u^l-1Ḃ̃̇_α,u(·)+
       B̃_0,u^xB̃_α,u(·)=
       2x^2a(u)^3(l(u)(Ḃ̃̇_0,u(x(u)))^2l(u)(B̃_α,u(·))+
       xa(u)^2l(u)((B̃_0,u+2Ḃ̃̇_0,uR̃_0,uḂ̃̇_0,u)(x(u)))l(u)(B̃_α,u(·))+
       2xa(u)^2l(u)(Ḃ̃̇_0,u(x(u))l(u)(Ḃ̃̇_α,u(·))+O(1)=O(1);

    
  * 3^ rd order:

    ∑_1 ≤ l ≤ xB̃_0,u^x-l⃛B̃_0,uB̃_0,u^l-1B̃_α,u(·)+
       3∑_1 ≤ l<k  ≤ xB̃_0,u^x-kB̃_0,uB̃_0,u^k-l-1Ḃ̃̇_0,uB̃_0,u^l-1B̃_α,u(·)+
       3∑_1 ≤ l<k  ≤ xB̃_0,u^x-kḂ̃̇_0,uB̃_0,u^k-l-1B̃_0,uB̃_0,u^l-1B̃_α,u(·)+
       6∑_1 ≤ l<k <m ≤ xB̃_0,u^x-mḂ̃̇_0,uB̃_0,u^m-k-1Ḃ̃̇_0,uB̃_0,u^k-l-1Ḃ̃̇_0,uB̃_0,u^l-1B̃_α,u(·)+
       3∑_1 ≤ l ≤ xB̃_0,u^x-lB̃_0,uB̃_0,u^l-1Ḃ̃̇_α,u(·)+
       6∑_1 ≤ l<k  ≤ xB̃_0,u^x-kḂ̃̇_0,uB̃_0,u^k-l-1Ḃ̃̇_0,uB̃_0,u^l-1Ḃ̃̇_α,u(·)+
       3∑_1 ≤ l ≤ xB̃_0,u^x-lḂ̃̇_0,uB̃_0,u^l-1B̃_α,u(·)+
       B̃_0,u^x⃛B̃_α,u(·)=O(n^2ϵ);

       
  * m^ th order for m ≥ 4: first, notice that
       
    d^m/du^m (B̃_0,u^xB̃_α,u(·) )=
       ∑_l=0^mmld^l/du^l(B̃_0,u^x) d^(m-l)/du^(m-l)(B̃_α,u(·)).

The term d^l(B̃_0,u^x)/du^l is a sum over all possible ways of distributing the derivatives among the factors B̃_0,u.

Then one can use the spectral decomposition of the B̃_0,u which have not been differentiated and glue together differentiated terms using R_0,u or turning them into products using the projection l(u)(·)x(u) as we did in the previous items. From Eq. (<ref>), one can see that the blocks with a single derivative of a single term B̃_0,u bring a growth of the order n^2ϵ^', the terms with two derivatives (B̃_0,u+2Ḃ̃̇_0,uR̃_0,uḂ̃̇_0,u) bring a growth of the order n^1/2+ϵ^' while all other blocks (with at least 3 elements), cause a growth of the order at most n. Notice that the highest growth is attained by making as many groups of three derivatives as possible and this will be used in obtaining the following estimate.

The term d^(m-l)(B̃_α,u)/du^(m-l) does not cause any reduction in the growth if m-l ≥ 2, while causes a decay equal to n^-1/2+ϵ^' if m=l+1 and n^-1+2ϵ^' if m=l.

Therefore,

    d^m/du^m (B̃_0,u^xB̃_α,u(·) )≲
       n^max{f(m-2), f(m-1)-1/2+ϵ^', f(m)-1+2ϵ^'},


where

    f(m)=a(m) + (1/2+ϵ^' )b(m)+2ϵ^' (m-(3a(m)+2b(m)))


    a(m):=⌊m/3⌋,    b(m):=⌊1/2 (m-3⌊m/3⌋ )⌋.

a(m) is the maximum number of groups with three derivatives and b(m) is the maximum number of groups with two derivatives that we can make with the derivatives left. Notice that if ϵ^' <1/6, then f(m) ≤ m/3: indeed,

    a(m) + (1/2+ϵ^' )b(m)+2ϵ^' (m-(3a(m)+2b(m)))=
       2ϵ^' m + (1-6ϵ^')a(m)+  ( 1/2-3ϵ^' ) b(m) ≤
       2ϵ^' m + (1-6ϵ^')a(m)+ 1/2 ( 1/2-3ϵ^' )(m-3a(m)) ≤
       1/2 ( ( 1/2+ϵ^' )m + ( 1/2-3ϵ^' )a(m)  )≤
       1/2 ( ( 1/2+ϵ^' )m + ( 1/6-ϵ^' )m  )≤m/3.

Therefore, the growth of the term in Eq. (<ref>) is upper bounded by

    n^max{m-2/3, m-1/3-1/2+ϵ^', m/3-1+2ϵ^'}.

This implies that the first two derivatives spent on a block of the form B̃_0,u^xB̃_α,u cause a growth equal to n^1/2-ϵ^', while the other m-2 only brings a growth at most equal to the term in Eq. (<ref>). Notice that it is more convenient to spend them in a way that every block of the form B̃_0,u^xB̃_α,u has two derivatives, since one has that

    (1/2-ϵ^' )(m-2) ≥m-2/3⇔ m ≥ 2,

moreover

    (1/2-ϵ^' )(m-2) ≥m-1/3-1/2+ϵ^'⇔ m ≥ 1

and

    (1/2-ϵ^' )(m-2) ≥m/3-1+2ϵ^'⇔ m ≥ 0.

Therefore the growth of

    ∑_x_1 +… +x_k+1 = n-K
    x_2,…,x_k≥ n^γ   d^2k+1/u^2k+1ρ̃^ ss (u/√(n) ) (B̃^x_1_0,uB̃_α_1,uB̃^x_2_0,u⋯.
        . .⋯B̃^x_k_0,uB̃_α_k,uB̃^x_k+1_0,u(1)) )  |_u=η,

is of the order of O(n^2ϵ^') uniformly in u and it is attained by the term where we spend at least two derivatives in each block of the form B̃_0,u^xB̃_α,u and the last derivative in any of such blocks. One can see that it is not convenient to use any derivative in the final term B̃_0,u^x(1), since it is better to contrast the decay induced by the other blocks. To conclude, the term in Eq. (<ref>) grows at most as n^(2k+3)ϵ^'-1/2=o(n^2kϵ^'). We proved Proposition <ref> and notice that Theorem <ref> follows considering ϵ^'=0.




















































§ PROOF OF COROLLARY <REF>
 

    Let us define the measurable space given by the set Ω=ℕ^ℕ together with the σ-field F generated by cylindrical sets; we can consider on (Ω,  F) the law ν_u of {N_α:α∈ P} and the law of {N_α(n): α∈ P}, which, with a slight abuse of notation, we still denote by ν_u,n as well. We know that for every finite set A of patterns, one has
    
    lim_n → +∞sup_|u|<C |ν_u,n(A)-ν_u(A)|=0.

    Notice that, for every ϵ >0, there exists a set A_ϵ of finitely many patter such that
    
    inf_|u|<Cν_u(A_ϵ) > 1-ϵ.

    Therefore, using Eq. (<ref>), one has that there exists N_ϵ such that ∀ n ≥ N_ϵ,
    
    inf_|u|<Cν_u,n(A_ϵ) > 1-ϵ,  and sup_|u|<C |ν_u,n(A_ϵ)-ν_u(A_ϵ)|<ϵ.
 
    Therefore, given a bounded function f:ℕ^ P→ℝ, for every ϵ>0, ∀ n ≥ N_ϵ, one has
    
    sup_|u| < C|𝔼_ν_u,n[f]-𝔼_ν_u[f]| ≤
       f_∞ (sup_|u|<C |ν_u,n(A_ϵ)-ν_u(A_ϵ)| + .
        .sup_|u|<Cν_u,n(A_ϵ^C)+ν_u(A_ϵ^C)  ) ≤
       3f_∞ϵ.

    For the arbitrariness of ϵ, we proved the first statement about the weak convergence.

    Consider the random process N_ tot(n) that counts all the occurrences of 1's in the output up to time n and notice that for every α∈ P, N_α(n) ≤ N_ tot(n). Therefore for every p ≥ 1 and m ∈ℕ, one has
    
    𝔼_ν_u,n[N_α(n)^p 1_{N_α(n) >m}] ≤𝔼_ν_u,n[N_ tot(n)^p 1_{N_ tot(n) >m}].

    If we show that the moments of every order of N_ tot(n) converge to some finite limit and that it converges in law to some limit random variable X_u, we obtain the second statement as well. Indeed, let us call C(p) the limit of the p-moment of N_ tot(n); notice that for every p ≥ 1, m ∈ℕ
    
    lim sup_n → +∞𝔼_ν_u,n[N_α(n)^p 1_{N_α(n) >m}] ≤
       lim sup_n → +∞𝔼_ν_u,n[N_ tot(n)^p 1_{N_ tot(n) >m}] ≤
       lim_n → +∞𝔼_ν_u,n[N_ tot(n)^pq]^1/qν_u,n(N_ tot(n) >m)^1/q^' =
       C(pq)^1/qℙ(X_u >m)^1/q^'.

    Notice that in the last inequality we made use of Hölder inequality for some pair of conjugate indices (q, q^'). Therefore, if we fix p≥ 1, for every ϵ>0, one can choose m_ϵ such that
    
    lim sup_n → +∞𝔼_ν_u,n[N_α(n)^p 1_{N_α(n) >m_ϵ}] ≤ϵ/2,
       𝔼_ν_u[N_α^p 1_{N_α >m_ϵ}] ≤ϵ/2

    and one gets
    
    lim sup_n → +∞|𝔼_ν_u,n[N_α(n)^p]-𝔼_ν_u[N_α^p]| ≤
       lim_n → +∞|𝔼_ν_u,n[N_α(n)^p1_{N_α(n) ≤ m_ϵ}]-𝔼_ν_u[N_α^p 1_{N_α≤ m_ϵ}]|+
       lim sup_n → +∞𝔼_ν_u,n[N_α(n)^p 1_{N_α(n) >m_ϵ}] +
       𝔼_ν_u[N_α^p 1_{N_α >m_ϵ}] ≤ϵ.

Since this holds for every ϵ>0, we proved the statement.

We need to show that, under ν_u,n, N_ tot(n) converges in law to a random variable X_u with finite moments of every order and that we have convergence of the moments as well. One can see that the Laplace transform of N_ tot(n) can be expressed as

    𝔼_ν_u,n[e^zN_ tot(n)]=ρ̃^ ss(u/√(n))(𝒯̃_u,z,n^n(1))    z ∈,

where

    𝒯̃_u,z,n(·)=   K̃^*_0(u/√(n))·K̃_0(u/√(n))+
       e^zK̃^*_1(u/√(n)) ·K̃^*_1(u/√(n)).

Notice that 𝒯̃_0,z:=𝒯̃_0,z,n is independent from n and is an analytic perturbation of 𝒯̃. If we pick z small enough in modulus, perturbation theory ensures that 𝒯̃_0,z has 1 as eigenvalue with maximum modulus with |χ^ ss⟩⟨χ^ ss| as left eigenvector. Let x_z be the corresponding right eigenvector such that 
Tr(ρ^ ss x_z) = ⟨χ^ ss|x_z |χ^ ss⟩=1, and let

    x_z=[ 1 a; b c ]

be its block matrix form with respect to 
the decomposition of ℂ^D into 
ℂ|χ^ ss⟩ and its orthogonal complement. Then one can prove that 

a=b=0 by using the fact that the Kraus operators are of the form

    K_0=[ 1 0; 0 β ],    K_1=[ 0 γ; 0 δ ]
 
for some blocks β, γ, δ 
such that |β|^2+|γ|^2+|δ|^2=1. Let us first fix z small enough, then for n big enough, 𝒯̃_u,z,n has a unique eigenvalue λ_z(u,n) of maximum modulus with corresponding left and right eigenvectors l_z(u,n), x_z(u,n) and one has that

    𝔼_ν_u,n[e^zN_ tot(n)] =
       λ_z(u,n)^n l_z(u,n)(1) ρ̃^ ss(u/√(n))(x_z(u))+o(n)

where o(n) is uniform in u (and z in compact sets). Notice that both l_z(u,n)(1) and ρ̃^ ss(u/√(n)) (x_z(u)) converge to 1 for n → +∞; regarding the behaviour of λ_z(u,n) consider the Taylor expansion up to second order in u around 0:

    λ_z(u,n)=1+u/√(n)λ^(1)_z+u^2/nλ^(2)_z + o(n^-1).

We can choose l_z(u,n) and x_z(u,n) such that (l_z(u,n)x_z(u,n))≡ 1, therefore differentiating (l_z(u,n)𝒯̃_z,u,n(x_z(u,n)))=λ_z(u,n) at 0 one gets

    λ^(1)_z=(l_z ∂_u𝒯̃_z|u=0 x_z)

which can be easily seen to be 0. Summing up, we proved that

    lim_n→ +∞𝔼_ν_u,n[e^zN_ tot(n)]=e^λ_z^(2)

uniformly in u and in z in compact small neighborhoods of 0. Since for every u, f_n(z):=𝔼_ν_u,n[e^zN_ tot(n)] are analytic functions around 0, then we can deduce that e^λ_z^(2) is analytic as well and that we have uniform convergence (on compact small neighborhoods of 0) of all the derivatives, which consists exactly in the convergence of moments of all orders and we are done.




§ ACHIEVABILITY OF THE QCRB UNDER ADDITIONAL ASSUMPTIONS 


 

In this section we present a proof of Theorem <ref> and we comment on the gap between the hypothesis that we need to assume and what we proved in Proposition <ref>.



















By hypothesis, we know that θ belongs to the (random) confidence interval 
    I_n = (θ̃_n-n^-1/2 +ϵ,  θ̃_n+n^-1/2 +ϵ)

with high probability. In order to prove the statement, it suffices to show that 

    |(𝔼_θ[e^ia √(n) (θ̂_n-θ)|θ̃_n]-e^-a^2/2f_θ)|χ_{θ∈ I_n}(θ̃_n)
 
can be upper bounded uniformly in θ̃_n by a sequence converging to 0. Indeed, notice that for every a ∈ℝ

    𝔼_θ[e^ia √(n) (θ̂_n-θ)]   =e^-a^2/2f_θℙ_θ(θ∈ I_n) 
       +∫_θ∈ I_np_θ(dθ̃_n)(𝔼_θ[e^ia √(n) (θ̂_n-θ)|θ̃_n]-e^-a^2/2f_θ)
       +∫_θ∉ I_np_θ(dθ̃_n)𝔼_θ[e^ia √(n) (θ̂_n-θ)|θ̃_n].

Since ℙ_θ(θ∉ I_n) goes to zero, the first term goes to e^-a^2/2f_θ and the third one vanishes. If we show that the second term vanishes then we obtain the convergence in distribution of √(n) (θ̂_n-θ) as in the statement.

First of all notice that for every a ∈ℝ, for every θ̃∈Θ

    𝔼[e^ia(Y_n,θ, θ̃-√(n)(θ-θ̃))]=
       e^λ_ tot(θ̃)(√(n)(θ-θ̃)+τ_n)^2 (e^ia/(2λ_ tot(θ̃) τ_n) -1)-ia  (τ_n/2+√(n)(θ̃-θ)  ).

Therefore for every a ∈ one has

    lim_n → +∞sup_θ̃:|θ̃-θ|<n^-1/2+ϵ |𝔼[e^ia(Y_n,θ, θ̃-√(n)(θ-θ̃))]-e^-a^2/f_θ |=0




We will denote by u=√(n)(θ-θ̃_n). The definition of θ̂_n implies that

    √(n)(θ̂_n-θ)=Y_n-u.

Therefore we obtain that

    sup_θ̃:|θ̃-θ|<n^-1/2+ϵ |𝔼_θ[e^ia(Y_n-u)|θ̃_n=θ̃]-e^-a^2/2f_θ | ≤
       sup_θ̃: |θ̃-θ|<n^-1/2+ϵ|𝔼_θ[e^iaY_n|θ̃_n=θ̃]-𝔼[e^iaY_n,θ, θ̃]|+
       sup_θ̃:|θ̃-θ|<n^-1/2+ϵ |𝔼[e^ia(Y_n,θ, θ̃-√(n)(θ-θ̃))]-e^-a^2/f_θ |.

Hence,

    lim_n → +∞sup_θ̃:|θ̃-θ|<n^-1/2+ϵ |𝔼_θ[e^ia(Y_n-u)|θ̃_n=θ̃]-e^-a^2/2f_θ |=0

and we are done.

Let us briefly comment on the relationship between the additional hypothesis we introduced (Eq. (<ref>)) and the result in Proposition <ref>; let us consider the following family of stochastic processes: for every n ∈ℕ, θ̃, θ∈Θ, consider the collection of independent random variables

    N_n,θ, θ̃,α∼Poisson(λ_α(θ̃ (√(n)(θ-θ̃)+τ_n)^2),   α∈𝒫

and their law ν_n, θ, θ̃ on ℕ^ P. (together with the σ-field of cylindrical sets). Notice that
∑_α∈ PN_n, θ,θ̃, α converges in mean square and has the same law as N_n, θ,θ̃.

Inspecting the proof, one can notice that the convergence in the statement of Proposition <ref> holds uniformly in a small neighborhood of the reference parameter θ_0, therefore we can restate it in the following way: for ϵ small enough and for every finite collections of excitation patterns counts 𝐦 one has

    lim_n → +∞sup_θ̃:|θ̃-θ|<n^-1/2+ϵ | ν_√(n)(θ-θ̃),n(B_𝐦(n))/ν_θ, θ̃,n(𝐦)-1 |=0.

If we were able to show that the previous result still holds integrating with respect to ν_θ, θ̃,n, i.e.

    lim_n → +∞sup_θ̃:|θ̃-θ|<n^-1/2+ϵ∑_𝐦ν_θ, θ̃,n(𝐦) | ν_√(n)(θ-θ̃),n(B_𝐦(n))/ν_θ, θ̃,n(𝐦)-1 |=
       lim_n → +∞sup_θ̃:|θ̃-θ|<n^-1/2+ϵ∑_𝐦 | ν_√(n)(θ-θ̃),n(B_𝐦(n))-ν_θ, θ̃,n(𝐦) |=0,

this would imply the condition in Eq. (<ref>) (it can be seen using the fact that ∑_α∈ PN_n, θ,θ̃, α and N_n, θ,θ̃ have the same law). Unfortunately, we are not able to prove this.

The last remark we make is that Eq. (<ref>) cannot be true unless

    lim_n → +∞sup_θ̃:|θ̃-θ|<n^-1/2+ϵν_θ, θ̃,n(G(n)^C)=0,

where G(n) is the set of all 𝐦's such that ν_√(n)(θ-θ̃),n(B_𝐦(n))>0. We can prove that this is indeed the case.

If ϵ is small enough, then

    lim_n → +∞sup_θ̃:|θ̃-θ|<n^-1/2+ϵν_θ, θ̃,n(G(n)^C)=0.


Notice that G(n) is the set of all those patterns counts m=(m_α^(1), …, m_α^(k)) such that the following conditions are satisfied


  * α^(i) does not contain more than n^γ consecutive 0s for every i=1,…, k;

  * ∑_i=1^km_α^(i)|α^(i)| + (k-1)n^γ≤ n.

We recall that |α| is the length of the pattern α.

Let us consider a positive number η<1-γ and notice that one has G̃(n) ⊆ G(n), where G̃(n) is the set of all those patterns counts m=(m_α^(1), …, m_α^(k)) such that


  * |α^(i)| ≤ηlog_2(n) for every i=1,…, k and

  * ∑_i=1^km_α^(i)|α^(i)|+(k-1)n^γ≤ n.

We denote by A(n) the set of patterns satisfying 1. and B(n) the set of patterns satisfying 2., hence G̃(n)=A(n) ∩ B(n). In order to prove the statement, it suffices to show that

    lim_n → +∞sup_θ̃:|θ̃-θ|<n^-1/2+ϵν_θ, θ̃,n(G̃(n)^C)=0.

Notice that G̃(n)^C=A(n)^C ⊔ (B(n)^C∩ A(n)); let us first show that

    lim_n → +∞inf_θ̃:|θ̃-θ|<n^-1/2+ϵν_θ, θ̃,n(A(n)) → 1.

Notice that, 

    lim_n → +∞inf_θ̃:|θ̃-θ|<n^-1/2+ϵν_θ, θ̃,n(A(n))=
       lim_n → +∞inf_θ̃:|θ̃-θ|<n^-1/2+ϵe^- (∑_|α| > ηlog_2(n)λ_α ) (√(n)(θ-θ̃)-τ_n)^2=1,

since, for every θ̃ such that |θ-θ̃|<n^-1/2+ϵ, one has

    (∑_|α| > ηlog_2(n)λ_α ) (√(n)(θ-θ̃)-τ_n)^2 ≲ n^ηlog_2(λ) + 3ϵ→ 0

if ϵ <-ηlog_2(λ)/3.
Let us now study the probability of B(n)^C ∩ A(n): first notice that

    B(n)^C ∩ A(n) ⊆
          { m :|α^(i)| ≤ηlog_2(n),  ∑_i=1^km_α^(i)|α^(i)|+2^ηlog_2(n)n^γ > n},

because 2^ηlog_2(n) upper bounds the cardinality of all the patterns of length smaller or equal than ηlog_2(n). Therefore

    ν_θ, θ̃,n(B(n)^C ∩ A(n)) ≤ν_θ, θ̃,n(C(n))
    
    
       ≤ν_θ, θ̃,n ( { m:∑_|α| ≤ηlog_2(n)m_α > n-n^η+ γ/ηlog(n)} ).

where C(n) is the set of pattern counts such that |α^(i)| ≤ηlog_2(n) and ∑_i=1^km_α^(i)ηlog_2(n)+n^η +γn > n.
The last term amounts to the probability that a Poisson random variable of parameter ∑_|α| ≤ηlog_2(n)λ_α (√(n)(θ-θ̃)-τ_n)^2 ≲ n^3ϵ is bigger than something that grows as
n/(ηlog(n)).
Such a probability goes to 0 uniformly in θ̃ if 3ϵ<1 and we are done.





    




§ PRIMITIVITY OF SYSTEM AND ABSORBER DYNAMICS



In this section we investigate the validity of the hypothesis that system and matching absorber generate primitive dynamics. Currently, it is unclear whether this property follows from the primitivity of the system dynamics, and if not, whether it holds for “generic” dynamics or fails for a larger class of systems.



To explore this question we generated random sets of Kraus operators for the system dynamics and  corresponding sets for the system and absorber and computed the spectral gaps of the system and system plus absorber transition operators.


We could start looking at the case 𝔥_S=𝔥_A=𝔥_E=ℂ^2.



  * Sampling a primitive quantum channel, which is the generic type (Generating random quantum channels, JMP, 2021);

  * K_0, K_1, ρ^ss=∑_i=0^1λ_i |i⟩⟨i| with λ_0∈ (0,1/2).

  * V=∑_i=0^1|i⟩_A ⊗|0⟩_E ⟨v_i|+ |i⟩_A ⊗|1⟩_E ⟨w_i| where

    |v_i⟩=∑_j,k=0^1√(λ_j/λ_i)⟨i| K_k |j⟩|j⟩_A ⊗|k⟩_E.

|w_i⟩ are free to be chosen (we could sample random 2×2 unitaries).

  * the Kraus operators of the evolution of system and absorber are given by

    K̃_i=∑_k=0^1 (W_ik⊗1_S )(1_A ⊗ K_k),

where W_ik=⟨i|_E V |k⟩_E, hence

    W_0,k=∑_i,j=0^1√(λ_j/λ_i)⟨j|K_k |i⟩|i⟩_A⟨j|_A

and

    W_1,k=∑_i^1|i⟩_A⟨w_i||k⟩_E.

Check that ∑_i=0^1K̃^*_i ·K̃_i is ergodic and aperiodic.

Does the assumption that one has a strong interaction between the system and the environment (we have d^2 l.i. Kraus operators) improve things?








apsrev4-1









