Quantum theory at the macroscopic scale
Miguel Gallego Borivoje Dakić



miguel.gallego.ballester@univie.ac.at
University of Vienna, Faculty of Physics, Vienna Center for Quantum Science and Technology, Boltzmanngasse 5, 1090 Vienna, Austria
University of Vienna, Vienna Doctoral School in Physics, Bolztmanngasse 5, 1090 Vienna, Austria

borivoje.dakic@univie.ac.at
University of Vienna, Faculty of Physics, Vienna Center for Quantum Science and Technology, Boltzmanngasse 5, 1090 Vienna, Austria

Institute for Quantum Optics and Quantum Information (IQOQI),
Austrian Academy of Sciences, Boltzmanngasse 3,
A-1090 Vienna, Austria



§ ABSTRACT 
The quantum description of the microscopic world is incompatible with the classical description of the macroscopic world, both mathematically and conceptually. Nevertheless, it is generally accepted that classical mechanics emerges from quantum mechanics in the macroscopic limit. In this letter, we challenge this perspective and demonstrate that the behavior of a macroscopic system can retain all aspects of the quantum formalism, in a way that is robust against decoherence, particle losses and coarse-grained (imprecise) measurements. This departure from the expected classical description of macroscopic systems is not merely mathematical but also conceptual, as we show by the explicit violation of a Bell inequality and a Leggett-Garg inequality.


Quantum theory at the macroscopic scale
    Borivoje Dakić
    September 9, 2024
=======================================





§ INTRODUCTION

Quantum mechanics is one of the most successful scientific theories, and it is generally accepted as more fundamental than classical mechanics. However, quantum behavior is not observed at larger scales, where classical physics provides a better description. To explain the macroscopic world we perceive in our everyday life, it is believed that there must exist a quantum-to-classical transition or, in other words, that classical mechanics must somehow emerge from quantum mechanics in the macroscopic limit (somewhat in the spirit of Bohr's correspondence principle <cit.>). The questions of when and how exactly this transition occurs, in spite of being active for almost a hundred years now <cit.>, are still debated today <cit.>. One way to explain the emergence of classicality is to introduce genuine non-quantum effects (thus modifying quantum theory), such as the dynamical <cit.> or gravitationally-induced collapse <cit.> of the wave function. Another standard way is to investigate the quantum-to-classical transition from within quantum theory (which is also the subject here). One of the most famous approaches in this direction is the decoherence mechanism <cit.>, which shows that macroscopic systems, being hard to isolate, lose coherence in their interaction with the environment. Consequently their description becomes effectively classical. Complementary to this approach is the coarse-graining mechanism <cit.>, which shows that outcomes of macroscopic measurements admit a classical description when their resolution is limited even for perfectly isolated systems. While the main focus of the decoherence mechanism is on the dynamics, the coarse-graining approach focuses on the kinematic aspect of the transition to classicality. One way or another, the common conclusion is that quantum effects disappear in the macroscopic limit. Here, we want to challenge this view and ask how much decoherence or coarse-graining is needed to observe classicality? Typically, one takes a “transition” parameter, such as the size of the system: for example, the number of microscopic constituents N of a large system. Then, the standard statement is rather qualitative, positing that for a measurement resolution much greater than √(N), one obtains an effective classical description <cit.>. Nevertheless, the precise mathematical meaning of what “much greater than” means is vague in a concrete experimental situation. Our goal here is to make such statements mathematically precise and to show a well-defined macroscopic scale at which large quantum systems can fully preserve a quantum description, in the sense of the typical ingredients of quantum theory such as the notion of Hilbert space, the Born rule and the superposition principle. Furthermore, we will show that these are not merely mathematical artifacts but genuine quantum phenomena by explicitly showing the violation of Bell <cit.> and Leggett-Garg <cit.> inequalities for such systems.  



§ COARSE-GRAINED MEASUREMENTS AND THE QUANTUM-TO-CLASSICAL TRANSITION

The concept of coarse-grained measurement in quantum mechanics appeared already long ago as an attempt to address Born's rule using the relative frequency operator (fuzzy or coarse-grained observables <cit.>). Continuing this line of inquiry, many subsequent works have followed, including discussions related to the weak and strong laws of large numbers <cit.>, as well as the quantum-to-classical correspondence (see <cit.> and references therein). A significant breakthrough in the operational understanding of the emergence of classicality through the coarse-graining mechanism came due to Kofler and Brukner <cit.>. In their works, the authors focus on classicality via the notion of macroscopic realism <cit.> or Bell's local realism <cit.>. Such a framework opens an operational route to study the observability of genuine quantum effects, signaled by the violation of Leggett-Garg <cit.> or Bell <cit.> inequalities. Their main result is that with a sufficient level of coarse-graining (much greater than √(N)) the outcomes of the experiment can be modeled by a classical distribution (at least for finite dimensional systems) <cit.>. In particular, outcomes of successive measurements on a single system satisfy macroscopic realism (i.e. satisfy all Leggett-Garg inequalities) <cit.>, and local measurements on a bipartite system satisfy local realism (i.e. satisfy all Bell inequalities) <cit.>. Similar results can be found in the context of more general (post-quantum) theories in <cit.>. On the other hand, if the level of coarse-graining is just right, namely of the order √(N), then we have the following facts: for independent and identically distributed (IID) pairs of (finite-dimensional) quantum systems, the coarse-grained quantum correlations satisfy Bell locality <cit.> (an analogous result can be shown for quantum contextuality <cit.>), while non-IID quantum states can exhibit nonlocal correlations as shown in our earlier works <cit.>. Furthermore, these works show that an entire family of (quantum) non-central limit theorems arises in such non-IID scenarios, raising an interesting situation in which, although being coarse-grained, macroscopic quantum systems exhibit quantum phenomena. We formalized this via the idea of macroscopic quantum behavior <cit.>, a property of a system in the macroscopic limit that retains the mathematical structure of quantum theory under the action of decoherence, particle losses, and coarse-graining. We have shown that it is possible to preserve the typical ingredients of the quantum formalism in the macroscopic limit, such as the Born rule, the superposition principle, and the incompatibility of the measurements. 

In this letter, we build on our previous work and develop a unified framework for quantum theory at the macroscopic scale, which describes the theory of successive concatenation of measurements in the macroscopic limit. We use the formalism of Kraus operators in the limit Hilbert space, which allows us to show a violation of a Leggett-Garg inequality in the macroscopic limit, which in turn shows the incompatibility of coarse-grained measurements (at the √(N) level) with a macrorealistic description of the large quantum system.



  §.§.§ Decoherence vs. coarse-graining

Before proceeding further, we would like to make some remarks on decoherence and its relation to coarse-graining. The decoherence mechanism stresses the role of dynamical loss of quantum coherence due to the (instantaneous) interaction with the environment followed by the einselection process of the pointer basis <cit.>. On the other hand, the coarse-graining mechanism focuses on a kinematical aspect of the transition to classicality by measuring collective coarse-grained observables.  Nevertheless, if decoherence is understood broadly as a mechanism of “classicalization" due to interaction with the environment, then coarse-graining can be seen as an instance of such a mechanism. Namely, suppose such a process is described by an interaction Hamiltonian of the type H=H_S⊗ H_E, with H_S=∑_i h_i describing the collective Hamiltonian of the large quantum system (here h_i refers to the operators associated to local, microscopic constituents). In that case, the decoherence model effectively reduces to the measurement model of collective coarse-grained observables. This will be precisely our model of measurement, which we will introduce in the next sections. Therefore, essential aspects of the standard decoherence mechanism are incorporated in our study through coarse-graining of the measurements, and this is the standard argument to draw the parallel between the two approaches to the quantum-to-classical transition <cit.>. Notice that that the system interacts collectively with the environment in that case. On the other hand, microscopic constituents can as well be independently subjected to a decoherence channel (such as the dephasing or depolarizing channel) <cit.>. We will also include this mechanism in our study and refer to it as local decoherence to distinguish it from the standard decoherence mechanism (a precise definition will be provided to avoid misunderstandings). Our aim is to show robustness of quantum phenomena in the macroscopic limit against both mechanisms.

The paper is organized as follows. In Section <ref>, we define the setting under consideration and specify all relevant assumptions. The notion of macroscopic quantum behavior is introduced in Section <ref>, encompassing the quantum properties of systems in the macroscopic limit, with concrete examples provided. In Section <ref>, we further analyze the properties of the system, demonstrating the genuineness of macroscopic quantum behavior through device-independent tests, including explicit violations of Bell and Leggett-Garg inequalities. Finally, we conclude with final remarks and open questions in Section <ref>.




§ SETUP


We consider a macroscopic quantum measurement scenario analogous to the one presented in <cit.> (see Figure <ref>). The setting consists of two parts: a quantum system 𝖲 and a quantum measurement apparatus 𝖬. In order to model a realistic situation in the absence of perfect control, we assume these satisfy certain assumptions. 



  §.§.§ Macroscopic system

First, we assume the system 𝖲 satisfies the following conditions:

    
  * Large N. The system is composed of a large number N of identical particles or subsystems with associated Hilbert space 𝔥. We describe the state of the system with a density matrix ρ_N ∈𝒟(𝔥^⊗ N), i.e. a positive, self-adjoint bounded linear operator on 𝔥^⊗ N satisfying ρ_N =1.
    

    
  * Local decoherence. The system is subject to independent, single-particle decoherence channels Γ, such as the depolarizing or the dephasing channel. The effective state thus becomes Γ^⊗ N ( ρ_N ).
    

    
  * Particle losses. Each individual particle has a probability p ∈ (0,1] of reaching the measurement apparatus, while 1-p is the probability of being lost (we assume p > 0 to avoid the trivial case where no particles reach the apparatus). Therefore, in each run of the experiment, only a number M ≤ N of particles reach the measurement apparatus with a probability f_N(M) = N  M p^M (1-p)^N-M. Consequently, the state received by the measurement apparatus is of the form _1 … N-M_̊N (where _i is understood as the partial trace over the Hilbert space 𝔥 of the i-th particle).
    

Given these assumptions, for an initial state of the system _̊N ∈𝒟(𝔥^ N), the effective state of the system can be written in Fock-like space ℱ_N = ⊕_M=0^N 𝔥^ M as  

    Ł_N(_̊N) = ⊕_M=0^N  f_N(M)  ∑_π∈𝔖_N_π(1) …π(N-M)[  ^⊗ N ( ρ_N )  ] /N!  ,

where 𝔖_N is the symmetric (permutation) group of N elements. 



  §.§.§ Coarse-grained measurements

In order to model the macroscopic measurements, we assume the measurement apparatus 𝖬 satisfies the following conditions:

    
    
  * Collective measurement. The measurement setting of the apparatus is given by a single-particle observable, i.e. a Hermitian bounded operator A ∈ℬ(𝔥). We denote by A |a⟩_A = a |a⟩_A the diagonalization of A, where a ∈ℝ are its eigenvalues and |a⟩_A ∈𝔥 are its eigenstates. We denote by 𝒜⊆ℬ(𝔥) the set of (experimentally) accessible single-particle observables.
    
    
    
  * Intensity measurement. Given a measurement setting A ∈𝒜, the measurement apparatus measures the intensity ∑_i a_i, i.e. the sum of individual outcomes. The corresponding observable in Fock-like space is therefore the intensity observable I_N(A) = ⊕_M=1^N ∑_i=1^M A_i, where A_i= 𝕀⊗…⊗ A⊗…⊗𝕀 is the operator that acts with A on the i-th particle and with the identity on the rest. 
    
    
    
  * Coarse-graining. The measuring scale for the intensity ∑_i a_i has a limited resolution of the order of √(N) (the square-root of the total number of particles), meaning that it cannot distinguish between values that differ by approximately less than √(N). 
    




  §.§.§ Measurement model

In order to implement these assumptions explicitly, we follow von Neumann's model of quantum measurements <cit.>. First, we assume that the measurement apparatus 𝖬 couples the system 𝖲 to an auxiliary system 𝖯 called the pointer, initially in a state |Φ⟩∈ L^2(ℝ) centered around zero in position basis and with a standard deviation of order √(N). For simplicity, we take this state to be a Gaussian with standard deviation √(N) for some > 0, i.e.

    Φ_N(x) = 1/(2 π N ^2)^1/4 e^-x^2/(4 N  ^2)  .

The coupling between the system and the pointer is described by the Hamiltonian H(t) = (t) ⊕_M=0^N ∑_i=1^M A_i  P, where (t) is a nonzero function only for a short time satisfying ∫ dt  (t) =1 and P is the momentum operator of the pointer. After unitary interaction, the position of the pointer is translated to a distance equal to the value of the system's intensity ∑_i a_i (i.e., an eigenvalue of the intensity operator I_N(A)). Finally, the pointer's position is measured, obtaining a value x ∈ℝ. For a single-particle observable A ∈𝒜 (which represents a measurement setting) and an initial state of the system _̊N ∈𝒟(𝔥^ N), we set X_N to be the random variable associated to the measured value x (the measurement result on 𝖯). We shall use the symbol “∼" to denote “distributed according to", and as shown in the Appendix <ref>, we have X_N ∼ P(x|A) where

    P_N(x |A) = [ K_N(x| A)  Ł(_̊N)    K_N^†(x| A) ].

Here, the Kraus operators K_N(x|A) are given by

    ⊕_M=0^N ∑_a_1 … a_MΦ_N ( x - ∑_i=1^M a_i )  Π_a_1 | A⊗…⊗Π_a_M | A  ,

with Π_a|A=|a⟩_A   _A⟨a| being the eigenprojectors of A. These Kraus operators define a positive operator-valued measure (POVM) with elements E_N (x | A) = K_N^† (x|A) K_N (x|A), normalized so that ∫ dx   E_N(x|A) = 𝕀. The corresponding (normalized) post-measurement state of the system is given by the standard expression

    K_N (x | A )  Ł(_̊N)   K_N^†(x | A ) /[ K_N(x | A)  Ł(_̊N)   K_N^†(x | A) ]   ,

To summarize, the effective state Ł(_̊N) describes the system 𝖲 under assumptions <ref>, <ref> and <ref>, while the Kraus operators K_N(x|A) describe the measurement apparatus 𝖬 under assumptions 
<ref>, <ref> and <ref>.



  §.§.§ Macroscopic limit

The macroscopic limit corresponds to the limit of an infinite number of particles, i.e., N→∞. Nevertheless, before proceeding further, let us consider possible scenarios in such a limit. The random variable X_N will generally not converge in distribution as N →∞. To illustrate this, let the initial state of the system be an independent and identically distributed (IID) state _̊N = ^̊ N for some ∈̊𝒟(𝔥). In this case, the distribution of the intensity ∑_i=1 a_i does not converge in general, unless one subtracts to it the mean value ⟨∑_i=1 a_i ⟩ and divides by √(N), just like in the central limit theorem <cit.>. Therefore, to ensure convergence, we take an affine transformation of X_N, namely we consider a family of random variables of the form ł_N X_N + _N, and choose the parameters ł_N, _N ∈ℝ (independent of the measurement setting and initial state of the system) in a way such that ł_N X_N + _N converges in distribution. Once this is fixed, the corresponding probability density function and the Kraus operators given in Equations (<ref>) and (<ref>) transform as

    P_N(x|A)    ↦ł_N^-1   P_N ( ł_N^-1(x-_N) |A )   ,

and

    K_N(x|A)    ↦ł_N^-1/2  K_N ( ł_N^-1(x-_N) | A )   ,

respectively.




  §.§.§ n consecutive measurements


Finally, we are ready to present the most general scenario where a number n of successive measurements are performed (see Figure <ref>). In this case, the system 𝖲, subjected to assumptions <ref> - <ref> as before, goes through measurement apparatuses 𝖬_1, ..., 𝖬_n, each of which satisfies assumptions <ref> - <ref>. For an initial state of the system _̊N ∈𝒟(𝔥^ N) and a sequence of n single-particle observables (A_1, … , A_n) (n measurement settings) let X⃗_N = (X_N^(1), …, X_N^(n)) be the random vector associated to the measurement outcomes (x_1, …, x_n). Then X⃗_N ∼ P(x_1, …, x_n | A_1, …, A_n), where the distribution P(x_1, …, x_n | A_1, …, A_n) is given by

    [ K_N^(n)Ł^(n)( … K_N^(1) Ł^(1)(_̊N)    K_N^(1) †…)   K_N^(n) †]   .

Here K_N^(j) = K_N(x_j |A_j) and Ł^(j) is the map (<ref>) extended to Fock-like space (defined by loss probability p_j and decoherence channel _j), i.e.,

    Ł^(j)( ⊕_M=0^N  _̊M  ) = ⊕_M=0^N Ł_M^(j) ( _̊M)   ,

where

    Ł_M^(j) (_̊M) = ⊕_J=0^M f_M^(j)(J) ∑_π∈𝔖_M_π(1) …π(M-J) [ _j^ M (_̊M)]/M!

and f_M^(j)(J) = M  Jp_j^J (1-p_j)^M-J. As before, the random vector X⃗_N may not converge in distribution; thus, we shall consider an affine transformation of the form ł⃗_N ⊙X⃗_N + _N, where ⊙ denotes the Kronecker product (e.g. (a,b) ⊙ (c,d) = (ac, bd)), and choose the vectors ł⃗_N and _N to facilitate convergence.





§ QUANTUM THEORY AT THE MACROSCOPIC SCALE


We now argue that, in the context of the above scenario, it is possible to define a joint notion of convergence for states and measurements that preserves the complete mathematical structure of quantum theory in the macroscopic limit. Furthermore, we will show how this formalism specifically applies to device-independent quantities such as correlations (both spatial, as in Bell experiments, and temporal, as in Leggett-Garg experiments). In other words, we will show that, in the limit N →∞, states _̊N ∈𝒟(𝔥^ N) can be mapped to states ρ∈𝒟(ℌ) in some “limit" Hilbert space ℌ and Kraus operators K_N(x|A) can be mapped to Kraus operators K(x|A) acting on ℌ in a way such that the essential ingredients of quantum mechanics, including the Born rule, the superposition principle and the incompatibility of measurements, are retained. To do this, let us introduce the concepts of macroscopic quantum representation and (robust) macroscopic quantum behavior of order n (MQB_n).



    Given a closed subspace ℌ_N ⊆𝔥^⊗ N and a subset 𝒜⊆ℬ( 𝔥 ), a macroscopic quantum representation is a limit of the form
    
    ( ℌ_N, _̊N, K_N(x|A) ) ( ℌ, ,̊ K(x|A) )

    for all _̊N ∈𝒟(ℌ_N) and for all A ∈𝒜, where
    
        
  * ℌ is a Hilbert space;
        
  * the limit state ∈̊𝒟 is a “linear" function of _̊N, in the sense that if the pure states |⟩_N and |⟩_N are mapped, respectively, to |⟩ and |⟩, then the linear combination |⟩_N + |̱%̱s̱⟩̱_N is mapped to the linear combination |⟩ + |̱%̱s̱⟩̱;
        
  * the limit Kraus operators K( x | A)  ∈ℬ(ℌ) form a non-compatible set of measurements. 
    



    A closed subspace ℌ_N ⊆𝔥^⊗ N and a subset 𝒜⊆ℬ( 𝔥 ) possess MQB_n for some n ∈ℕ if there exists a macroscopic quantum representation
    
    ( ℌ_N, _̊N, K_N(x|A) ) →( ℌ, ,̊ K(x|A) )

    such that for every state ρ_N ∈𝒟 (ℌ_N) and for every sequence of measurement settings (A_1, …, A_n ) ∈𝒜^ n, the random vector 
    
    X⃗_N ∼[ K_N^(n)… K_N^(1) _̊N    K_N^(1) †… K_N^(n) †]

     (or an affine transformation thereof, i.e. ł⃗_N ⊙X⃗_N + _N for some suitably chosen ł⃗_N, _N ∈ℝ^n), where K_N^(j) = K_N(x_j |A_j), converges in distribution as N →∞ to some random vector
    
    X⃗∼[  K^(n)… K^(1) ρ   K^(1) †… K^(n) †]   ,
  
    where K^(j) = K(x_j | A_j) (as given by the macroscopic quantum representation).


In references <cit.>, only the case of a single measurement was considered (n=1), and the definition of “MQB" given there corresponds to MQB_1 as defined here. Now we define a stronger notion, where the macroscopic quantum representation is robust against decoherence and losses as introduced in assumptions <ref> and <ref> respectively.


    A closed subspace ℌ_N ⊆𝔥^⊗ N and a subset 𝒜⊆ℬ( 𝔥 ) possess robust MQB_n for some n ∈ℕ if there exist a macroscopic quantum representation
    
    ( ℌ_N, _̊N, K_N(x|A) ) →( ℌ, ,̊ K(x|A) )

    and ϵ > 0 such that for every state ρ_N ∈𝒟 (ℌ_N), for every sequence of measurement settings (A_1, …, A_n ) ∈𝒜^ n and for every sequence of channels (Ł^(1), …, Ł^(n) ) of the form (<ref>) with max_j Ł^(j) - Id≤ϵ, the random vector 
    
    X⃗_N ∼[ K_N^(n)Ł^(n)( … K_N^(1) Ł^(1)(_̊N)    K_N^(1) †…) K_N^(n) †]   ,

     (or an affine transformation ł⃗_N ⊙X⃗_N + _N for some suitably chosen ł⃗_N, _N ∈ℝ^n), where K_N^(j) = K_N(x_j |A_j), converges in distribution as N →∞ to some random vector
    
    X⃗∼[  K^(n)… K^(1) ρ   K^(1) †… K^(n) †]   ,
  
    where K^(j) = K(x_j | A_j) (as given by the macroscopic quantum representation).




  §.§.§ An example

To illustrate these ideas, consider the case where 𝔥 = ℂ^2 (thus, particles or subsystems are qubits). In this case, the single-particle projective measurements have two possible outcomes, which we label +1 and -1. Let us define the N-particle Dicke states <cit.>

    |N,k⟩ := 1/√(N  k)( |1…1_k 0…0 ⟩ + permutations)   ,

the subspace of (ℂ^2)^ N generated by the first d Dicke states 

    𝔇_N := {|N,k⟩  ,   k = 0 , 1 , …, d-1  }

and the set of non-diagonal observables on ℂ^2

    𝒩𝒟 := { A ∈ℬ(ℂ^2) : A^† = A   and  ⟨ 0 | A | 1 ⟩≠ 0 }  .

Then we have the following result:

    The Dicke subspace 𝔇_N with dimension d ≪ N (in the sense that lim_N →∞ d/ N = 0, which holds, for instance, if d is fixed) and the set 𝒩𝒟 possess robust MQB_1.



    See Appendix <ref>.

In particular, as shown in the Appendix <ref>, the MQB_1 is given by the macroscopic quantum representation
    
    ( 𝔇_N, |N,k⟩, K_N(x|A) ) →( L^2(ℝ), |k⟩, K(x|A) )   .

    Here, L^2(ℝ) is the space of square-integrable functions, |k⟩ are number states (energy eigenstates of the quantum harmonic oscillator) and
    
    K(x|A) =  e^-(X_ȷ-x)^2/(2 ^̱2)/(π^̱2)^1/4  ,

    where X_ȷ = X cosȷ + P sinȷ are phase space observable in terms of position X and momentum P observables and angle ȷ = ⟨ 0 | A | 1 ⟩ and
    
    ^̱2 = ^2 + p 0^†(A^2)0 - p^2 0^†(A)0^2/p^2 |0Γ^†(A)1 |^2 - 1   ,

    in terms of the probability p and decoherence channel  defined in Eq. (<ref>). We conjecture that this macroscopic quantum representation  for the considered system constitutes a robust MQB_n for all n:

    The Dicke subspace 𝔇_N (with dimension d ≪ N) and the set 𝒩𝒟 possess robust MQB_n for all n ∈ℕ.


We prove a weaker result that, together with Theorem <ref>, supports Conjecture <ref>:

    The Dicke subspace 𝔇_N (with dimension d ≪ N) and the set 𝒩𝒟 possess MQB_n for all n ∈ℕ.



    See Appendix <ref>.

Moreover, as shown in Appendix <ref>, all MQB_n are given by the same macroscopic quantum representation as before

    ( 𝔇_N, |N,k⟩, K_N(x|A) ) →( L^2(ℝ), |k⟩, K(x|A) )

with =̱ (i.e., the value of $̱ is given by Eq. (<ref>) withp=1and= Id).§ DEVICE-INDEPENDENT TESTS OF MACROSCOPIC NON-CLASSICAL BEHAVIORS

We have shown that the above system, consisting of Dicke states𝔇_Nand collective measurements (as long as they are not represented by diagonal hermitian operators), possesses  MQB_nfor alln ∈ℕas well as robust MQB_1, which lead us to conjecture that it does indeed possess robust MQB_nfor alln ∈ℕ. These results strongly hint that the quantum nature of the system can be observed at the macroscopic scale. In order to make this statement concrete, we use device-independent tests that witnesses the non-classical nature of our system at the macroscopic scale. In particular, we will see that our system possesses nonlocal correlations (in the sense of Bell <cit.>), ruling out a local realistic description of the outcome statistics in the macroscopic limit. We also show a violation of a Leggett-Garg inequality <cit.>, ruling out a macroscopic realistic description of the statistics.





  §.§.§ Violation of a Bell inequality

Consider the bipartite macroscopic measurement scenario depicted in Figure <ref>: a system𝖲, subject to assumptions <ref> - <ref>, is divided in two parts which are sent to measurement apparatuses𝖬_Aand𝖬_B, each of which satisfies assumptions <ref> - <ref>. Suppose the system is in a state of the form∑_k =0^d-1 c_k |N,k⟩_A |N,k⟩_B ∈𝔇_N 𝔇_N, and suppose that Alice selects a single-particle observableA ∈𝒩𝒟obtaining an outcomex ∈ℝ. Likewise, Bob selectsB ∈𝒩𝒟obtainingy ∈ℝ. Then, by applying Theorem <ref> to each party locally, the limit bipartite distribution is given by

    P(x,y|A,B) = ⟨ψ|  E(x|A)  E(y|B) |ψ⟩  ,

where|ψ⟩ = ∑_k=0^d-1 c_k |k⟩_A |k⟩_B ∈L^2(ℝ) L^2(ℝ)andE(x|A) = K(x|A)^†K(x|A)withK(x|A)given by (<ref>). This distribution does not admit a local hidden variable model, as we showed by explicit violation of a Bell-CHSH inequality in <cit.>.





  §.§.§ Violation of a Leggett-Garg inequality

We now consider a macroscopic Leggett-Garg experiment as depicted in Figure <ref>: a system𝖲, subject to assumptions<ref>-<ref>, is consecutively sent into two measurements apparatuses𝖬_Aand𝖬_B, which satisfy conditions <ref> - <ref>. Suppose the system is initially in the state∑_k=0^d-1 c_k |N,k⟩ ∈𝔇_N. Next, suppose that during the first measurement, defined by a single-particle observableA ∈𝒩𝒟, an outcomex ∈ℝis obtained. Similarly, for the second measurement, we have the associated observableB ∈𝒩𝒟resulting in an outcomey ∈ℝ. Then, applying Theorem <ref> (namely the MQB_2of the system), the limit bipartite distribution is given by

    P(x,y|A,B) = ⟨ψ| K(x|A)^† K(y|B)^† K(y|B) K(x|A) |ψ⟩  ,

where|ψ⟩ = ∑_k=0^d-1 c_k |k⟩ ∈L^2(ℝ)andK(x|A)are given by (<ref>). Now consider the following Leggett-Garg CHSH inequality <cit.>
    C = ⟨ a_1 b_1 ⟩ + ⟨ a_1 b_2 ⟩ + ⟨ a_2 b_1 ⟩ - ⟨ a_2 b_2 ⟩≤ 2    ,

wherea_i = (x|A_i)andb_i = (y|B_i). Then, as we show in the Appendix <ref>, the state 

    |ψ⟩ = √(1/2 - 577/2 √(1244179)) |0⟩ + √(1/2 + 577/2 √(1244179)) |2⟩  .

gives

    C = 2/675 π( 577 + √(1244179) + 2700 arctan1/3) ≃ 2.42

forA_i = _x cosȷ_i + _y sinȷ_iandB_i = _x cos_i + _y sin_iwith the following set of angles

    ȷ_1 = π/4  ,   ȷ_2 = 3 π/4  ,   _1 = π/2  ,   _2 = 0   .

This violation of a Leggett-Garg inequality rules out a macroscopic realistic description of the correlations obtained.




§ OUTLOOK AND OPEN QUESTIONS

Our results shed new light on the question of the quantum-to-classical transition, suggesting that genuine quantum phenomena might be more robust than previously thought. There are several interesting questions to be addressed in the future:

  *  Question of scale. Our results show that genuine quantum behavior can be visible through measurements with a precision of the order of √(N), even in the presence of (single-particle) decoherence and losses. The relevant parameter that defines the scale of these quantum effects is, therefore, the resolution of the measurements as a function of the system's size N. An open question is whether genuine quantum effects exist at a scale larger than √(N), thus surviving even more coarse-graining than the system we consider. The results of Kofler and Brukner <cit.> indicate that this is not possible, showing classicality for a resolution much larger than √(N). But their result only applies to finite-dimensional systems, and the case of infinite dimensional systems (in our language above, the case where the single-particle Hilbert space 𝔥 is infinite-dimensional) is still to be investigated. 

    
  *  Macroscopicity measures. The question of macroscopic quantum states dates back to Schrödinger <cit.>, and still today work is done to characterize the “macroscopicity” of quantum states (see <cit.> and references therein).  An example of a quantum state that is typically thought to be macroscopically quantum is the Greenberger-Horne-Zeilinger (GHZ) <cit.> state (|0⟩^ N + |1⟩^ N)/2 (also called cat-like state). However, such a state is extremely fragile, since the loss of coherence of a single particle destroys the coherence of the global state, collapsing it into a classical mixture. For this reason, the GHZ state is not useful for our purposes. Another state typically associated with macroscopic quantumness is the W state <cit.>, which in the language of this work corresponds to the Dicke state |N,1⟩. These two examples show that quantum macroscopicity does not necessarily result in non-classicality in the macroscopic limit as defined in our sense, and robustness seems to be a key requirement. More precise relations are to be left for future considerations.
    
    
  *  Classical limit and infinite tensor products. There is a recent proposal that suggests that focusing on type I operator algebras might be the source of the problem and considering instead quantum mechanics on type II operator algebras might provide a framework that encompasses the classical macroscopic limit in a natural way (see <cit.>). This seems to provide another formalism to arrive at the macroscopic limit, and an interesting question to be investigated is how these findings relate to our result.
    
    
  *  Experimental implementations. Of particular interest are experimental considerations to test our findings. Some potential experimental settings that seem to provide the appropriate characteristics are atomic memories and Bose-Einstein condensates (see e.g. <cit.>). While our results are derived in the explicit limit N→∞, one has to derive concrete bounds for finite N or at least provide numerical simulations.
    

Acknowledgments.—  We would like to thank Joshua Morris for helpful comments. This research was funded in whole, or in part, by the Austrian Science Fund (FWF) [10.55776/F71] and [10.55776/P36994]. M.G. also acknowledges support from the ESQ Discovery programme (Erwin Schrödinger Center for Quantum
Science & Technology), hosted by the Austrian Academy
of Sciences (ÖAW). For open access purposes, the author(s) has applied a CC BY public copyright license to any author accepted manuscript version arising from this submission. 
apsrev4-1

§ KRAUS OPERATORS

Let the initial joint state of the system and pointer be described by the density matrix_̊𝖲 _̊𝖯where_̊𝖲is the initial state of the system in Fock-like space⊕_M=0^N 𝔥^Mand_̊𝖯 = |Φ⟩⟨Φ|is the initial state of the pointer. It is convenient to write_̊𝖲in the eigenbasis of theA_i's, i.e.

    _̊𝖲 = ⊕_M=0^N  ∑_[ a_1 … a_M; b_1 … b_M ] C_[ a_1 … a_M; b_1 … b_M ] |a_1… a_M⟩_s   _s⟨b_1 … b_M|  .

Then, after unitary interactionU = exp{ - i ⊕_M=1^N ∑_i=1^M A_i(s) P }, the joint state of system and pointer is

    U  _̊𝖲_̊𝖯   U^†    = ⊕_M=0^N ∑_[ a_1 … a_M; b_1 … b_M ] C_[ a_1 … a_M; b_1 … b_M ]   e^-i ∑_i=1^M A_i(s)  P |a_1… a_M⟩_s   _s⟨b_1 … b_M||Φ⟩⟨Φ|  e^i ∑_i=1^M A_i(s)  P
        = ⊕_M=0^N ∑_[ a_1 … a_M; b_1 … b_M ] C_[ a_1 … a_M; b_1 … b_M ]   e^-i ( ∑_i=1^M a_i )  P |a_1… a_M⟩_s   _s⟨b_1 … b_M||Φ⟩⟨Φ|  e^i ( ∑_i=1^M b_i ) P
        = ⊕_M=0^N ∑_[ a_1 … a_M; b_1 … b_M ] C_[ a_1 … a_M; b_1 … b_M ]|a_1… a_M⟩_s   _s⟨b_1 … b_M| e^-i ( ∑_i=1^M a_i )  P |Φ⟩⟨Φ|  e^i ( ∑_i=1^M b_i ) P  .

If we measure the position of the pointer obtaining the valuex, then the state of the system is projected to

    _𝖯x U  _̊𝖲_̊𝖯    U^†x_𝖯    = ⊕_M=0^N ∑_[ a_1 … a_M; b_1 … b_M ] C_[ a_1 … a_M; b_1 … b_M ]|a_1… a_M⟩_s   _s⟨b_1 … b_M| xe^-i ( ∑_i=1^M a_i )  P |Φ⟩⟨Φ|  e^i ( ∑_i=1^M b_i ) Px
        = ⊕_M=0^N ∑_[ a_1 … a_M; b_1 … b_M ] C_[ a_1 … a_M; b_1 … b_M ]|a_1… a_M⟩_s   _s⟨b_1 … b_M| Φ( x - ∑_i=1^M a_i )  Φ^* ( x - ∑_i=1^M b_i ) 
        = K_N(x|s)  _̊𝖲 K_N(x|s)^†  ,

where 

    K_N(x|s) = ⊕_M=0^N ∑_a_1 … a_MΦ( x - ∑_i=1^M a_i )  |a_1… a_M⟩_s   _s⟨b_1 … b_M|  ,

and the probability of obatining such outcomexis[ K_N(x|s) _̊𝖲 K_N^†(x|s) ], proving Equations (<ref>), (<ref>) and (<ref>) in the main text.




§ PROOF OF THEOREM <REF>.

Consider the random variable X_N with distribution

    P_N(x) = [K_N(x|A)  Ł(_̊N) K_N^† (x|A) ]   ,

where _̊N = ∑_k,l=0^d_N-1 c_kl|N,l⟩⟨N,k|, d_N satisfies lim_N →∞ d_N/N=0 and

    K_N(x|A) = ⊕_M=0^N ∑_a_1 … a_MΦ( x - ∑_i=1^M a_i )   Π_a_1 | A⊗…⊗Π_a_M | A  .

By Lévy's continuity theorem, in order to show that X_N converges in distribution it is sufficient to show that its characteristic function χ_N(t)
converges pointwise to some function χ(t) continuous at t=0. We have

    χ_N(t)     =  ∫_-∞^+∞ dx    e^itx P_N(x) 
        =  ∫_-∞^+∞ dx    e^itx [K(x|A) Ł (_̊N )   K_N^† (x|A) ] 
        = [ Ł (_̊N)  ∫_-∞^+∞ dx   e^itx  K_N^† (x|A) K_N(x|A) ] 
        = [ Ł(_̊N) ⊕_M=0^N ∑_a_1 … a_M∫_-∞^+∞ dx   e^itxΦ^2 ( x - ∑_i=1^M a_i )   Π_a_1 | A⊗…⊗Π_a_M | A]   .

Using that 

    ∫_-∞^+∞ dx    e^itx Φ^2 ( x - ∑_i=1^M a_i )     =  ∫_-∞^+∞ dx    e^itx e^-(x- ∑_i=1^M a_i)^2/(2 N ^2)/√(2 π N ^2)
        = e^i t ∑_i=1^M a_i - N ^2 t^2 /2  ,

we have

    χ_N(t)     =  e^- N ^2 t^2 /2 [ Ł(_̊N) ⊕_M=0^N ∑_a_1 … a_M  e^i t ∑_i=1^M a_i Π_a_1 | A⊗…⊗Π_a_M | A] 
        = e^- N ^2 t^2 /2 [ Ł(_̊N)  ⊕_M=0^N ( ∑_a_1 e^i t a_1Π_a_1|A) …( ∑_a_M e^i t a_MΠ_a_M|A) ] 
        = e^- N ^2 t^2 /2 [ Ł(_̊N)  ⊕_M=0^N ( e^i t A)^ M] 
        =   e^- N ^2 t^2 /2 [ ⊕_M=0^N   N  M p^M (1-p)^N-M 𝒮{_N-M[ ^⊗ N  (_̊N)  ] }( e^itA)^ M]  
        = e^- N ^2 t^2 /2 ∑_M=0^N N  M p^M (1-p)^N-M_M [_N-M[ ^⊗ N (_̊N) ] ( e^itA)^ M] 
        = e^- N ^2 t^2 /2 ∑_M=0^N N  M p^M (1-p)^N-M_N [ ^⊗ N (_̊N)  ( e^itA)^ M𝕀^ (N-M)] 
        =e^- N ^2 t^2 /2 [ ^⊗ N (_̊N)  ∑_M=0^N N  M p^M (1-p)^N-M ( e^itA)^ M𝕀^ (N-M)] 
        = e^- N ^2 t^2 /2 [ ^⊗ N (_̊N)  ( 1-p + p   e^itA)^ N] 
        = e^- N ^2 t^2 /2 [  _̊N   ^†( 1-p + p   e^itA)^ N]   .

We now perform an affine transformation X_N ↦ł_N X_N + _N of the random variable with ł_N = 1/√(2 N p^2 |G_01|^2) and _N = -  G_00√(N) / √(2 |G_01|^2 ), where G_ij := iΓ^†(A)j. The characteristic function of the new random variable is

    χ_N(t)     = e^- i t  G_00√(N)/√(2 |G_01|^2) - ^2  t^2 / (4 p^2 |G_01|^2) [  _̊N   Γ^†(1-p+p   e^i t A /√(2 N p^2 |G_01|^2))^ N]   .

Defining 𝒢 = Γ^†(1-p+p   e^i t A /√(2 N p^2 |G_01|^2)), the matrix element N,k𝒢^⊗ NN,l in the case k ≥ l is:

    ⟨N, k| 𝒢^ N |N, l⟩    = 1/√(NkNl)( 1…1_k 0…0 + perm.) 𝒢^ N( 1…1_l 0…0 +  perm.) 
        = √(Nk/Nl) 1…1_k 0…0 𝒢^ N( 1…1_l 0…0 +  perm.) 
        =  √(Nk/Nl) ∑_m=0^l   kmN-kl-m 1…1_k 0…0 𝒢^ N1…1_m0…0_k1..1_l-m 0…0_N-k
        = √(Nk/Nl) ∑_m=0^l  kmN-kl-m 1𝒢1^m 1𝒢0^k-m 0𝒢1^l-m 0𝒢0^N-k-l+m
        =∑_m=0^l  √(k!   l!   N^k+l-2m )/ m! (k-m)! (l-m)! [ 1 + O(1/N) ]  1𝒢1^m 1𝒢0^k-m 0𝒢1^l-m 0𝒢0^N-k-l+m  .

In the second equality we have used permutational invariance; in the third equality we have gathered all the terms that contribute equally, multiplied by their combinatorial multiplicity; in the last line we have used Stirling's formula. Expanding 𝒢 as

    𝒢    = ^†( 1-p + p   e^i t A /√(2 N p^2 |G_01|^2)) 
        = ^†( 1 +  p i t A/√(2 N p^2 |G_01|^2) -p   t^2 A^2/4 N p^2 |G_01|^2 + O(N^-3/2) ) 
        = 1 +  i t/√(2 N |G_01|^2)^† (A)-   t^2/4 N p|G_01|^2^†(A^2) + O(N^-3/2)    ,

we can see that 1𝒢1^m = 1 + O(1/√(N)) and 0𝒢0^m = 1 + O(1/√(N)), while 1𝒢0^k-m = ( it /√(2 N |G_01|^2)  G_10)^k-m[ 1 + O(1/√(N)) ] and 0𝒢1^l-m = (  it/√(2 N |G_01|^2)  G_01)^l-m[ 1 + O(1/√(N)) ]. These first order contributions of the off-diagonal matrix elements cancel the overall factor of √(N^k+l-2m), while higher order corrections are suppressed. On the other hand,

    0𝒜0^N-k-l    = exp{ (N-k-l) log0𝒢0}
        = exp{ (N-k-l) log[ 1 + it /√(2N |G_01|^2)0^†(A)0 - t^2/4 N p |G_01|^20^†(A^2)0 + O(N^-3/2) ] }
        = exp{ (N-k-l) [ i t G_00/√(2 N |G_01|^2) - t^2 G^(2)_00/4 N p |G_01|^2 +  t^2 G_00^2 /4  N |G_01|^2 + O(N^-3/2) ] }
        = exp{i t G_00√(N)/√(2 |G_01|^2)  - t^2/4 p |G_01|^2( G_00^(2) - p G_00^2   ) +O(1/√(N)) }  .

Then, defining G = ^†(A) and G^(2) = ^†(A^2), the matrix element reads

    N,k𝒜^ NN,l    = e^i t G_00√(N) / √(2 |G_01|^2)   -t^2 (G_00^(2) - p G_00^2) / (4 p |G_01|^2)  + O(1/√(N))
               ·∑_m=0^l √(k!   l!)/m! (k-m)! (l-m)!(  i t G_10/√(2 |G_01|^2))^k-m( i t G_01/√(2 |G_01|^2))^l-m[ 1 +O(1/√(N)) ]   .

For the case l>k all we have to do is exchange k and l, so that the sum only runs until the smallest of the two, and also exchange G_10 and G_01. Then, defining ^2 = ^2/(p^2 |G_01|^2) +  (G^(2)_00 - p G_00^2)/(p |G_01|^2) and G_01=|G_01| e^i ȷ, the characteristic function in the limit reads

    lim_N →∞χ_N(t)     = e^-^2 t^2/4∑_k,l e^i l ȷ c_kl e^-ikȷ∑_m=0^min (k,l)√(k!   l!)/m! (k-m)! (l-m)!( i t/√(2))^k+l-2m  .

Since this is a continuous characteristic function, we conclude that the random variable ł_N X_N + _N converges in distribution. In order to compute its distribution, we take the Fourier transform of the above expression:

    P(x)     = 1/2 π∫_-∞^+∞ dt   e^-itx χ(t) 
        =   ∑_k,l e^i l ȷ c_kl e^-ikȷ∑_m=0^min (k,l)√(k!   l!)/m! (k-m)! (l-m)!(1/√(2))^k+l-2m1/2 π∫_-∞^+∞ dt  ( i t  )^k+l-2m  e^-itx- ^2 t^2/4
        =  ∑_k,l e^i l ȷ c_kl e^-ikȷ∑_m=0^min (k,l)√(k!   l!)/m! (k-m)! (l-m)!( 1/√(2))^k+l-2m (-1)^k+l-2md^k+l-2m/dx^k+l-2m1/2 π∫_-∞^+∞ dt    e^-itx- ^2 t^2/4
        =   ∑_k,l e^i l ȷ c_kl e^-ikȷ∑_m=0^min (k,l)√(k!   l!)/m! (k-m)! (l-m)!(1/√(2))^k+l-2m (-1)^k+l-2md^k+l-2m/dx^k+l-2me^-x^2/ ^2/√(π^2)  .

Using Rodrigues' formula for Hermite polynomials <cit.>
    H_n(x) = (-1)^n e^x^2d^n/dx^n e^-x^2

we have

    P(x)     = ∑_k,l  e^i l ȷ c_kl e^-ikȷ∑_m=0^min (k,l)√(k!   l!)/m! (k-m)! (l-m)!( 1/√(2))^k+l-2me^- x^2/^2/^k+l-2mH_k+l-2m ( x/)/√(π^2)
        = e^- x^2/^2/√(π^2 )∑_k,l e^i l ȷ c_kl e^-ikȷ√(k!   l!)∑_m=0^min (k,l)1/m!k+l-2m  l-m( 1/)^k+l-2mH_k+l-2m ( x/)/√(2^k+l-2m)  (k+l-2m)!  ,

where we have introduced the binomial coefficient for convenience. Now, the above sum over Hermite polynomials can be written as a product of two Hermite polynomials of order k and l by virtue of the following lemma:

For any constants , $̱ andsatisfying^2=^̱2+^2, and for any non-negative integerskandlwithk ≥ l, the following identity of Hermite polynomials holds:

    e^- x^2/^2/√(π^2)∑_m=0^l     1/m! k+l-2m  l-m( /)^k+l-2m H_k+l-2m(x/ )/√(2^k+l-2m)  (k+l-2m)!  = 
                = ∫_-∞^+∞ dx'  e^-( x-x')^2/^̱2 /√(π^̱2)e^-(x')^2/^2/√(π^2 ) H_k ( x'/)/√(2^k)  k!H_l (x'/)/√(2^l)  l!  .

See <cit.>.

Then, the limit distribution may be written as

    P(x)     = ∑_k,l  e^i l ȷ c_kl e^-ikȷ√(k!   l!)∫_-∞^+∞ dx'  e^-( x-x')^2/^̱2 /√(π^̱2)e^-(x')^2/√(π)  H_k (x' )/√(2^k)  k! H_l (x' )/√(2^l)  l!
        = ∑_k,l  e^i l ȷ c_kl e^-ikȷ∫_-∞^+∞ dx'  e^-( x-x')^2/^̱2 /√(π^̱2)[  e^-(x')^2/2  H_k ( x') /√(2^k   k!  √(π))] [ e^-(x')^2/2 H_l (x' ) /√(2^l   l!  √(π))] 
        = ∑_k,l  e^i l ȷ c_kl e^-ikȷ∫_-∞^+∞ dx'  e^-( x-x')^2/^̱2 /√(π^̱2)⟨k|x'⟩⟨x'|l⟩  ,

where^̱2 = ^2 - 1and we have introduced the wave-functions

    ⟨x|k⟩ =  1/√(2^k   k! √(π))  e^- x^2/2   H_k ( x )

of a one dimensional harmonic oscillator. In conclusion, we can write the limit probability distribution as

    P(x) = [  K(x|A)   K^†(x|A) ]    ,

where=̊∑_kl c_kl|l⟩⟨k|and

    K(x|A) =  ∫_-∞^+∞ dx'  e^-(x-x')^2/(2 ^̱2)/(π^̱2)^1/4 |x'⟩_ȷ  _ȷ⟨x'|  ,

where 

    ^̱2 = ^2 + p 0^†(A^2)0 - p^2 0^†(A)0^2/p^2 |0Γ^†(A)1 |^2 - 1

and

    ȷ = 0^†(A)1  .


§ PROOF OF THEOREM <REF>.

Now consider the random vectorX⃗_Nwith distribution

    P_N(x_1, …, x_n | A_1 , … , A_n) = [ K_N(x_n|A_n) … K_N(x_1|A_1) _̊N K_N^†(x_1|A_1) … K_N^†(x_n|A_n) ]   .

We want to show thatX⃗_Nconverges in distribution to a random vectorX⃗with distribution

    P(x_1, …, x_n | A_1 , … , A_n) = [ K(x_n|A_n) … K(x_1|A_1) K̊^†(x_1|A_1) … K^†(x_n|A_n) ]   .

For this, it is sufficient to show that

    ⟨ N , k | K̃_N^(1)…K̃_N^(n)K̃_N^(n+1) | N,l ⟩= ∑_m=0^∞( ⟨ N,k | K̃_N^(1)…K̃_N^(n) | N , m ⟩) ( ⟨ N,m | K̃_N^(n+1) | N , l ⟩)   ,

where

    K̃_N^(j)    = K̃_N(x_j | A_j) 
        = 1/√(ł_N^(j))   K_N ( x_j - _N^(j)/ł_N^(j)  | A_j ) 
        = 1/√(ł_N^(j))⊕_M=0^N ∑_a_1 … a_Me^-( (x_j - _N^(j))/ł_N^(j) - ∑_i=1^M a_i )^2/(4 N ^2) /(2 π N ^2)^1/4Π_a_1 | A_j…Π_a_M | A_j
        = 1/( 2 π N ^2 (ł_N^(j))^2 )^1/4⊕_M=0^N ∑_a_1 … a_M√( N ^2 (ł_N^(j))^2/π)∫ dp   e^i p x_j  e^-N ^2 (ł_N^(j))^2 p^2 - i p (_N^(j) + ł_N^(j)∑_i=1^M a_i ) Π_a_1 | A_j…Π_a_M | A_j
        = ( N ^2 (ł_N^(j))^2 /2 π^3)^1/4∫ dp   e^i p x_j  - N ^2 (ł_N^(j))^2 p^2
               ·⊕_M=0^N e^-ip_N^(j) (1-M/N)( ∑_a_1 e^-i p ( ł_N^(j) a_1 + _N^(j)/N )Π_a_1|A_j) …( ∑_a_M e^-i p (ł_N^(j) a_M + _N^(j)/N)Π_a_M|A_j)

for some constantsł_N^(j)and_N^(j). Choosing as beforeł_N^(j) = (2 N |(A_j)_01|^2)^-1/2and_N^(j) = - (A_j)_00√(N) / √(2 | (A_j)_01|^2)and defining𝒜_j = ∑_a e^-i p (ł_N^(j) a+ _N^(j) /N)Π_a|A_j, we have

    K̃_N^(j)    = ( ^2  /4 π^3 |(A_j)_01|^2)^1/4∫ dp   e^i p x_j - ^2 p^2 / (2 |(A_j)_01|^2) ⊕_M=0^N e^-ip_N^(j) (1-M/N)𝒜_j^ M  .

Then the left hand side of (<ref>) is

    LHS    = ( 1/4 π^3)^(n+1)/4√(_1 …_n+1/ |(A_1)_01| … |(A_n+1)_01|)
               ·∫ dp_1 … dp_n+1 e^i ∑_j=1^n+1 x_j p_j  - 1/2∑_j=1^n+1^2 p_j^2 / |(A_j)_01|^2 ⟨ N,k | ( 𝒜_1 …𝒜_n+1)^ N | N, l ⟩  ,

while the right hand side is

    RHS    = ∑_m=0^∞{( 1/4 π^3)^n/4√(_1 …_n/ |(A_1)_01| … |(A_n)_01|)
                   ·∫ dp_1 … dp_n e^i ∑_j=1^n x_j p_j - 1/2∑_j=1^n^2 p_j^2 / |(A_j)_01|^2 ⟨ N,k | ( 𝒜_1 …𝒜_n)^ N | N, m ⟩}
             ·{( 1/4 π^3)^1/4√(_n+1/ |(A_n+1)_01|)∫ dp_n+1 e^i  x_n+1 p_n+1 - 1/2_n+1^2 p_n+1^2 / |(A_n+1)_01|^2 ⟨ N,m |  𝒜_n+1^ N | N, l ⟩}  .

It is then sufficient to show thatLHS' = RHS', where

    LHS' = ⟨ N,k | ( 𝒜_1 …𝒜_n+1)^ N | N, l ⟩

and

    RHS'     = ∑_m=0^∞( ⟨ N,k | ( 𝒜_1 …𝒜_n)^ N | N, m ⟩)  ( ⟨ N,m |  𝒜_n+1^ N | N, l ⟩)   .

Expanding

    𝒜_1 …𝒜_n     = ( 1 - i p_1 A_1-(A_1)_00/√(2 N |(A_1)_01|^2) - p_1^2/2( A_1-(A_1)_00)^2/2 N |(A_1)_01|^2 + O(N^-3/2) ) …
           …( 1 - i p_n A_n-(A_n)_00/√(2 N |(A_n)_01|^2) - p_n^2/2( A_n-(A_n)_00)^2/2 N |(A_n)_01|^2 + O(N^-3/2) ) 
        = 1 - i/√(2 N)∑_j=1^n p_j A_j - (A_j)_00/|(A_j)_01| - 1/4N∑_j=1^n p_j^2 ( A_j - (A_j)_00)^2/|(A_j)_01|^2 - 1/2N∑_i<j^n p_i p_j A_i - (A_i)_00/|(A_i)_01|A_j - (A_j)_00/|(A_j)_01| + O(N^-3/2)   ,

we have that 

    ⟨ 1 | 𝒜_1 …𝒜_n |1 ⟩    = 1 + O(N^-1/2)   , 
    ⟨ 1 | 𝒜_1 …𝒜_n |0 ⟩    = - i/√(2N)∑_j=1^n p_j e^-i ȷ_j+ O(N^-1)   ,
    ⟨ 0 | 𝒜_1 …𝒜_n |1 ⟩    = - i/√(2N)∑_j=1^n p_j e^i ȷ_j+ O(N^-1)   ,
    ⟨ 0 | 𝒜_1 …𝒜_n |0 ⟩    = 1 - 1/4N∑_j=1^n p_j^2 - 1/2N∑_i<j^n p_i p_j e^i (ȷ_i - ȷ_j)+ O(N^-3/2)   ,

where we have defined⟨ 0 | A_j | 1 ⟩ = |(A_j)_01| e^i ȷ_j, so that using (<ref>) (from the proof in the previous appendix) we have

    ⟨ N,k | 𝒜_1 …𝒜_n | N,l ⟩    = e^-1/4∑_j=1^n p_j^2 - 1/2∑_i<j^n p_i p_j e^i(ȷ_i-ȷ_j)
           ·∑_r=0^min(k,l)√(k! l!)/r! (k-r)! (l-r)!( -i/√(2)∑_j=1^n p_j e^-i ȷ_j)^k-r( -i/√(2)∑_j=1^n p_j e^i ȷ_j)^l-r  .

Therefore, defining

    a     = - i/√(2)∑_j=1^n p_j e^i ȷ_j  , 
    
        b     = - i/√(2)∑_j=1^n p_j e^-i ȷ_j  ,
    
        a'     = - i/√(2)  p_n+1 e^i ȷ_n+1  , 
    
        b     = - i/√(2) p_n+1 e^-i ȷ_n+1  ,

we have

    LHS'     = e^-1/4∑_j=1^n+1 p_j^2- 1/2∑_i<j^n+1 p_i p_j e^i(ȷ_i-ȷ_j)∑_r=0^min(k,l)√(k! l!)/r! (k-r)! (l-r)!( b+b')^k-r( a+a' )^l-r
        = √(k!   l!)  e^-1/4∑_j=1^n+1 p_j^2- 1/2∑_i<j^n p_i p_j e^i(ȷ_i-ȷ_j) f_kl(a,b,a',b')   ,

where we have defined the function

    f_kl (a, b, a', b')     = e^ab' (a+a')^l (b+b')^k ∑_r=0^min(k,l) (a + a')^-r ( b+b')^-r/r! (k-r)! (l-r)!
        = e^ab' (a+a')^l-min(k,l)  (b+b')^k-min(k,l)∑_r=0^min(k,l) (a + a')^min(k,l)-r  ( b+b')^min(k,l)-r/r! (k-r)! (l-r)!
        = e^ab' (a+a')^l-min(k,l)  (b+b')^k-min(k,l)∑_q=0^min(k,l) (a + a')^q  ( b+b')^q/( min(k,l)-q )!  (max(k,l)- min(k,l)+q )!   q!
        = e^ab'(a+a')^l-min(k,l)  (b+b')^k-min(k,l)/max(k,l)!∑_q=0^min(k,l)max(k,l) min(k,l)-q (a + a')^q  ( b+b')^q/ q!
        =  e^ab'(a+a')^l- min(k,l) (b+b')^k-min(k,l)/max(k,l)! L_min(k,l)^|k-l|(-(a + a') ( b+b') )   ,

in terms of the generalized Laguerre polynomials

    L_k^m(x) = ∑_q=0^k k+m  k-q(-x)^q/q!  .

On the other hand, we have that

    RHS'     =  ∑_m=0^∞{ e^-1/4∑_j=1^n p_j^2- 1/2∑_i<j^n p_i p_j e^i(ȷ_i-ȷ_j)∑_s=0^min(k,m)√(k! m!)/s! (k-s)! (m-s)! b^k-s a^m-s}
               ·{ e^-1/4  p_n+1^2∑_t=0^min(m,l)√(m! l!)/t! (m-t)! (l-t)! b'^m-t a'^l-t}
        = √(k!   l!)  e^-1/4∑_j=1^n+1 p_j^2- 1/2∑_i<j^n p_i p_j e^i(ȷ_i-ȷ_j) g_kl(a,b,a',b')

where we have defined the function

    g_kl(a, b, a', b')     = (a')^l b^k ∑_m=0^∞ m!   (ab')^m ∑_s=0^min(k,m)(a b)^-s/s! (k-s)! (m-s)!∑_t=0^min(m,l)(a'b')^-t/t! (m-t)! (l-t)!
        = (a')^l b^k ∑_m=0^∞ m!   (ab')^m   (ab)^-min(k,m)∑_s=0^min(k,m)(ab)^min(k,m) -s/s! (k-s)! (m-s)!  (a' b')^-min(m,l)∑_t=0^min(m,l)(a'b')^min(m,l)-t/t! (m-t)! (l-t)!
        = (a')^l b^k ∑_m=0^∞ m!   (ab')^m   (ab)^-min(k,m)∑_u=0^min(k,m)(ab)^u/( min(k,m)-u )!   ( max(k,m) - min(k,m) + u )!   u!
                           ·  (a' b')^-min(m,l)∑_v=0^min(m,l)(a'b')^v/( min(m,l)-v )!   ( max(m,l) - min(m,l) + v )!   v!
        = (a')^l b^k ∑_m=0^∞ m!   (ab')^m  (ab)^-min(k,m)/max(k,m)!∑_u=0^min(k,m)max(k,m) min(k,m)-u(ab)^u/u!
                           ·(a' b')^-min(m,l)/max(m,l)!∑_v=0^min(m,l)max(m,l) min(m,l)-v(a'b')^v/v!
        = (a')^l b^k ∑_m=0^∞ m!   (ab')^m  (ab)^-min(k,m)/max(k,m)! L_min(k,m)^|k-m| (-ab)  (a' b')^-min(m,l)/max(m,l)! L_min(m,l)^|m-l| (-a'b')   .

Therefore a sufficient condition for the identityLHS=RHSis thatf_kl(a,b,a',b') = g_kl(a,b,a',b')for allk,l ≥ 0and for alla, b, a', b' ∈ℂ, which holds by virtue of the following lemma:

    Let
    
    f_kl(a,b,a',b')     = e^ab'(a+a')^l-min(k,l)  (b+b')^k-min(k,l)/max(k,l)! L_min(k,l)^|k-l|( - (a+a') (b+b') )

    and
    
    g_kl (a,b,a',b')     = (a')^l b^k ∑_j=0^∞ j! (ab')^j (ab)^-min(k,j)/max(k,j)! L_min(k,j)^|k-j| (-ab) (a'b')^-min(l,j)/max(l,j)! L_min(l,j)^|l-j| (-a'b')   .

    Then f_kl(a,b,a',b') = g_kl(a,b,a',b') for every non-negative integers k and l and for every a,a',b,b' ∈ℂ. 

    The structure of the proof is as follows. First, seeing that f_00=g_00 is easy. Then we prove that, in the case l=0, both f and g satisfy the same recurrence relation, namely
    
    f_k+1,0 = b+b'/k+1 f_k0    and     g_k+1,0 = b+b'/k+1 g_k0  ,
        REC1

    which by induction on k implies that f_k0=g_k0 for all k ≥ 0. Next, for l≠ 0, we prove that, in the case k ≥ l+1, the following recurrence relations are satisfied:
    
    f_k,l+1=a+a'/l+1 f_kl + 1/l+1 f_k-1,l    and     g_k,l+1=a+a'/l+1 g_kl + 1/l+1 g_k-1,l  .
        REC2

    This, together with the previously established identity f_k0 = g_k0, implies by induction on l that f_k,l+1=g_k,l+1 for k ≥ l+1, so that we have f_kl=g_kl for k≥ l. Finally, since the identity is symmetric with respect to the exchange of k and l, a and b and a' and b', the identity is proven for all k and l. Therefore, it remains to prove equations (<ref>) and (<ref>). Before proceeding further, let us recall some useful identities of Laguerre polynomials:
    
    L_n^m(x)     = L_n^m+1(x) - L_n-1^m+1(x)   ,
        ∗
    
        n L_n^m(x)     = (n+m) L_n-1^m(x) - x L_n-1^m+1(x)   , 
        ∗∗

    (with the convention that L_n^m(x) = 0 if n < 0).
Step 1: proof of (<ref>). For l=0 we have
    
    f_k0    = e^ab'(b+b')^k/k!    and     g_k0  = b^k ∑_j=0^∞ (ab')^j (ab)^-min(k,j)/max(k,j)! L_min(k,j)^|k-j|(-ab)   .

    It is easy to see that the recurrence relation f_k+1,0 = b+b'/k+1 f_k0 is satisfied. Similarly,
    
    g_k+1,0    = b^k+1{∑_j=0^k (ab')^j (ab)^-j/(k+1)! L_j^k+1-j(-ab) + ∑_j=k+1^∞ (ab')^j (ab)^-k-1/j! L_k+1^j-k-1 (-ab)  }
        = b^k+1{1/(k+1)!∑_j=0^k ( b'/b)^j  L_j^k+1-j(-ab) + ∑_j=k+1^∞ (ab')^j (ab)^-k-1/j! j L_k^j-k-1(-ab) + ab L_k^j-k(-ab)/k+1}
        = b^k+1/k+1{1/k!∑_j=0^k ( b'/b)^j L_j^k+1-j(-ab)   
                      + ab' (ab)^-1∑_j=k+1^∞ (ab')^j-1(ab)^-k/(j-1)! L_k^j-k-1 (-ab)  + ∑_j=k+1^∞ (ab')^j(ab)^-k/j! L_k^j-k (-ab) }
        = b^k+1/k+1{1/k!∑_j=0^k ( b'/b)^j L_j^k+1-j(-ab)   
                    + b'/b∑_j=k^∞ (ab')^j(ab)^-k/j! L_k^j-k (-ab) + g_k0/b^k - ∑_j=0^k (ab')^j(ab)^-j/k! L_j^k-j (-ab) }
        = b/k+1 g_k0 +  b^k+1/k+1{1/k!∑_j=0^k ( b'/b)^j L_j^k+1-j(-ab)  
                                + b'/b[ g_k0/b^k - ∑_j=0^k-1 (ab')^j (ab)^-j/k! L_j^k-j(-ab) ]  - 1/k!∑_j=0^k ( b'/b)^j L_j^k-j (-ab) }
        = b+b'/k+1 g_k0 +  b^k+1/(k+1)!{∑_j=0^k ( b'/b)^j L_j^k+1-j(-ab)  
                                - b'/b∑_j=0^k-1( b'/b)^j L_j^k-j(-ab) - ∑_j=0^k ( b'/b)^j L_j^k-j (-ab) }
        =: b+b'/k+1 g_k0 +  b^k+1/(k+1)! R   ,

    where in the second equality we have used (<ref>) and in the last equality we have defined
    
    R     := ∑_j=0^k ( b'/b)^j [  L_j^k+1-j(-ab) - L_j^k-j(-ab) ] - b'/b∑_j=0^k-1( b'/b)^j L_j^k-j(-ab) 
        = ∑_j=1^k ( b'/b)^j  L_j-1^k+1-j(-ab)  -  ∑_j=0^k-1( b'/b)^j+1 L_j^k-j(-ab) 
        = 0   ,

where in the second equality we have used (<ref>). 
Step 2: proof of (<ref>). For l ≠ 0 and k≥ l we have

    f_kl    = e^ab' (b+b')^k-l/k! L_l^k-l(-(a+a')(b+b'))   ,

and for k ≥ l+1 we have the following recurrence relation for f:

    f_k,l+1    = e^ab'(b+b')^k-l-1/k! L_l+1^k-l-1(-(a+a')(b+b')) 
        = e^ab'(b+b')^k-l-1/k! k L_l^k-l-1(-(a+a')(b+b')) + (a+a')(b+b') L_l^k-l(-(a+a')(b+b'))/l+1
        = 1/l+1 e^ab'(b+b')^k-l-1/(k-1)! L_l^k-l-1(-(a+a')(b+b')) + a+a'/l+1 e^ab'(b+b')^k-l/k! L_l^k-l(-(a+a')(b+b')) 
        = 1/l+1 f_k-1,l + a+a'/l+1 f_kl  ,

where in the second equality we have used (<ref>). For g we have

    g_k,l+1    = (a')^l+1 b^k ∑_j=0^∞ j! (ab')^j (ab)^-min(k,j)/max(k,j)! L_min(k,j)^|k-j| (-ab) (a'b')^-min(l+1,j)/max(l+1,j)! L_min(l+1,j)^|l+1-j|(-a'b') 
        = (a')^l+1 b^k {∑_j=0^l j! (ab')^j (ab)^-j/k! L_j^k-j (-ab) (a'b')^-j/(l+1)! L_j^l+1-j(-a'b')  
                         + ∑_j=l+1^∞  j! (ab')^j (ab)^-min(k,j)/max(k,j)! L_min(k,j)^|k-j| (-ab) (a'b')^-l-1/j! L_l+1^j-l-1(-a'b') }
        = (a')^l+1 b^k {1/k!   (l+1)!∑_j=0^l j!  (a'b)^-j L_j^k-j (-ab)  L_j^l+1-j(-a'b')  
                      +  ∑_j=l+1^∞ j! (ab')^j (ab)^-min(k,j)/max(k,j)! L_min(k,j)^|k-j| (-ab)  (a'b')^-l-1/j! j L_l^j-l-1 (-a' b') + a' b' L_l^j-l(-a'b')/l+1}
        = (a')^l+1 b^k/l+1{1/k!   l!∑_j=0^l j! (a'b)^-j L_j^k-j (-ab)  L_j^l+1-j(-a'b')  
                      + (a'b')^-l-1∑_j=l+1^∞   (ab')^j (ab)^-min(k,j)/max(k,j)! L_min(k,j)^|k-j| (-ab)     j    L_l^j-l-1 (-a' b')   
                      +  ∑_j=l+1^∞  j! (ab')^j (ab)^-min(k,j)/max(k,j)! L_min(k,j)^|k-j| (-ab)  (a' b')^-l/j!   L_l^j-l(-a'b')}
        = (a')^l+1 b^k/l+1{1/k!   l!∑_j=0^l j! (a'b)^-j  L_j^k-j (-ab) L_j^l+1-j(-a'b')  
                      + (a'b')^-l-1∑_j=l+1^k  (ab')^j (ab)^-j/k! L_j^k-j (-ab)    j    L_l^j-l-1 (-a' b')   
                      + (a'b')^-l-1∑_j=k+1^∞   (ab')^j (ab)^-k/j! L_k^j-k (-ab)   j   L_l^j-l-1 (-a' b')   
                      + g_kl/(a')^l b^k - ∑_j=0^l j! (ab')^j (ab)^-j/k! L_j^k-j (-ab) (a'b')^-j/l! L_j^l-j(-a'b') }

    = a'/l+1 g_kl + (a')^l+1 b^k/k!   (l+1)!∑_j=0^l j! (a'b)^-j  L_j^k-j (-ab)  ( L_j^l+1-j(-a'b')  - L_j^l-j(-a'b') ) 
                  + b^k (b')^-l-1/k!   (l+1)∑_j=l+1^k j ( b'/b)^j L_j^k-j (-ab) L_l^j-l-1(-a' b') 
                  +a^-k+1 (b')^-l/l+1∑_j=k+1^∞(ab')^j-1/(j-1)! L_k^j-k(-ab) L_l^j-l-1(-a'b') 
        = a'/l+1 g_kl + (a')^l+1 b^k/k!   (l+1)!∑_j=1^l j! (a'b)^-j  L_j^k-j (-ab) L_j-1^l+1-j(-a'b') 
                  + b^k (b')^-l-1/k!   (l+1)∑_j=l+1^k  ( b'/b)^j j L_j^k-j (-ab) L_l^j-l-1(-a' b') 
                  +a^-k+1 (b')^-l/l+1∑_j=k^∞(ab')^j/j! L_k^j+1-k(-ab) L_l^j-l(-a'b')    
        = a'/l+1 g_kl + (a')^l+1 b^k/k!   (l+1)!∑_j=1^l j! (a'b)^-j  L_j^k-j (-ab)  L_j-1^l+1-j(-a'b') 
                  + b^k (b')^-l-1/k!   (l+1)∑_j=l+1^k  ( b'/b)^j( k L_j-1^k-j(-ab) + ab L_j-1^k-j+1(-ab) ) L_l^j-l-1(-a' b') 
                  +a^-k+1 (b')^-l/l+1∑_j=k^∞(ab')^j/j!( L_k^j-k (-ab) + L_k-1^j-k+1 (-ab) ) L_l^j-l(-a'b')    
        = a'/l+1 g_kl + (a')^l+1 b^k/k!   (l+1)!∑_j=1^l j! (a'b)^-j L_j^k-j (-ab)  L_j-1^l+1-j(-a'b') 
                  + b^k (b')^-l-1/(k-1)!   (l+1)∑_j=l+1^k  ( b'/b)^j  L_j-1^k-j(-ab)  L_l^j-l-1(-a' b') 
                  + a b^k+1 (b')^-l-1/k!   (l+1)∑_j=l+1^k  ( b'/b)^j  L_j-1^k-j+1(-ab) L_l^j-l-1(-a' b') 
                  +a^-k+1 (b')^-l/l+1∑_j=k^∞(ab')^j/j!  L_k^j-k (-ab) L_l^j-l(-a'b')    
                  +a^-k+1 (b')^-l/l+1∑_j=k^∞(ab')^j/j!  L_k-1^j-k+1 (-ab)  L_l^j-l(-a'b')   ,

where we have used (<ref>) in the third and eighth equalities and (<ref>) in the seventh and eighth. On the other hand we have 

    g_kl    = (a')^l b^k ∑_j=0^∞ j! (ab')^j (ab)^-min(k,j)/max(k,j)! L_min(k,j)^|k-j| (-ab) (a'b')^-min(l,j)/max(l,j)! L_min(l,j)^|l-j| (-a'b') 
        = (a')^l b^k ∑_j=0^k-1 j! (ab')^j (ab)^-j/k! L_j^k-j (-ab) (a'b')^-min(l,j)/max(l,j)! L_min(l,j)^|l-j| (-a'b') 
           + (a')^l b^k ∑_j=k^∞ j! (ab')^j (ab)^-k/j! L_k^j-k (-ab) (a'b')^-l/j! L_l^j-l (-a'b') 
        =  (a')^l b^k /k!∑_j=0^k-1 j! ( b'/b)^j  L_j^k-j (-ab) (a'b')^-min(l,j)/max(l,j)! L_min(l,j)^|l-j| (-a'b') 
           + a^-k (b')^-l∑_j=k^∞(ab')^j/j! L_k^j-k (-ab) L_l^j-l (-a'b')

and 

    g_k-1,l    = (a')^l b^k-1∑_j=0^∞ j! (ab')^j (ab)^-min(k-1,j)/max(k-1,j)! L_min(k-1,j)^|k-1-j| (-ab) (a'b')^-min(l,j)/max(l,j)! L_min(l,j)^|l-j| (-a'b')  
        = (a')^l b^k-1∑_j=0^k-1 j! (ab')^j (ab)^-j/(k-1)! L_j^k-1-j (-ab) (a'b')^-min(l,j)/max(l,j)! L_min(l,j)^|l-j| (-a'b')  
           +  (a')^l b^k-1∑_j=k^∞ j! (ab')^j (ab)^-k+1/j! L_k-1^j-k+1 (-ab) (a'b')^-l/j! L_l^j-l (-a'b')  
        = (a')^l b^k-1/(k-1)!∑_j=0^k-1 j! ( b'/b)^j  L_j^k-1-j (-ab) (a'b')^-min(l,j)/max(l,j)! L_min(l,j)^|l-j| (-a'b')  
           +  a^-k+1 (b')^-l∑_j=k^∞(ab')^j/j! L_k-1^j-k+1 (-ab)  L_l^j-l (-a'b')    .

Therefore,

    g_k,l+1    = a'/l+1 g_kl + (a')^l+1 b^k/k!   (l+1)!∑_j=1^l j! (a'b)^-j L_j^k-j (-ab)  L_j-1^l+1-j(-a'b') 
                  + b^k (b')^-l-1/(k-1)!   (l+1)∑_j=l+1^k  ( b'/b)^j  L_j-1^k-j(-ab)  L_l^j-l-1(-a' b') 
                  + a b^k+1 (b')^-l-1/k!   (l+1)∑_j=l+1^k  ( b'/b)^j  L_j-1^k-j+1(-ab) L_l^j-l-1(-a' b') 
                  + a/l+1[ g_kl - (a')^l b^k/k!∑_j=0^k-1 j! ( b'/b)^j L_j^k-j (-ab) (a'b')^-min(l,j)/max(l,j)! L_min(l,j)^|l-j|(-a'b') ]    
                  + 1/l+1[ g_k-1,l - (a')^l b^k-1/(k-1)!∑_j=0^k-1 j! ( b'/b)^j L_j^k-j-1(-ab) (a'b')^-min(l,j)/max(l,j)! L_min(l,j)^|l-j| (-a'b') ]   
        =: 1/l+1 g_k-1,l + a+a'/l+1 g_kl + 1/k!   (l+1) R   ,

where we have defined

    R  :=    (a')^l+1 b^k/l!∑_j=1^l j! (a'b)^-j L_j^k-j (-ab)  L_j-1^l+1-j(-a'b') + k b^k (b')^-l-1∑_j=l+1^k  ( b'/b)^j  L_j-1^k-j(-ab)  L_l^j-l-1(-a' b')  
         + a b^k+1 (b')^-l-1∑_j=l+1^k  ( b'/b)^j  L_j-1^k-j+1(-ab) L_l^j-l-1(-a' b') - a (a')^l b^k ∑_j=0^l-1 j! ( b'/b)^j L_j^k-j (-ab) (a'b')^-j/l! L_j^l-j(-a'b') 
        - a (a')^l b^k ∑_j=l^k-1 j! ( b'/b)^j L_j^k-j (-ab) (a'b')^-l/j! L_l^j-l(-a'b') - k (a')^l b^k-1∑_j=0^l-1 j! ( b'/b)^j L_j^k-j-1(-ab) (a'b')^-j/l! L_j^l-j (-a'b')
         - k (a')^l b^k-1∑_j=l^k-1 j! ( b'/b)^j L_j^k-j-1(-ab) (a'b')^-l/j! L_l^j-l (-a'b') 
    
        =    (a')^l+1 b^k/l!∑_j=1^l j! (a'b)^-j L_j^k-j (-ab)  L_j-1^l+1-j(-a'b') + k b^k-1 (b')^-l∑_j=l^k-1( b'/b)^j  L_j^k-j-1(-ab)  L_l^j-l(-a' b')
        + a b^k (b')^-l∑_j=l^k-1( b'/b)^j  L_j^k-j(-ab) L_l^j-l(-a' b') - a (a')^l b^k/l!∑_j=0^l-1 j!  (a'b)^-j L_j^k-j (-ab)  L_j^l-j(-a'b') 
         - a  b^k (b')^-l∑_j=l^k-1( b'/b)^j L_j^k-j (-ab)  L_l^j-l(-a'b') - k (a')^l b^k-1/l!∑_j=0^l-1 j!  (a'b)^-j L_j^k-j-1(-ab)  L_j^l-j (-a'b')
         - k  b^k-1 (b')^-l∑_j=l^k-1( b'/b)^j L_j^k-j-1(-ab)  L_l^j-l (-a'b')   .

Here the second term cancels the last and the third cancels the fifth, so we have

    R     = (a')^l b^k-1/l!{ a' b ∑_j=1^l j! (a'b)^-j L_j^k-j (-ab)  L_j-1^l+1-j(-a'b') 
                    - ab ∑_j=0^l-1 j!  (a'b)^-j  L_j^k-j (-ab)  L_j^l-j(-a'b')  - k ∑_j=0^l-1 j! (a'b)^-j  L_j^k-j-1(-ab)  L_j^l-j (-a'b') }
        = (a')^l b^k-1/l!{ a' b ∑_j=1^l j! (a'b)^-j L_j^k-j (-ab)  L_j-1^l+1-j(-a'b') -  ∑_j=0^l-1 j!  (a'b)^-j (j+1) L_j+1^k-j-1 (-ab)  L_j^l-j(-a'b') }
        = 0   ,

where in the second equality we have used (<ref>).


§ LEGGETT-GARG CHSH INEQUALITY VIOLATION

Here we compute the maximal violation of the Leggett-Garg CHSH inequality (<ref>) using states and measurements as obtained in the macroscopic limit (see main text). For convenience, we will work with phase-space representation of operators. In particular, we define

    W_ρ(x,p)     := 1/π∫_-∞^+∞ ds   e^2 i psx-sρx+s  , 
    
        E(x,p)     := 2 ∫_-∞^+∞ ds   e^-2 ips|x-s⟩⟨x+s|  ,

so thatρ = ∫ dx   dp   W_ρ(x,p) E(x,p). DefiningK_ȷ() := K( | cosȷ _x + sinȷ _y)corresponding to the case where Alice measures spin in the(cosȷ , sinȷ , 0)direction, we have

    K_0() ρ K_0()^†    = ∫_-∞^+∞ dx   dp   W_ρ(x,p) K_0() E(x,p) K_0()^†
        = ∫_-∞^+∞ dx   dp   W_ρ(x,p) · 2 ∫_-∞^+∞ ds   e^-2ip s K_0() |x-s⟩⟨x+s| K_0()^†
        = 2 ∫_-∞^+∞ dx   dp   ds   W_ρ(x,p)   e^-2ip s∫_-∞^+∞ dy  e^-(y-)^2/(2 ^2)/(π^2)^1/4 |y⟩⟨y|x-s⟩∫_-∞^+∞ dy' e^-(y'-)^2/(2 ^2)/( π^2)^1/4⟨x+s|y'⟩⟨y'|
        =2/√(π^2)∫_-∞^+∞ dx   dp   ds   W_ρ(x,p)   e^-2ip s  e^-(x-s-)^2/(2 ^2)  e^-(x+s-)^2/(2 ^2) |x-s⟩⟨x+s|
        = 2/√(π^2)∫_-∞^+∞ dx   dp   ds   W_ρ(x,p)     e^-2ip s  e^-(x-)^2/^2  e^-s^2/^2 |x-s⟩⟨x+s|  .

Therefore

    P_ȷ(,)     = [ K_ȷ() K_0() ρ K_0()^† K_ȷ()^†] 
        = 2/√(π^2)∫_-∞^+∞ dx   dp   ds   W_ρ(x,p)     e^-2ip s  e^-(x-)^2/^2  e^-s^2/^2 [  K_ȷ()  |x-s⟩⟨x+s| K_ȷ()^†] 
        = 2/√(π^2)∫_-∞^+∞ dx   dp   ds   W_ρ(x,p)     e^-2ip s  e^-(x-)^2/^2  e^-s^2/^2 ∫_-∞^+∞ dy  e^-(y-)^2/^2/√(π^2)⟨x+s|y⟩_ȷ  _ȷ⟨y|x-s⟩  .

Now, using that <cit.>
    ⟨x|y⟩_ȷ = 1/√(2 π |sinȷ |)exp{ - i cosȷ/2 sinȷ( x- y/cosȷ)^2 }  ,

the above expression reads

    P_ȷ (,)     = 1/π^2  ^2 | sinȷ |∫_-∞^+∞ dx   dp   ds   dy   W_ρ(x,p)   e^(x,p,s,y)  ,

where

    (x,p,s,y)     =  -2ip s - (x-)^2/^2 -  s^2/^2 - (y-)^2/^2   -i cosȷ/2 sinȷ( x + s - y/cosȷ)^2  + i cosȷ/2 sinȷ( x - s - y/cosȷ)^2  
        =   -2ip s - (x-)^2/^2 -  s^2/^2 - (y-)^2/^2 - 2 i cosȷ/sinȷ( x - y/cosȷ) s    .

Integrating insgives

    P_ȷ(,)     = 1/√(π^3 ^2) |sinȷ |∫_-∞^+∞ dx   dp   dy   W_ρ(x,p)   e^(̱x,p,y)  ,

where 

    (̱x,p,y)     = - (x-)^2/^2  - (y-)^2/^2 - ^2/4[ 2 p + 2 cosȷ/sinȷ( x - y/cosȷ) ]^2   
        = - (x-)^2/^2  - y^2/^2 - ^2/^2 + 2  y/^2 - ^2 p^2  - ^2 cos^2 ȷ/sin^2 ȷ( x - y/cosȷ)^2 - 2 ^2 p cosȷ/sinȷ( x - y/cosȷ) 
        = - (x-)^2/^2  - y^2/^2  - ^2/^2 + 2  y/^2 - ^2 p^2  -  ^2 x^2 cos^2 ȷ/sin^2 ȷ  - ^2 y^2/sin^2 ȷ + 2 ^2 x y cosȷ/sin^2 ȷ - 2 ^2 p x cosȷ/sinȷ  +   2 ^2 p y /sinȷ
        = - (x-)^2/^2  - ^2/^2  - ^2/sin^2 ȷ p^2 sin^2 ȷ  -  ^2/sin^2 ȷ x^2 cos^2 ȷ   - 2 ^2/sin^2 ȷ   x cosȷ  p sinȷ
          - ^4 + sin^2 ȷ/^2 sin^2 ȷ y^2 + ( 2/^2 + 2 ^2 x cosȷ/sin^2 ȷ + 2 ^2 p/sinȷ) y  .

Integrating inygives

    P_ȷ(,)     = 1/π√(^4 + sin^2 ȷ)∫_-∞^+∞ dx   dp   W_ρ(x,p)   e^(x,p)  ,

where

    (x,p)     =  - (x-)^2/^2  - ^2/^2  - ^2/sin^2 ȷ (x cosȷ + p sinȷ)^2 + 1/^2sin^2 ȷ/^4 + sin^2 ȷ( + ^4/sin^2 ȷ (x cosȷ + p sinȷ) )^2 
        =  - (x-)^2/^2  - 1/^2( 1 - sin^2 ȷ/^4 + sin^2 ȷ) ^2 - ^2/sin^2 ȷ( 1- ^4/^4 + sin^2 ȷ) (x cosȷ + p sinȷ)^2 
           + 2 ^2/^4 + sin^2 ȷ (x cosȷ + p sinȷ ) 
        = - (x-)^2/^2  - ^2/^4 + sin^2 ȷ^2 - ^2/^4 + sin^2 ȷ (x cosȷ + p sinȷ)^2 + 2 ^2/^4 + sin^2 ȷ (x cosȷ + p sinȷ ) 
        = - (x-)^2/^2 - ^2/^4 + sin^2 ȷ( x cosȷ + p sinȷ - )^2   .

So we have

    P_ȷ (,) = 1/π√(^4 + sin^2 ȷ)∫_-∞^+∞ dx   dp   W_ρ (x, p )  exp{ - (x-)^2/^2 - ^2/^4 + sin^2 ȷ( x cosȷ + p sinȷ - )^2 }  .

Then, the correlator of the random variables()and()is

    () ()_ȷ    = ∫_-∞^+∞ d  d () () P_ȷ(, ) 
        = 1/π√(^4 + sin^2 ȷ)∫_-∞^+∞ dx   dp   W_ρ (x, p )  ∫_-∞^+∞ d () e^ - (x-)^2/^2∫_-∞^+∞ d () e^ - ^2 (  - x cosȷ - p sinȷ  )^2/^4 + sin^2 ȷ
        = 1/π√(^4 + sin^2 ȷ)∫_-∞^+∞ dx   dp   W_ρ (x, p )  √(π^2) ( x/) √(π^4 + sin^2 ȷ/^2) ( x cosȷ + p sinȷ/√(^4 + sin^2 ȷ/^2)) 
        = ∫_-∞^+∞ dx   dp   W_ρ (x, p )  ( x/)  ( x cosȷ + p sinȷ/√(^4 + sin^2 ȷ/^2))   .

Now let us compute the Wigner function of the stateρ = ∑_k,l c_kl|l⟩⟨k|:

    W_ρ (x,p)     = 1/π∑_k,l  c_kl∫_-∞^+∞ ds   e^2ips⟨x-s|l⟩⟨k|x+s⟩
        =  1/π√(π)∑_k,l c_kl/√(2^k   k!   2^l   l!)∫_-∞^+∞ ds   e^2ips e^-(x-s)^2/2 H_l(x-s) e^- (x+s)^2/2 H_k (x+s) 
        =  e^- x^2/π√(π)∑_k,l c_kl/√(2^k   k!   2^l   l!)∫_-∞^+∞ ds   e^-s^2 +  2ips  H_l (x-s)  H_k(x+s)   
        = e^-x^2/π√(π)∑_k,lc_kl/√(2^k   k!   2^l   l!)∫_-∞^+∞ ds   e^-(s - ip)^2 - p^2     H_l(x - s )   H_k( x+s)    .

Defining := s - ipwe have

    W_ρ (x,p)     =  e^- x^2 - p^2/π√(π)∑_k,l c_kl/√(2^k   k!   2^l   l!)∫_-∞-ip^+∞-ip d  e^-^2     H_l ( x- - ip)   H_k ( x+ +ip )   .

Since the integrand is holomorphic, we can set to zero the imaginary shift in the integration contour. Then, using the property <cit.>
    ∫_-∞^+∞ dx   e^-x^2     H_k (x + y)   H_l ( x +z) = 2^max(k,l)√(π) min(k,l)!    y^k-min(k,l) z^l-min(k,l) L_min(k,l)^|k-l| (-2yz)   ,

we can write

    W_ρ (x,p)     =   e^- x^2 - p^2/π∑_k,l  c_kl  (-1)^l√(2^max(k,l)min(k,l)!/2^min(k,l)max(k,l)!)
                           ·( -x+ip )^l-min(k,l)(x + ip)^k-min(k,l) L_min(k,l)^|k-l|(-2(-  x+ip)(x+ip) ) 
        =  e^-x^2 - p^2/π{∑_k≥ l  c_kl (-1)^l √(2^k   l!/2^l   k!) (x+ip)^k-l  L_l^k-l( 2 x^2 + 2 p^2 ) 
                            +∑_k< l c_kl (-1)^l √(2^l   k!/2^k   l!) (- x+ip)^l-k  L_k^l-k( 2 x^2 + 2 p^2 ) }  .

Plugging this back into the correlator (<ref>) yields

    ⟨ ()  () ⟩_ȷ =    1/π∑_k≥ l  c_kl (-1)^l √(2^k   l!/2^l   k!)∫_- ∞^+ ∞ dx dp    e^-x^2 - p^2 (x+ip)^k-l
                                     · L_l^k-l( 2 x^2 + 2 p^2 )  ( x/)  ( x cosȷ + p sinȷ/√(^4 + sin^2 ȷ/^2)) 
         + 1/π∑_k< l c_kl (-1)^l √(2^l   k!/2^k   l!)∫_- ∞^+ ∞ dx dp    e^-x^2 - p^2 (- x+ip)^l-k
                                     · L_k^l-k( 2 x^2 + 2 p^2 ) ( x/)  ( x cosȷ + p sinȷ/√(^4 + sin^2 ȷ/^2))   .

Let us restrict to the 3-dimensional subspace spanned by the first three excitations in the Hilbert space so that the sums overkandlonly run until 2. In this case we can analytically perform the integrals using the identities <cit.>
    ∫_-∞^+∞ dx   e^-x^2   (ax+b)     = √(π) ( b/√(1+a^2))   , 
    ∫_-∞^+∞ dx   x  e^-x^2   (ax+b)     = a/√(1+a^2)exp( -b^2/1+a^2)   , 
    ∫_-∞^+∞ dx   x^2m e^-a x^2  (b x)  (c x)     = 2 (-1)^m/√(π)∂^m/∂ a^m[ 1/√(a)arctan( b c/√(a^2 +a (b^2 +c^2))) ]    ,    (a > 0   ,     m=0, 1, … )   ,
    ∫_-∞^+∞ dx   x^2   e^-x^2  (a x+b)     = √(π)/2( b/√(1+a^2)) - a^2 b/(1+a^2)^3/2exp( - b^2/1+a^2)   ,
    ∫_-∞^+∞ dx   x^3   e^-x^2  (a x)     = a/√(1+a^2)( 1 + 1/21/1+a^2)    ,
    ∫_-∞^+∞ dx   x^4   e^-x^2  (a x +b)     = 3 √(π)/4( b/√(1+a^2)) - 3/2a^2 b/(1+a^2)^3/2[1 + 2 a^2 b^2/3(1+a^2)^2 + 1/1+a^2 ]  exp( - b^2/1+a^2)   .

The results are compact expressions, but too long to show here. Finally, the Leggett-Garg CHSH parameter is

    C = ⟨ ()  () ⟩_ȷ_A^(1)-ȷ_B^(1) + ⟨ ()  () ⟩_ȷ_A^(1)-ȷ_B^(2) +  ⟨ ()  () ⟩_ȷ_A^(2)-ȷ_B^(1) -⟨ ()  () ⟩_ȷ_A^(2)-ȷ_B^(2)  .

MaximizingCover the four angles and the coefficientsc_klwe have the following violation of the inequalityC≥ 2:

    C = 2/675 π( 577 + √(1244179) + 2700 arctan (1/3) ) ≃ 2.416   ,

obtained for the angles 

    ȷ_A^(1) = π/4  ,   ȷ_A^(2) = 3 π/4  ,   ȷ_B^(1) = π/2  ,   ȷ_B^(2) = 0   ,

and the state 

    |ψ⟩ = √(1/2 - 577/2 √(1244179)) |0⟩ + √(1/2 + 577/2 √(1244179)) |2⟩  .