Paper Title: Capacities of quantum Markovian noise for large times.



Vera Verto: Multimodal Hijacking Attack
    
Minxing Zhang1  Ahmed Salem2  Michael Backes1  Yang Zhang1




1CISPA Helmholtz Center for Information Security    
2Microsoft

    
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§ ABSTRACT 
Given a quantum Markovian noise model, we study the maximum dimension of a classical or quantum system that can be stored for arbitrarily large time. We show that, unlike the fixed time setting, in the limit of infinite time, the classical and quantum capacities are characterized by efficiently computable properties of the peripheral spectrum of the quantum channel. In addition, the capacities are additive under tensor product, which implies in the language of Shannon theory that the one-shot and the asymptotic i.i.d. capacities are the same. We also provide an improved algorithm for computing the structure of the peripheral subspace of a quantum channel, which might be of independent interest.





§ INTRODUCTION





Consider a quantum system that we would like to use for storing quantum or classical information. This system is affected by noise that we assume is Markovian. It is natural to ask what is the minimum error that can be achieved for storing log D (qu)bits of information for some fixed time t. This is a typical question studied in Shannon theory, but here we focus on the limit t →∞, i.e., the information should remain in the system for arbitrarily long times.





 Building a quantum system able to store quantum information for large times is one of the important goals of quantum information theory and it has been studied from different angles. Quantum error correction gives a mechanism to actively preserve the quantum information in a system undergoing local noise <cit.>. Such methods can achieve much more than a memory and can be used for fault-tolerant quantum computation <cit.>. A related important area of research is the study of self-correcting (or passive) quantum memories <cit.>, i.e., physical systems that are robust to different forms of imperfections including thermal noise.
 
 In this paper, we study the question of quantum memory from an abstract Shannon theory perspective where given a noise model, the objective is to characterize the fundamental limits that can be achieved without placing computational restrictions on the encoding/decoding operations. As an example, we would like to characterize how many qubits can be stored reliably for time t as a function of the noise model and t? Such questions were studied in the recent work <cit.> and they obtained, among other results about scrambling, conditions for the classical capacity to be zero as well as lower bounds on the classical capacity of ergodic channels.



  
Our results In this work, we focus on the setting of arbitrarily large time, i.e., t →∞ and we characterize both the classical and quantum capacities for a fixed error δ in terms of the peripheral spectrum of the noise model (Theorem <ref>). In addition, we show that such capacities are additive for the tensor product of channels (Theorem <ref>) and can be computed in polynomial-time in the dimension (Theorem <ref>). This algorithm for computing the structure of the peripheral subspace (or the fixed point subspace) of a quantum channel improves on previous works and might of independent interest. We note that the fact that we can efficiently characterize the infinite-time capacities of noisy channels is in contrast with other settings for which capacities or optimal success probabilities correspond to hard problems, such as maximum independent set <cit.> or maximum coverage problems <cit.>.



§ CAPACITIES OF QUANTUM NOISE MODELS


For a Hilbert space H, we use the notation B(H) for the bounded linear operators on H. When we do not need to make the Hilbert space explicit and when it has dimension d, we denote the algebra of linear operators (or d × d complex matrices) by M_d(C) or simply M_d for short. We recall that a quantum channel E :  M_D(C) → M_d(C) is a completely positive and trace preserving linear map. We refer to <cit.> for more detailed definitions and properties of quantum channels.

We start with the standard definition of an error correcting code for a noisy quantum channel T. Note that throughout the paper d ≥ 1 is an integer and T is a quantum channel from _d( C) to _d( C).
    
        A quantum channel E:  M_D( C)→ M_d( C) is a (D,δ) classical code for a channel T:  M_d( C)→ M_d( C) if there exists a recovery channel R:  M_d( C)→ M_D( C) such that the average fidelity of the channel R∘ T∘ E over all diagonal density matrices in M_D is at least 1-δ, i.e.,
        
    1/D∑_i=1^D  i|  R∘ T∘ E(|i i|) |i≥ 1-δ,

        where {|i⟩}_i=1^D is a fixed orthonormal basis of C^D. We say that E is a (D,δ) quantum code if there exist channel R such that the entanglement fidelity of R∘ T∘ E is at least 1-δ, i.e.,
        
    F_E( R∘ T∘ E) = ⟨Φ^+|(I⊗ ( R∘ T∘ E) )(|Φ^+Φ^+|) |Φ^+⟩≥ 1-δ,

        where |Φ^+ = 1/√(D)∑_i=1^D |i⟩⊗|i⟩∈ ( C^D)^⊗ 2 and I is the identity quantum channel on the reference system of dimension D.
    
    
    Then, we introduce the capacity as the dimension of the largest code for the channel T for a fixed error parameter δ.
    
        The (one-shot) classical capacity for a channel T is defined as 
                
    C_δ(T) = sup_Elog D,

        where the supremum is over all possible (D,δ) classical codes E for T. Similarly, the (one-shot) quantum capacity for the channel T is defined as 
        
    Q_δ(T) = sup_Elog D,
 
        where the supremum is taken over all possible (D,δ) quantum codes.
    

    We now see  as the noise that is applied for one time unit. Thus, by the Markov assumption, after t time steps, the noise that is applied to the system is given by composing  t times, which we denote by ^t = ∘⋯∘. As mentioned in the introduction, in this paper, we are interested in the limit of large time t, which motivates the following definition.
     
        The infinite-time classical or quantum capacity of a quantum channel T is defined as
        
        
    C_δ^∞ ( T)= lim_t→∞ C_δ( T^t)   and    Q_δ^∞( T) = lim_t→∞ Q_δ( T^t).

    
    We make a few remarks about this definition.
    
    In this definition, the encoding could in principle depend on the time t. However, all the achievability results (i.e., lower bounds of the capacities) we obtain, the same encoder  works for ^t for all times t ≥ 1. In contrast, it is important for the results to hold that the recovery channel  does depend on the time t.
    
    
    This definition is concerned about passive error correction where no recovery is allowed at regular intervals. One could define an active capacity as follows:
    
    Q_δ^active, t( T) = log{max D : ∃, _1, …, _t s.t.  F_E(_t ∘∘⋯∘∘_1 ∘∘) ≥ 1-δ},

    where : _D →_d and _i : _d →_d for 1 ≤ i ≤ t-1 and _t : _d →_D are quantum channels. Note that the active capacity can be larger than the passive one; in particular, it is simple to see that for any t ≥ 1, Q_0^active, t( T) = Q_0( T) by choosing the recovery maps _i to decode and re-encode information. The works <cit.> have studied some Shannon-theoretic aspects of active error correction.
    
    
    Note that in general T^t does not have a limit as t →∞, but for the capacities C_δ ( T^t) and Q_δ ( T^t), the limits exist as they are nonnegative nonincreasing sequences.        
    

    The characterization of both the classical and quantum capacities will be in terms of the peripheral subspace χ_ T of the quantum channel T. The following proposition summarizes the definition and key properties about the peripheral subspace that will be used in this paper. A proof of these statements and more properties can be found in <cit.>.
    
    
    The peripheral subspace of a quantum channel T is defined by
    
    χ_T = span{ X ∈M_d(C) : T(X) = λ X,  for some  |λ| = 1}.

    It satisfies the following:
    
    
  * There exists a direct sum decomposition of the Hilbert space C^d = H_0 ⊕⊕_k=1^K (H_k,1⊗ H_k,2) for some nonnegative integer K such that 
    
    χ_T = 0 ⊕⊕_k=1^K M_d_k(C) ⊗ω_k,

    where M_d_k(C) is the full matrix algebra on H_k,1, with d_k =  H_k,1, and ω_k is a density operator on H_k,2.
    
  * There exists unitaries U_k acting on H_k,1 and a permutation π which permutes systems {1, …, K} having the same dimension such that for every X ∈χ_T of the form X = 0 ⊕⊕_k=1^K x_k ⊗ω_k, we have
    
    T(X) = 0 ⊕⊕_k=1^K U_k x_π^-1(k) U_k^†⊗ω_k.

    
    
    
    
    We are now ready to state our main theorem.
    
        Let T:  M_d → M_ d be a quantum channel and let K and {d_k}_k=1^K be the integers from the peripheral subspace decomposition in (<ref>).
        
        
        For any δ∈ [0,1), the infinite time classical capacity of T is given by
        
    C_δ^∞(T) = log(⌊∑_k d_k/1-δ⌋),

        and the quantum capacity can be bounded as
        
    log(max_k d_k/√(1-δ)) ≤  Q_δ^∞ ( T) ≤log(max_k d_k)+log(1/1-δ).

        
        In addition, as shown in Theorem <ref> below, the values {d_k}_k determining the structure of χ_ can be computed in time polynomial in d given the description of .
    
    We note that, in independent work, the special case δ = 0 of (<ref>) and (<ref>) were proved in <cit.>. Before proving the Theorem, we make a few remarks.

    
    For the quantum capacity, we do not have an exact expression for the capacity but upper and lower bounds that differ by roughly 1/2log(1-δ). The upper and lower bounds we establish in the proof is slightly stronger and it is simpler to express in terms of the optimal error δ for a fixed code size D:
    
    max_(γ_k)_k∑_k=1^Kγ_k^2/D^2≤sup{1- δ : ∃ (D, δ)  quantum code for }≤max_(γ_k)_k∑_k=1^K d_k γ_k/D^2

    where the supremum is taken over integers γ_k satisfying γ_k ≤ d_k and ∑_k=1^K γ_k ≤ D. Note that it is simple to see that the optimal choice for γ_k is simply to take γ_1 = d_1, …, γ_s = d_s and γ_s+1 = D - ∑_k=1^s γ_k where s = { s : ∑_k=1^s d_k < D }. When D is the sum of the largest r elements of (d_k)_k, then the upper and lower bounds match and we obtain an exact characterization. Observe that (<ref>) follows from (<ref>) by observing that max_(γ_k)_k∑_k=1^Kγ_k^2 ≥min(D^2, max_k d_k^2) and 
    ∑_k=1^K d_k γ_k≤ (max_k d_k) D. We leave the problem of computing the exact optimal fidelity as an open problem.
    

    
    Note that a classical stochastic d × d matrix M can be seen as a special case of a quantum channel ( |i⟩⟨i'|) = 0 if i ≠ i' and ( |i⟩⟨i|) = ∑_j M_j,ij. It is straightforward to verify that the eigenvalues of M are the eigenvalues of  and that the quantity ∑_k=1^K d_k for such a channel is the number of eigenvalues of M of modulus 1 counted with multiplicity or the dimension of the peripheral subspace. This quantity even has a combinatorial interpretation as the sum of the periods of the bottom strongly connected components of the directed graph associated with the Markov chain M, as shown in <cit.>. Note that this combinatorial interpretation holds more generally for nonnegative matrices and such a decomposition into bottom strongly connected components is sometimes called the Frobenious normal form of the matrix <cit.>, see also e.g., in <cit.> for the period of each component. Using the combinatorial interpretation, one can find an algorithm running in linear time in the size of the graph for computing the capacity by using e.g., Tarjan's algorithm <cit.> to find the bottom strongly connected components and then find the periods of each component using <cit.>.
    
    
    
    
    
    
    
    
    
    
    

    
    
    We start with the achievability statement, i.e., lower bounds on the capacities. Using the decomposition of the space in Proposition <ref>, C^d = H_0 ⊕⊕_k=1^K (H_k,1⊗ H_k,2), let {|e_k,j⟩}_j=1^d_k be orthonormal bases of H_k,1. For different H_k,1 having the same dimension, we will identify the orthonormal bases.


    We define the classical code E as follows.  Let σ : {1, …, ∑_k=1^K d_k}→{(k,j) : k ∈{1, …, K}, j ∈{1, …, d_k}} be an arbitrary bijection, then 
    
    E(i) = 
        {[              e_σ(i)⊗ω_k  if  1 ≤ i ≤∑_k=1^K d_k;                 e_1⊗ω_k if ∑_k=1^K d_k < i ≤ D. ].

    For 1 ≤ i ≤∑_k=1^K d_k, we have (i) = e_k,j⊗ω_k ∈ B(H_k,1⊗ H_k,2) with (k,j) = σ(i). As a result, applying , we get
    
    ((i)) = U_π(k)e_k,j⊗ω_π(k) U_π(k)∈ B(H_π(k),1⊗ H_π(k),2),

    using Proposition <ref>.
    Composing t times the map , we get  
    
    ^t((i)) = U_π^t(k)⋯ U_π(k)e_k,j U^†_π(k)⋯ U^†_π^t(k)⊗ω_π^t(k)∈ B(H_π^t(k),1⊗ H_π^t(k),2).

    
    Note that we always have ^t((i)) ∈χ_. For a time t, we choose a recovery map  defined by 
    
    (X) = ∑_k=1^K V_k U^†_π(k)⋯ U^†_π^t(k)_H_π^t(k),2(P_π^t(k) X P_π^t(k)) U_π^t(k)⋯ U_π(k) V_k^†,

    where P_k is the orthogonal projection onto H_k,1⊗ H_k,2 and V_k = ∑_j=1^d_k|σ^-1(k,j)⟩⟨e_k,j|. Clearly, an error occurs only if i > ∑_i=1^K d_K and as such the success probability is given by
    
    1/D∑_i=1^D ⟨i|∘^t ∘( i ) |i⟩ = min(1, ∑_k=1^D d_k /D).

    As a result, if we choose D = ∑_k=1^K d_k/1-δ, we obtain the desired achievability.

    Now let us construct a quantum code of dimension D satisfying the condition max_(γ_k)_k{∑_k=1^Kγ_k^2/D^2}≥ 1-δ.
    
    Let γ_k achieve the maximum. It is convenient to label the basis of C^D by elements in S ∪ S', where S = {(k,j) : k ∈{1, …, K}, j = 1 ∈{1, …, γ_k}} and S' = {1, …, D-∑_k=1^Kγ_k}. By the condition ∑_k = 1^K γ_k ≤ D, the set S is of size at most D.
    
    
    In order to define our encoding map , we first define the Kraus operators E_k = ∑_j=1^γ_k|e_k,j⟩⟨(k,j)| for k = 1 to K, and F_i = |e_(1,1)⟩⟨i| for i ∈ S'.
    
    Then for X ∈_D(C) we define (X) = ∑_k=1^K (E_k X E_k^†) ⊗ω_k + ∑_i ∈ S' F_i X F_i^†⊗ω_1. As a result, we have
    
    (|(k,j)⟩⟨(k,j')|) = |e_k,j⟩⟨e_k,j'|⊗ω_k,

    and (|(k,j)⟩⟨(k',j')|) = 0 for k ≠ k'. For i, i' ∈ S' we get
    
    (|(k,j)⟩⟨i|) = 0,

    
    (|i⟩⟨i|) = |e_1,1⟩⟨e_1,1|⊗ω_1    and   (|i⟩⟨i'|) = 0  for  i≠ i'.
        
    Thus, applying ^t, we get
    
    (^t ∘)(|(k,j)⟩⟨(k,j')|) = U_π^t(k)⋯ U_π(k)|e_k,j⟩⟨e_k,j'| U_π(k)^†⋯ U_π^t(k)^†⊗ω_π^t(k).

    Note that ^t ∘(|(k,j)⟩⟨(k',j')|) ∈χ_. We choose  as
    
    (X) = ∑_k=1^K V_k  U^†_π(k)⋯ U^†_π^t(k)_H_π^t(k),2(P_π^t(k) X P_π^t(k)) U_π^t(k)⋯ U_π(k) V_k^†,

    where P_k is the orthogonal projection onto H_k,1⊗ H_k,2 and V_k = ∑_j=1^γ_k|(k,j)⟩⟨e_k,j|.
    

    
    
    
    Then we can compute the entanglement fidelity as
    
    ⟨Φ^+|(I⊗ R∘ T^t ∘ E )(|Φ^+Φ^+|) |Φ^+⟩
       ≥1/D^2∑_(k,j), (k',j') ∈ S⟨(k,j)|∘^t ∘( |(k,j)⟩⟨(k',j')| ) |(k',j')⟩
       = 1/D^2∑_k=1^K∑_j,j'=1^γ_k 1 
       = ∑_k=1^K γ_k^2 /D^2
       ≥ 1-γ,

    which proves the desired result.




    











    
    
    
    We now move to the converse bound. For this, it is convenient to consider the peripheral projection channel T_P <cit.> which satisfies the following properties:  
        <cit.>
        
            
  * T_P( M_d)= χ_ T
            
  * there exist an increasing sequence {t_i} such that lim_i→∞ T^t_i =  T_P
            
  * T_P(X) = X for any X ∈χ_ T.
        

    Note that if  is a (D, δ) code for , and - '_♢≤η, where the diamond norm is defined as - ' _♢ = sup_ρ(I⊗ ( - '))(ρ) _1, then  is also a (D, δ + η) code for '. 
    
    Consider a sequence t_i such that lim_i →∞^t_i = _P and let η_i = ^t_i - _P _♢. Let _i be a (D, δ) code for the channel ^t_i. Then _i is also a (D, δ + η_i) code for the channel _P. Thus C_δ(^t_i) ≤ C_δ + η_i(_P).
    
    Taking the limit i →∞, we have that sup_δ' < δ C_δ'(_P)≤ C^∞_δ() ≤inf_δ' > δ C_δ'(_P). The exact same result holds for the quantum capacity as well. It now suffices to find upper bounds on the capacities of the channel _P.

    
    First, let us show that the classical and quantum capacities of _P and ∑_k=0^K _k ∘_P are the same, where we define _k as follows.
    Define P_0 to be the orthogonal projector onto H_0 and _0(X) = P_0 X P_0. Then define P_k to be the orthogonal projector onto H_k,1⊗ H_k,2 and _k : B(C^d) → B(H_k,1) by _k(X) = _H_k,2(P_k X P_k). Then ∑_k=0^K _k ∘_P is clearly a quantum channel. As _P(_d) = χ_, ∑_k=0^K _k ∘_P = ∑_k=1^K _k ∘_P so  ∑_k=1^K _k∘_P is a quantum channel mapping B(C^d) to ⊕_k=1^K B(H_k,1).  
    The inequality C_δ(∑_k=1^K _k ∘_P) ≤ C_δ(_P) is clear. For the other inequality, let  be a code for _P and  be a corresponding recovery map. We define the recovery map ' = ∑_k=1^K ∘_k where _k : B(H_k,1) → B(H_k,1⊗ H_k,2) and maps _k(x_k) = x_k ⊗ω_k. It is easy to see that
    
    ' ∘∑_k=1^K _k ∘_P ∘ = ∘_P ∘,

    which proves the desired statement. For this reason, in what follows, we assume that H_0 = 0 and H_k,2 = 1 and the space decomposes as C^d = ⊕_k=1^K H_k,1.
    

    Let E be a (D,δ) classical code for T_P and R a corresponding recovery channel. Note that because _P(M_d) = χ_, for any density operator ρ∈ B(C^d), there exists positive operators ρ_k on H_k,1, _P(ρ) = ⊕_k=1^K ρ_k. In addition, as _P is trace-preserving ∑_k(ρ_k) = 1 and ρ_k ≤ I_H_k,1, where I_H_k,1 is the identity on H_k,1. As a result, _P(ρ) ≤⊕_k=1^K I_H_k,1. Thus,
    
    1/D∑_i=1^D⟨i| R ∘_P ∘ E(i) |i⟩   ≤1/D∑_i=1^D ⟨i|( ⊕_k=1^K I_H_k,1) |i⟩
       = 1/D( ⊕_k=1^K I_H_k,1)
        = ∑_k=1^K d_k/D.

    Thus, for any (D, δ) code, we should have ∑_k=1^K d_k/D≥ 1-δ which implies D ≤∑_k=1^K d_k/1-δ. This implies that 
    C_δ(_P) ≤∑_k=1^K d_k/1-δ. As a result C^∞_δ() ≤∑_k=1^K d_k/1-δ' for any δ' > δ which gives the desired result.

    Let us now move to the quantum capacity. Let E be a (D,δ) classical code for T_P and R a corresponding recovery channel. 
    
    
    In order to compute the entanglement fidelity, recall that the  entanglement fidelity of channel E with Kraus operators {E_i} is given by F_E( E) = ∑_i |(E_i)|^2. Let us introduce Kraus operators for {E_j}_j for _P ∘, {R_i}_i for . As _P(_d) = χ_, the operator {P_k E_j}_k, j are also Kraus operators for the map _P ∘, where we recall that P_k is the projector onto H_k,1.
    
    

    Then we have
            
    F_E( R ∘_P ∘ E)    = 1/D^2∑_i,j,k|(R_i P_k E_j)|^2.

    Let us denote α_k^2 = ∑_i,j|(R_iP_k E_j)|^2. We show two properties on α_k: α_k^2 ≤ d_k β_k with ∑_kβ_k = D and α_k^2 ≤ d_k^2. We start with the first property
            
    α_k^2    ≤∑_i,j( R_i P_k P_k^† R_i^† ) (E_j^† P_k^† P_k E_j)
       = (P_k) ∑_j( E_j^† P_k E_j  ).
    
For the first inequality, we used the Cauchy-Schwarz inequality |(A^† B)|^2 ≤(A^† A) (B^† B). Then we used the fact that  is trace preserving. Now note that because _P ∘ is a quantum channel, we have ∑_k,j(  E_j^† P_k^† P_k E_j ) = D. Calling β_k = ∑_j( E_j^† P_k^† P_k E_j ), we proved the first claimed inequality. 

To show the second inequality α_k^2 ≤ d_k^2, we write
            
    α_k^2    ≤∑_i,j( E_j R_i P_k P_k^† R_i^† E_j^† ) ( P_k^† P_k )
       = (P_k) d_k 
       = d_k^2 .
 
where we used again the Cauchy-Schwarz inequality and the fact that _P ∘∘ is trace-preserving. Defining γ_k = min(d_k, β_k), we have that ∑_k=1^K γ_k ≤ D and γ_k ≤ d_k and the entanglement fidelity can be bounded as 

    F_E( R ∘_P ∘ E) 
                       ≤1/D^2∑_k=1^K α_k^2 
       ≤1/D^2∑_k=1^K d_k γ_k ,

which concludes the proof of (<ref>).
    

    

 §.§ Continuous-time dynamical system


    We now consider a quantum Markovian evolution with a continuous time parameter. Such an evolution is called a quantum dynamical semigroup or quantum Markov semigroup and is a family of quantum channels (_t)_t ≥ 0 satisfying the property _t+s = _t∘_s for t, s ≥ 0. As we show below, the infinite-time capacities of such evolutions can be easily related to the discrete time case considered above. We note that a quantum Markov semigroup (_t)_t ≥ 0 is characterized by a linear map L (called the generator) satisfying T_t= e^ L t: (_t)_t ≥ 0 forms a quantum Markov semigroup if and only if there exists a Hermitian matrix H and matrices {L_j}_j such that <cit.><cit.>
    
    L(ρ)= i[ρ, H]+∑_j L_j ρ L_j^† -1/2( L_j^† L_j ρ + ρ L_j^† L_j ).

    
    
    

    

     
     Let (_t)_t ≥ 0 be a quantum Markov semigroup. Then we have
     
    lim_t →∞ C_δ(_t) = C^∞_δ(_1) 
    lim_t →∞ Q_δ(_t) = Q^∞_δ(_1).

     In addition, the peripheral subspace χ__1 can be expressed in terms of the spectrum of generator L as follows:
             
    χ__1 = span{ X ∈M_d(C) | ∃θ∈ R ,   L(X) = iθ X }.
 
     

     

    
            Let ℰ be a (D,δ) classical (or quantum) code for 𝒯_t, and ℛ be its corresponding recovery channel. If t' ≤ t, then ℰ is a (D,δ) classical (or quantum) code for 𝒯_t' with recovery operator ℛ∘𝒯_t-t'. Therefore, we have the following inequalities:
            
    C_δ(𝒯_⌊ t ⌋) ≥ C_δ(𝒯_t) ≥ C_δ(𝒯_⌈ t ⌉)

            and
            
    Q_δ(𝒯_⌊ t ⌋) ≥ Q_δ(𝒯_t) ≥ Q_δ(𝒯_⌈ t ⌉).

            In the limit as t →∞, both C_δ(𝒯_⌊ t ⌋) and C_δ(𝒯_⌈ t ⌉) converge to C_δ^∞(𝒯_1). Therefore, we have
            
    lim_t →∞ C_δ(𝒯_t) = C_δ^∞(𝒯_1)
                  and  lim_t →∞ Q_δ(𝒯_t) = Q_δ^∞(𝒯_1).

        
            Next, we examine the peripheral subspace of  𝒯_1. It is straightforward to see that if ℒ(X) = iθ X for some operator X and real number θ, then  𝒯_t(X) = e^tℒ(X) = e^iθ tX, which implies X ∈χ_ T_t. Consequently, if X is spanned by eigenoperators of ℒ corresponding to imaginary eigenvalues, then X belongs to χ_𝒯_1.
            
                
           
    


      To show the converse, we use the “super-operator form" which represents the linear map ℰ as a matrix Ê acting on the vector space of operators. Specifically, for a quantum channel ℰ and a density matrix ρ, the action of ℰ can be written as vec(ℰ(ρ)) = Êvec(ρ), where vec(·) denotes the column-stacking vectorization of a matrix.
            Let  L be the super-operator form of ℒ, with a Jordan decomposition given by
            
    L = V ( ⊕_ℓ=1^s J_ℓ(λ_ℓ) ) V^-1,

            where J_ℓ(λ_ℓ) is a Jordan block corresponding to an eigenvalue λ_ℓ of L, i.e., J_ℓ(λ_ℓ) = λ_ℓ I + J_ℓ(0) with J_ℓ(0) = [ 0 1 0 … 0; 0 0 1 … 0; ⋮ ⋮ ⋱ ⋮ ⋮; 0 … … … 0 ]∈_d_ℓ, where d_ℓ is the size of the block.
            Thus the super-operator of 𝒯_1 = e^ℒ has the form
            
    e^L = V ( ⊕_ℓ=1^s e^J_ℓ(λ_ℓ)) V^-1,

           with 
              













    e^J_ℓ(λ_ℓ) = e^λ_ℓ·[          1          1       1/2!          ⋯ 1/(d_ℓ-1)!;          0          1          1          ⋯ 1/(d_ℓ-2)!;          0          0          1          ⋯ 1/(d_ℓ-3)!;          ⋮          ⋮          ⋮          ⋱          ⋮;          0          0          0          ⋯          1 ].

Thus the eigenvalues of _1 are {e^λ_ℓ}_ℓ. Now let X be an eigenvector of _1 with eigenvalue e^λ_ℓ with |e^λ_ℓ| = 1. We know from <cit.> that for such eigenvalues d_ℓ = 1. Therefore, X is also an eigenvector of L with eigenvalue λ_ℓ and λ_ℓ is pure imaginary.








        

    

    
    
    
    
    
        
        

        
        
        
        

        
         
        
        
        
        
        

        
        
    



§ TENSOR PRODUCT OF QUANTUM CHANNELS


In this Section, we show the the infinite-time capacities are basically additive under tensor product. To show that, we start with the following lemma about the peripheral subspace.
    
        Consider two quantum channels, T:  M_d( C)→ M_d( C) and S: M_d'( C)→ M_d'( C), with their respective peripheral projections denoted by T_P and S_P, and their respective peripheral subspaces denoted by χ_ T and χ_ S. For the tensor product of these channels, T ⊗ S, the peripheral projection is given by T_P ⊗ S_P, and the corresponding peripheral subspace is χ_ T ⊗χ_ S.
    

    
        By definition  there exist increasing sequences {m_i}, {n_i}, and {k_i} such that
        
    lim_i→∞ T^m_i =  T_P,  lim_i→∞ S^n_i =  S_P, and  lim_i→∞ ( T⊗ S)^k_i = ( T⊗ S )_P.
 
        
        Since any power of a peripheral projection is itself, we have
        
    lim_i→∞ T^m_in_ik_i =  T_P,  lim_i→∞ S^m_in_ik_i =  S_P, and  lim_i→∞ ( T⊗ S)^m_in_ik_i = ( T⊗ S )_P.

        Therefore, (  T⊗ S)_P =  T_P ⊗ S_P.

        Next, we consider the peripheral subspaces. The peripheral subspace is the fixed-point subspace of the peripheral projection. If X∈χ_ T⊗χ_ S, then it can be written as X= ∑_i Y_i⊗ Z_i where Y_i∈χ_ T and Z_i∈χ_ S. So we have
        
    T_P⊗ S_P (X)= ∑_i  T_P(Y_i)⊗ S_P(Z_i)= ∑_i Y_i⊗ Z_i=X,

        showing that X is a fixed point of  T_P⊗ S_P . Therefore χ_ T ⊗χ_ S ⊆χ_ T⊗ S.

        For the other direction, let X∈ M_d( C)⊗ M_d'( C) belongs to peripheral subspace of T⊗ S, and decompose X as X= ∑_i Y_i⊗ Z_i  where Y_i∈ M_d( C) and Z_i∈ M_d'( C). Since X is a fixed point of T_P ⊗ S_P, we have 
        
    X=  T_P ⊗ S_P(X)= ∑_i  T_P(Y_i) ⊗ S_P(Z_i).

        Because T_P(Y_i)∈χ_ T and S_P(Z_i)∈χ_ S ,  it follows that  X∈χ_ T ⊗χ_ S. Thus  χ_ T⊗ S ⊆χ_ T ⊗χ_ S. 
        Therefore, we conclude that χ_ T⊗ S = χ_ T ⊗χ_ S.
    

    

    
    
    
        For any two quantum channels T and S, the infinite-time zero-error classical and quantum capacities are additive under the tensor product. Specifically, we have:
        
    C_0^∞( T ⊗ S)    =  C_0^∞( T) +  C_0^∞( S), 
    
                  Q_0^∞( T ⊗ S)    =  Q_0^∞( T) + Q_0^∞( S).

       
    

    
        By Lemma <ref>, the  peripheral subspace of T⊗ S is the tensor product of the peripheral subspace of T and S. So if χ_ T= 0 ⊕⊕_k=1^K M_d_k(C) ⊗ω_k for the decomposition C^d = H_0 ⊕⊕_k=1^K H_k,1⊗ H_k,2 and χ_ S= 0 ⊕⊕_k'=1^K'M_d'_k'(C) ⊗ω'_k' for the decomposition C^d' = H'_0 ⊕⊕_k'=1^K' H'_k',1⊗ H'_k',2, then we can decompose
        
    C^d⊗C^d' = H_0 ⊕⊕_k ∈{1, …, K}, k' ∈{1, …, K'} H_k,1⊗ H'_k',1⊗ H_k,2⊗ H'_k',2,

        where 
        
    H_0 = (H_0 ⊗ H'_0) ⊕(H_0 ⊗ (⊕_k'=1^K' H'_k',1⊗ H'_k',2) ) ⊕((⊕_k=1^K H_k,1⊗ H_k,2) ⊗ H'_0)

        and get for this decomposition
        
    χ_ T⊗ S= 0 ⊕⊕_k,k'M_d_k× d'_k'(C) ⊗ω_k⊗ω'_k'.


        By Theorem <ref>, C_0^∞(⊗S) = log(∑_k,k' d_k d'_k') = C_0^∞() + C_0^∞(S) and Q_0^∞(⊗S) = log(max_k,k' d_k d'_k') = Q_0^∞() + Q_0^∞(S).
        
        
    


    


    A simple corollary is that when taking tensor powers of a channel, as is common in Shannon theory, the classical and quantum capacities are given by the zero-error capacities at infinite time.
    
        Let δ∈ [0,1) and T:  M_d → M_d be a quantum channel. Then 
        
    lim_m→∞1/m C_δ^∞( T^⊗ m)= C_0^∞( T),

        and
        
    lim_m→∞1/m Q_δ^∞( T^⊗ m)= Q_0^∞( T).

    

    

    
By Theorem <ref> and Theorem <ref>, we have
        
    C_δ^∞ (  T^⊗ m)
               = log(⌊2^C_0^∞ (  T^⊗ m)/1-δ⌋) 
       = log(⌊2^mC_0^∞ (  T)/1-δ⌋).

        But 
        
    m C_0^∞() + log(1/1-δ) - 1 ≤log(⌊2^m C_0^∞ (  T)/1-δ⌋) ≤ m C_0^∞() + log(1/1-δ),

    which proves the desired result.

    For the quantum capacity, the same argument gives
        
        
        
        
        
        
        
    m Q_0^∞()+ 1/2log(1/1-δ)-1 
                ≤  Q_δ^∞( T^⊗ m) 
                ≤ m Q_0^∞()+ log(1/1-δ),

        which proves the desired result.  
        
        
        
        
    

    




    
    





    












    




    
    








        

    



§ AN EFFICIENT ALGORITHM FOR THE STRUCTURE OF THE PERIPHERAL SUBSPACE


    In this section, we describe an algorithm to efficiently determine the structure of the peripheral subspace of a quantum channel. An algorithm using O(d^8) operations for this task was previously described in <cit.> and a related algorithm in <cit.>. Here, we improve it and show that it can be done in O(d^6 log d). The paper <cit.> also describes an algorithm to represent the peripheral subspace as a linear span but relies on <cit.> for the structure, and for the latter, it is not clear whether it is efficient. 
    Note that here, we work under a simplifying assumption that standard operations on a d × d matrix can be done in time O(d^3) (e.g., matrix multiplication, spectral decomposition, Jordan normal form). We leave for future work a more detailed complexity analysis for finding the structure of the peripheral subspace.

    
    
        Let 𝒯: ℳ_d(ℂ) →ℳ_d(ℂ) be a quantum channel, and let χ_𝒯 = 0 ⊕⊕_k=1^K ℳ_d_k(ℂ) ⊗ω_k be the decomposition of its peripheral subspace as described in Proposition <ref>. Algorithm <ref>, takes the super-operator form of 𝒯 as input and returns a representation of the Hilbert space decomposition and the fixed density matrices ω_k in time O(d^6 log d). 
        
        
    

    
    
    
    
    
    
        An implementation of this algorithm 
        with examples can be found in <cit.>.
    


    

    The first step is to compute the peripheral projection channel _P. Let T̂∈_d^2 be the super-operator form of T. Using the Jordan normal form, we can express T̂ as
    
    
    
    
    T̂ = ∑_ℓ=1^s λ_ℓ P_ℓ +N_ℓ, where λ_ℓ are the eigenvalues of T̂, P_ℓ are projections and N_ℓ are nilpotent. The super-operator form of T_P is given by <cit.>
    
    T̂_P= ∑_ℓ: |λ_ℓ|=1 P_ℓ.

    Thus, T̂_P can be computed using the Jordan normal form of T̂ which can be obtained in time O((d^2)^3) = O(d^6).
    
    

    Recall that χ_ = Fix(_P), where Fix() is the fixed point space of a map , i.e., Fix() = span{X ∈_d : (X) = X }. The rest of the algorithm computes the structure of the fixed point space of the quantum channel _P.
    
    
    
     
    

    
    
    
    
    
    
    


    


    Let χ_𝒯 = Fix(_P) = 0 ⊕⊕_k=1^K ℳ_d_k⊗ω_k for the Hilbert space decomposition 
    
    C^d = H_0 ⊕⊕_k=1^K H_k,1⊗ H_k,2.

    Instead of working directly with χ_𝒯, it is convenient to work with the matrix algebra ⊆ B(C^d) which is defined as
        
    𝒜 = ⊕_k=1^K ℳ_d_k⊗ I_d_k'.

    for the Hilbert space decompostion (<ref>) with d_k =  H_k,1 and d'_k =  H_k,2. Then, χ_𝒯 is called a distortion of the matrix algebra 𝒜 = ⊕_k=1^K ℳ_d_k⊗ I_d_k', i.e., there is a completely positive map  such that () = χ_𝒯.
    
    

    In <cit.>, an algorithm is given to find operators A_1, …, A_N ∈_d such that = span(A_1, …, A_N). This is done by computing left and right eigenvectors of T̂ and thus can be done in time O(d^6).

    

    What remains is now to determine the structure of . Let us call P_k the orthogonal projector on H_k,1⊗ H_k,2. The first step is to compute the projectors P_k and the second step is to find the dimensions d_k and d'_k.
    
    
    We start with the first step, i.e., computing P_k. For that we use that the projectors P_k are the minimal projections in the center of . Recall that the center Z() = { X ∈ : X Y = YX  ∀ Y ∈} and that Z() = ∩' where ' is the commutant of  defined by ' = { X ∈ B(C^d) : XY = YX   ∀ Y ∈}. 

    Note that for the matrix algebra  in (<ref>), we have ' = ⊕_k=1^K I_H_k,1⊗_d_k' and Z() = ⊕_k=1^K C I_H_k,1⊗ I_H_k,2. Given an algebra , a minimal projection in  is an orthogonal projection P such that P  P = C P <cit.>. The minimal projections of Z(), that are also called the minimal central projections, are exactly the projectors P_k we are looking for.
    
    
    Thus, in order to compute the projectors {P_k}_k=1^K, we first compute a representation of Z() as a linear span. We do this by computing a representation of the commutant ' as the linear span of some operators B_1, …, B_N', which can be done by computing the kernel of the matrix Γ described in Lemma <ref>. Then, the center can be computed by taking the intersection of subspaces  and '; see Algorithm <ref>. Then Algorithm <ref> described a general algorithm to find all minimal projections of an algebra in B(C^d) in time O(N d^3 log d), where N is the number of operators describing the algebra as a linear span. Note that the center Z() has dimension at most d and so we may assume N ≤ d. As such, we have computed all the {P_k}_k=1^K in time O(d^5 log d).

    Now, the second step is within each block k, we want to compute d_k =  H_k,1 and d'_k =  H_k,2. For that we now compute minimal projections P_k,j in  satisfying P_k,j P_k = P_k,j in algebra  by using Algorithm <ref> with inputs P_k and {A_1, …, A_N}. From the form of the algebra , we know that any minimal projection Q with QP_k = Q is of the form Q = ψ⊗ I_H_k,2 for some |ψ⟩∈ H_k,1. As a result, there exists orthogonal unit vectors |e_k,j⟩ such that P_k,j = e_k,j⊗ I_H_k,2. We have that d'_k = (P_k,j) and d_k is the number of minimal projections found {P_k,j}_j. For each k, the runtime for finding the minimal projections {P_k,j}_j=1^k O(d_k × d^5 log d) (as the dimension N of the algebra  can be up to d^2). As ∑_k=1^K d_k ≤ d, the runtime of this step is O(d^6 log d).

    
    
         
    
    
    








    





    





    

    









    












   



         


    
Below we provide a lemma that was used in the above proof. For this lemma we need the concept of operator-vector correspondence: given a matrix X∈_n(ℂ), it represents a vector,  |X⟩⟩∈ℂ^n⊗ℂ^n.     


    
            Let 𝒜 be a matrix algebra generated by {A_1, ⋯, A_N}, and 𝒜' be the commutant of 𝒜. Then X belongs to 𝒜' if and only if |X⟩⟩ belongs to the kernel of Γ, where 
            
    Γ = ∑_i=1^N ( A_i ⊗ I - I ⊗ A_i^T )^†( A_i ⊗ I - I ⊗ A_i^T ).

    

    
        By the fact that A⊗ B|C = |ACB^T (see <cit.>) applies if the dimensions of A, B, and C indicate ACB^T is a valid matrix. So X⊗ I |Y= |XY and I⊗ X^T|Y =|YX. Therefore if we define 
        
    Γ_A_i:=( A_i ⊗ I - I ⊗ A_i^T )^†( A_i ⊗ I - I ⊗ A_i^T ),

         then an operator X commutes with A_i if and only if Γ_A_i|X= 0. As Γ_A_i is positive,  the kernel of Γ is the intersection of kernels of all Γ_A_i. Thus, X ∈' if and only if | X ∈Γ.
         
    





    



    
    
































    

    


    




    
    




    


























    
    




































































        






                
























            







                    




        











        






                

    




                    

    




    


    

§ CONCLUDING REMARKS


    In summary, we have shown that in the setting of infinite time, the classical and quantum capacities are essentially given by the zero-error capacities and can be efficiently computed, unlike the fixed time setting. We see this result as a first step towards understanding channel capacities for quantum evolutions. It would be interesting to study the behaviour of capacities for finite t both for active and passive error correction.







    

§ ACKNOWLEDGEMENTS

    We acknowledge support from the European Research Council (ERC Grant AlgoQIP, Agreement No. 851716). MR is supported by the Marie Skłodowska- Curie Fellowship from the European Union’s Horizon Research and Innovation programme, grant Agreement No. HORIZON-MSCA-2022-PF-01 (Project number: 101108117).
    We would like to thank Nilanjanna Datta for a talk at the “Quantum information" conference at the SwissMAP Research Station in Les Diablerets on <cit.> which stimulated this work.
    




    plain

    







 Paper Title: Slow decay rate of correlations induced by long-range extended Dzyaloshinskii-Moriya interactions.





Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute,  Chhatnag Road, Jhunsi, Prayagraj - 211019, India


§ ABSTRACT 
    We examine the impact of long-range  Dzyaloshinskii-Moriya (DM) interaction in the extended XY model on the phase diagram as well as the static and dynamical properties of quantum and classical correlation functions. It is known that in the nearest-neighbor XY model with DM interaction, the transition from the gapless chiral phase to a gapped one occurs when the strengths of the DM interaction and anisotropy coincide. We exhibit that the critical line gets modified with the range of interactions which decay according to power-law.   Specifically, instead of being gapless in the presence of a strong DM interaction,  a gapped region emerges which grows with the increase of the moderate fall-off rate (quasi-long range regime) in the presence of a transverse magnetic field. The gapless chiral phase can also be separated from a gapped one by the decay patterns of quantum mutual information and classical correlation with distant sites of the ground state which are independent of the fall-off rate in the gapless zone. We observe that the corresponding critical lines that depend on the fall-off rate can also be determined from the effective central charge involved in the scaling of entanglement entropy.  We illustrate that in a non-equilibrium setting, the relaxation dynamics of classical correlation,  the decay rate of total correlation, and the growth rate of entanglement entropy can be employed to uncover whether the evolving Hamiltonian and the Hamiltonian corresponding to the initial state are gapped or gapless. 

 Slow decay rate of correlations induced by long-range extended Dzyaloshinskii-Moriya interactions
    Tanoy Kanti Konar, Leela Ganesh Chandra Lakkaraju, Aditi Sen(De)
    
==================================================================================================









    
    



§ INTRODUCTION


Long-range order across quantum systems is desirable for a variety of quantum technological tasks, as it facilitates the sharing of correlation among distant parts of the system. Historically, such long-range order has been observed in many-body systems at quantum criticality or when the spectrum becomes gapless. In the case of the nearest-neighbor (NN) XY model with nonvanishing anisotropy parameter, it has been demonstrated that the classical correlations decay with the distance between the spins  exponentially (polynomially) when it is away from (at) the criticality.
Beyond NN models, long-range interacting (LR) systems with power-law decay in the range of interactions have been shown to exhibit polynomial decay of correlations, even when the spectrum is not gapless. These systems become an intense topic of research in the last few years, especially in the context of building analog quantum simulator<cit.> and due to their natural occurrence in several physical platforms such as the Rydberg atom arrays <cit.>, dipolar systems <cit.>, polar molecules <cit.>, trapped-ion setups <cit.>, and cold atoms in cavities <cit.>. Moreover, several established results that hold for nearest-neighbor systems came into the light of investigation in these LR systems in terms of Lieb-Robinson bound <cit.>, area law <cit.>, novel phases of matter  <cit.>,  dynamical phases <cit.> and from the perspectives of usefulness in quantum technologies like  quantum metrology <cit.> and quantum computation <cit.>. 








The nearest-neighbor Dzyaloshinskii-Moriya (DM) interaction, on the other hand,  involves an asymmetric exchange of spins caused by spin-orbit coupling <cit.>.

It has been widely explored in a variety of solid-state compounds, including Cu(C_6D_5COO)_2. 3D_2O <cit.>, Yb_4As_3<cit.>, Crl_3 <cit.> etc, exhibiting fascinating magnetic properties, like gapless chiral phase when the DM interaction strength is stronger than the anisotropy parameter in the transverse Ising <cit.>,  XY <cit.> and the gamma model <cit.>. It was also shown that the DM interaction can host a ground state in the gapless phase that contains logarithmic entanglement behavior, similar to the LR interacting systems which can enhance the fidelity in quantum teleportation <cit.>, thermal entanglement <cit.> and performance in quantum engines <cit.>.  



In the non-equilibrium domain, the DM interaction has also been examined in the context of non-equilibrium thermodynamics <cit.>, dynamical quantum phase transitions <cit.>, topological phase transition <cit.> and quantum speed limit <cit.>. 



 In this paper, we explore the long-range extended XY model in the presence of long-range extended DM interactions, which decay according to a power-law.  Specifically, we aim to analyze the responses of both long-range and DM interactions in equilibrium and non-equilibrium physics. In the former scenario, we present the phase-classification utilizing conventional order parameters, the law for the decreasing slopes of classical and quantum correlations, and the scaling of entanglement entropy (EE) with the corresponding central charge. In contrast to the NN model <cit.>, we show that given a fixed fall-off rates, there exists a gapped region in which the DM interaction is stronger than the anisotropy parameter. However, we establish that whenever the system is gapless, it is chiral. 
 Notably, in the static case, when the ground state belongs to the chiral gapless region, the classical correlation and quantum mutual information decay with constant decreasing exponent while this is not the case in the gapped regime.   In this model, we observe that the effective central charge associated with EE exhibits interesting behaviors: in the presence of LR and DM interactions, the central charge acts nonlinearly with multiple kinks that are absent from the extended Ising model without DM interactions.

  In comparison to the equilibrium scenario, the dynamical phases could not be efficiently characterized as they depend on the  initial state, evolving operator,  time and system parameters. 
   In order to distinguish between the gapless and gapped phases, we resort to a recently proposed relaxation dynamics of two-point classical correlations <cit.> and entanglement entropy of the evolved state in the transient regimes, as well as mutual information and classical correlation in the steady state domains
  (cf. <cit.> for other dynamical quantities). In our case, the initial state is  prepared by tuning the parameters of the Hamiltonian which have either gapless or gapped energy spectra while  evolution happens when the range of interactions is suddenly quenched, resulting in a gapless or gapped Hamiltonian. Specifically, in the transient  regime,  we show that the long-range DM interactions modify the exponent in the relaxation time of the two-point classical correlation between modes  and establish a generic  scaling law for the growth of entanglement entropy over time. Further, we  observe that  the behavior of mutual information and classical correlation between two arbitrary sites in the steady state differs depending on whether the Hamiltonians corresponding to the initial and dynamical states are in the same (gapped or gapless) phase or in different phases. 

  
  

  
  

 













 

The paper is organized as follows. In Sec. <ref>, we introduce the long-range extended XY model with an additional long-range DM term and find the energy spectra and eigenvectors. While 

we identify the conventional phases in Sec. <ref>, we indicate the decay of classical and quantum correlations in Sec. <ref>. The behavior of  entanglement entropy and the corresponding central charge with respect to system parameters are studied in Sec. <ref>. We then move on to the investigation of features in dynamical states, namely the relaxation dynamics of classical correlations, and the rate of entropy growth in the transient regime in Secs. <ref> and  <ref> respectively while we examine the decay of mutual information with distance between the spins  in the steady state  in Sec. <ref>. The results are summarized  in Sec. <ref>.



§ EMERGENT GAPPED PHASE WITH STRONG DM INTERACTIONS: PHASE DIAGRAM



The  critical lines  for the long-range interacting XY spin models in the presence of Coloumb-like interaction, described by a parameter α and for the nearest-neighbor transverse XY model with DM interaction  are known in the literature <cit.>. However,  the effects of long-range DM interactions has never been addressed before. In this situation, a possible query can be  – can the introduction of long-range DM interactions change the critical lines and phases of the LR Ising models? We answer this question affirmatively by first presenting the analytical prescription of this model for obtaining energy spectra and the eigenvectors, thereby leading to the modified phase-portrait of the extended long-range XY model <cit.> having long-range extended Dzyaloshinskii-Moriya interaction both in the thermodynamic limit and for finite systems.  In particular, we report the gapless chiral phase and its dependence on the fall-off rate, α. In addition, we identify appropriate order parameters that can reveal α-dependence in the modified magentic phase diagram. From the decay patterns of classical correlation including Landau parameters and quantum correlations in terms of quantum mutual information and block entanglement entropy, we again separate the gapless region from the gapped one in this model. 



 §.§ Energy spectrum of the long-range XY and DM interactions


The Hamiltonian, describing long-range extended XY model having long-range extended DM interaction in the presence of transverse magnetic field, reads as


    H   =   ∑_j=1^N∑_r=1^N/2 -J_r^'/𝒩 [1+γ/4σ_j^xℤ_r^zσ_j+r^x+1-γ/4σ_j^yℤ_r^zσ_j+r^y
          +D'/4(σ_j^xℤ_r^zσ_j+r^y-σ_j^yℤ_r^zσ_j+r^x) ]-h'/2∑_j=1^Nσ_j^z,

where ℤ_r^z = ∏_l=j+1^j+r-1σ_l^z, with ℤ_1^z=𝕀
and σ^k(k=1,2,3) is the Pauli matrix, J_r'=J/r^α with α being the strength of power-law decay of the model, γ is the anisotropy parameter, D is the strength of the DM interaction, 𝒩=∑_r=1^N/21/r^α is called the Kac-scaling factor <cit.> which ensures extensivity of the energy in the case of finite-size systems α<1 in the finite-size limit and h' is the strength of the external magnetic field. To make the analysis dimensionless, we redefine the magnetic field as h=h'/J and DM interaction strength as  D=D'/J. We consider the periodic boundary condition (PBC), i.e., σ_N+1≡σ_1.  This model can be solved analytically by mapping spins into free fermions under the following Jordan-Wigner transformation <cit.>:

    σ^x_n    =  ( c_n + c_n^†)
     ∏_m<n(1-2 c^†_m c_m) 
    σ^y_n    =i( c_n - c_n^†)
     ∏_m<n(1-2 c^†_m c_m) 
    and  σ^z_n   =1-2 c^†_n  c_n,

where c_m^†(c_m) is creation (annihilation) operator of spinless fermions and they follow fermionic commutator algebra. The corresponding free fermionic version of the Hamiltonian takes the form 

    H   =   ∑_n∑_rJ_r/2 ((1+iD)c_n^† c_n+r+γ c_n^† c_n+r^† + h.c )
          +h(c_n^† c_n-1/2),

where J_r=1/r^α. One can observe that the Hamiltonian which is quadratic in fermionic operators have long-range interaction both in hopping and superconducting terms. In addition, an extra hoping term emerges due to the DM interaction which also induces a complex phase. It is worthwhile to mention here that in the absence of DM interaction for γ=0, the model is 𝕌(1) symmetric while it is ℤ_2 symmetric with γ 0, D=0. The introduction of  DM interaction leads to a competition between these two symmetries.   In the short-range (SR) models, it is known that the conformal sector changes due to the shift of the symmetry sector <cit.>, responsible for a non-trivial phenomenon. 


Due to the presence of translational invariance in the system, momentum is a good quantum number. Hence, we perform a Fourier transform of the form, c_n=1/√(N)∑ e^-iϕ_kc_k where ϕ_k=2π k/N ∀ k∈[-N/2,N/2] and the corresponding Hamiltonian in the momentum basis can be written as H=k⊕ H_k with 

    H_k    =   ∑_k>0- [∑_r(Re(J_k^α)+DIm(J_k^α))c_k^† c_k
           +(Re(J_k^α)-DIm(J_k^α))c_-k^† c_-k
          -iγIm(J_k^α) (c_k^† c_-k^†- c_-k c_k)]
           +h(c_k^† c_k+c_-k^† c_-k)-h,

and J_k^α = ∑_r = 1^N/2J_r e^iϕ_kr. This Hamiltonian can be diagonalized by applying the Bogoliubov transformation,

    [    c_k; c_-k^† ]=[    u_k -v_k^*;    v_k  u_k^* ][    τ_k; τ_-k^† ],

where u_k=cosθ_k, v_k=isinθ_k and θ_k is called Bogoliubov angle, given by cosθ_k=-h+Re(J_k^α)-λ/√(2(λ^2-(Re(J_k^α)-h)^2)) and λ=√((h-Re(J_k^α))^2+(γ Im(J_k^α))^2). The eigenvalue corresponding to each momentum, k, is given as

    ϵ_± k=-D Im(J_k^α)  ±λ.

Note first that for D 0, although the Bogoliubov basis, (u_k, v_k)^T, are independent of DM interactions,  ϵ_kϵ_-k,  making the analysis non-trivial.

Further, when D=0, this model shows criticality at h=1 and at h=-1+2^1-α for α>1  corresponding to the both ends of the Brillouin zone, i.e., ϕ_k=0 and ϕ_k=π respectively.  In the case of α<1, the energy of the system diverges in the thermodynamic limit, hence, we consider large but finite system size <cit.>. With the introduction of DM, new gapless region emerges, which changes with α for D>γ. Such a region is known for the  nearest-neighbour XY model when D>γ <cit.>  and  gapless to gapped transition occurs at D= γ. However, we now establish that even when D>γ, long-range interactions can induce a gapped phase instead of a gapless one, which implies the modification of the critical lines in presence of LR interactions.






















 §.§ The gapless chiral phase depending on LR interactions


Apart from quantum phase transitions at zero temperature in which the energy gap vanishes, the LR Ising models possess another three distinctive regimes with respect to power-law fall-off rate α, namely, non-local (0 < α < 1), quasi-local (1 < α < 2) and local (α > 2) according to different scaling law of correlation length <cit.>. It is important to note here that in the absence of DM interaction, gap-closing never occurs by tuning the parameter α for h>0 except h_c=1. Further, we know that α≫ 2, the system with NN interactions becomes gapless for D>γ depending on h and the chiral phase emerges along with paramagnetic and ferromagnetic phases. Due to the long-range DM interactions, a competition between the range of interactions, α and the strength of DM interactions surfaces which becomes responsible for the change in condition for the gapless  to gapped transition. More specifically, for moderate values of α(<2), D>γ does not guarantee chiral gapless phase. We find the following in case of LR model. 




    For a given value of the magnetic field h (both in h≥ 0 and h<0), a gapped to gapless transition occurs when  D>γ depending upon the value of α instead of D=γ, known for the NN model.



    The proof is done by analyzing the dispersion relation in Eq. (<ref>). A system is said to be

gapped when the expression, Δ <cit.>, defined as

    Δ=max{min{ϵ_k},0}>0;  ∀ϕ_k∈ [-π, π],
 
otherwise, the spectrum is gapless (with Δ = 0). The exact point, where the gap-closing takes place, depends on the parameters of the Hamiltonian, and the corresponding momentum, k_F, called the Fermi point, can be obtained as a solution of ϵ_kk=0. However, due to the presence of long-range order, it is cumbersome to find the Fermi point analytically. By differentiating numerically, we observe that the gapless phase never occurs when D<γ while for D>γ, the gapless phase occurs although it is not ubiquitous, and changes with the variation of h,D,γ and α. In particular, when h<0, with the decrease of α, the system is gapped even when D>γ and two distinctive gapped regions emerge when 1≤α<2 and 0<α<1 which depend on the strength of D, thereby inducing (destroying) gapped (gapless) regions. However, for α>2, D > γ guarantees gapless phases and the gapless to gapped transition occurs at D=γ. On the other hand, we find, that when 1≤α<2, strong DM interactions are required to obtain the gapless phase and the desired asymmetric interaction strength for obtaining the gapless phase increases with the decrease of α (see Fig. <ref>). Note, further, that the maximum D necessary for the gapless phase does not happen exactly at α=1, but at a nearby point which can be due to finite size analysis.  



It is now evident that the interplay of α and D along with γ and h can change the critical lines of the phase diagram. Therefore, we now examine the phases in the (h,γ)- and (α, D)-planes.






α-dependent chiral phase. In the nearest-neighbor  XY model with DM interactions, the gapless phase gives rise to a chiral order that measures local current flow between the nearest neighbor sites. It can be established, by computing <cit.> the order parameter,

    𝒞ℋ= 1/4N∑_i=1^N σ^x_iσ^y_i+1-σ^y_iσ^x_i+1

between any two neighboring sites, i and i+1 in the ground state which describes the helical alignment of spins or chiral order in the z-direction. 


In the nearest-neighbor case, i.e., for the NN XY model with D > γ and suitable h values, it was shown that 𝒞ℋ > 0 when Δ = 0. 

In the LR model, we find that for a fixed h, α <2 and D > γ, whenever Δ = 0, i.e., the system is gapless, it possesses chiral order with order parameter 𝒞ℋ > 0 (as shown in Fig. <ref>). Interestingly, however, there exists region in which for D> γ, Δ>0 (as shown in the above proposition), and the corresponding chiral order parameter vanishes, i.e., 𝒞ℋ =0. 








Ferromagnetic and paramagnetic phases. We now calculate the long-range magnetic order parameter, C^xx_N/2 =  σ^x_1σ^x_N/2. When C^xx_N/2 0, the system is said to be in the ferromagnetic-x phase (FM_x) while paramagnetic when it vanishes. On one hand, we show that the FM_x phase occurs when the system is gapped and 0<h<h_c^1=1, 1<α<2, and D < γ (see Fig. <ref>). On the other hand, when h is negative, the system becomes ferromagnetic, i.e., C^xx_N/2 > 0  when h > h_c^2=-1+2^1-α with 1<α<2. When the parameters  are neither chiral nor ferromagnetic, the system is in a paramagnetic phase.




  













 §.§ Constant decaying exponent of correlations with DM interactions



From the above analysis, it is evident that the transition from the chiral gapless phase to the other magnetic gapped phase heavily depends on the fall-off rate α. Let us now investigate how the scaling of classical correlations and quantum mutual information between two spins (a measure of total correlations containing both quantum and classical correlations components) <cit.> can recognize the α-dependent phases discussed above. 

The classical correlation (CC) between two sites, separated by a distance R,  can be represented as C^xx_i,i+R = ⟨σ_i^x σ_i+R^x ⟩ while the quantum mutual information is defined as 
ℐ_i:i+R=S(ρ_i)+S(ρ_i+R)-S(ρ_i, i+R), where S(σ) = -(σlog_2σ) indicates the von-Neumann entropy of the state σ, ρ_i and ρ_i+R are the reduced density matrices of the joint state ρ_i,i+R. As we impose periodic boundary condition, the site index i can be ignored and both the classical and total correlation can be studied as C^xx_R and ℐ_R, where R = {1,2,…N/2}. In typical one-dimensional (1D) quantum spin models, the CC decays exponentially when it is away from criticality, whereas, at the criticality, the polynomial decay with R is observed. However, the extended Ising model deviates from this norm by showing different scaling laws near the critical points and away from it <cit.>. Moreover, in the long-range Kitaev chain (see Eq. (<ref>) with D = 0), quantum mutual information (QMI) persists between two distant segments of the chain, provided the system is in the non-local regime, i.e., α < 1, thereby  predicting the correlation between two distant regions of the system  <cit.>.

In our study, we expect to have a notable scaling law due to the presence of long-range and additional DM interactions which is indeed the case. Firstly, we observe that the trends of C_R^xx and ℐ_R with R is qualitatively different when the system is in the chiral phase and when it is not. Secondly, a counter-intuitive observation is that if one tunes the parameters 

 D, γ, h and α in such a way that 𝒞ℋ 0, the decay exponents of mutual information and classical correlation become constant as depicted in Fig. <ref>. Specifically, we observe that irrespective of the details of the power-law exponent and the strength of the magnetic field, the decay exponent of CC and QMI falls off as 

    ℐ_R    ∝ R^-1  and
    
        C^xx_R    ∝ R^-0.45,

which are independent of α provided the ground state belongs to the gapless chiral region (see Fig. <ref> and Table <ref>). This behavior of ℐ_R and C^xx_R highlights a more complex and highly correlated patterns between α and D within the gapless region. It indicates that although the range of interactions can control the critical lines in the (D,γ)-plane, differentiating gapless and gapped phases, the decay rate of CC and QMI in the gapless phase remains unaltered  and the chiral phase favors the sharing of both quantum and classical correlations between distant sites. More importantly, such α-dependent scaling behavior holds both for quantum and classical correlations (see Fig. <ref>).  

On the other hand,  when the system is in the non-chiral (which we refer to as achiral) phase, the ground state displays the decay exponent of both classical and quantum correlations which vary with α, i.e., depending on the internal descriptions of the model (see Fig. <ref> and Table <ref>). 

Through these observations, we elucidate the nuanced role of long-range DM interactions in modulating correlation properties across both chiral and achiral regions leading to the following proposition.


    In the chiral region of the LR XY model with DM interactions, the scaling exponents of CC and QMI are constant (independent of the fall-off rate, α) while their exponents depend on α in the achiral region.











 §.§ Amendment in scaling of entanglement entropy and central charge owing to long-range DM interactions



In a many-body system, an insight of universality classes can be obtained from the generic features of the ground state. A prominent physical quantity that serves the purpose is the block entanglement entropy  of block-size l <cit.>. In particular, after partitioning the N-party ground state into two blocks, containing l and N-l sites (with l ≪ N), we compute the block entanglement of the ground state as S_l = S(ρ_l) with ρ_l being the reduced density matrix of the N-party ground state.  The l-block entropy can be calculated from the two-point correlation functions between modes, m and n, given as

    C_mn=⟨Ψ|c_m^† c_n | Ψ⟩,    F_mn=⟨Ψ|c_m^† c_n^† |Ψ⟩,

where |Ψ⟩ is the ground state and the corresponding correlation matrix, consisting of correlation functions, can be written as

    ℂ_l=[ 𝕀-C   F; F^†   C ].

If λ_ls are the eigenvalues of ℂ_l, the von-Neumann entropy of |Ψ⟩ in the bipartition l:N-l is given as S_l=-∑_l=1^2Nλ_llog_2λ_l.

If a system is gapped which typically happens away from the criticality and when the range of interactions is relatively local, i.e., the norm of the Hamiltonian does not increase as the system size increases, S_l follows the area law, which implies S_l^NC∝ l^d-1, where d is the spatial dimension of the system and NC stands for “not at criticality" while it deviates from the area law at criticality <cit.>. Hence from the scaling behavior of S_l, one can detect the transition from a gapped to a gapless phase which is interesting since a simple scalar quantity can describe the essential properties of the Hamiltonian instead of a complete microscopic description. 

On the other hand, when the system becomes gapless, representing the critical point, e.g., in the transverse NN XY model,  S_l scales logarithmically, i.e., for a translationally invariant spin chain, at critical points, S_l^C= c/3log_2[N/πsinπ l/N]+a <cit.>, where c is the conformal or effective central charge in the conformal field theory and a is a non-universal constant. Further, it was shown that the central charge carries the signatures of an underlying symmetry. For example, c takes value 1/2 for the transverse-field Ising-like Hamiltonian at the critical point for which the Hamiltonian adheres to the ℤ_2-symmetry, while with the addition of asymmetric nearest-neighbor DM interaction, the Hamiltonian becomes  gapless region having c=1 and gapped with c=1/2. Hence EE turns out to be a powerful tool to separate a gapped phase from a gapless one.


It was also recently shown that if one considers long-range interactions instead of short-range ones <cit.>, where magnetic criticality depends on the value of α, EE depends upon α and the free-fermionic version of long-range Ising spin shows fractal entanglement apart from the volume and area law <cit.> and the scaling of entropy at critical points may not follow S_l^C. In this case, we can define an effective conformal charge, c_eff, to describe the universal properties of the system.

We are interested in finding how the scaling of S_l in the LR Ising Hamiltonian gets altered in the presence of DM interactions. Specifically, we will show the revision of a central charge c_eff, which occurs in S_l^C. Before presenting the effects of DM interaction on the central charge, we observe that at criticality, i.e., h_c^1 = 1 of the extended Ising model, c_eff monotonically increases when 1 < α < 2 and saturates to 1/2 at α≥ 2, (which is the scaling known for the NN model) as shown in Fig. <ref>(a). When α < 2, c_eff varies non-linearly with α which points out that due to the presence of long-range interaction, conformal symmetry breaks down. 

On the other hand, the introduction of long-range  DM interaction makes more modifications in c_eff –  (i) the saturation value changes to 1 when α→α_c = 2, with α_c being the point which mimics the NN transverse Ising model with NN DM interactions; (ii) for α<α_c, the conformal symmetry breaks down whereas when α > α_c,  c_eff→ 1 (see Fig. <ref>(b)). 
Our study reveals that even with the introduction of long-range asymmetric interaction, the conformal symmetry is not restored for all the values of α although c_eff indicating the conformal invariance begins for smaller values of α_c<2 which is found in the absence of DM interactions. This behavior can again be attributed to the fact that when the DM interaction is present, the system becomes gapless upon varying the power-law decaying factor α.











§ DECAY OF DYNAMICAL CORRELATION: UNCOVERING CHIRAL AND ACHIRAL PHASES



Let us now study the trends of physical quantities including quantum correlations in the time-evolved state in the transient and the steady state regimes, starting from the product state or the ground state which evolves according to the LR Hamiltonian with DM interactions. Our aim is to demonstrate that the scaling of correlations with time or with distant sites can indicate the transition from the chiral phase to the achiral ones. 



 §.§ Relaxation behavior of dynamical correlations






Let us first present the method to capture the relaxation of dynamical correlation <cit.> which is shown to comprehend distinct phases in equilibrium. The initial state and the evolution of the system in the momentum basis can be represented by the Bogoliubov–de Gennes (BdG) equation, given as


    ψ_k(0)=[u_k(0), v_k(0)]^T  and  ψ_k(t)=e^-iH_ktψ_k(0).

The corresponding dynamical correlation (DC) is defined as C_mn(t)=c_m^† c_nψ(t) which is a fermionic correlation between two modes. The relaxation of this correlation can be measured by taking the difference between correlation in arbitrary time and time in which the system reaches the steady state, given by δ C_mn(t)=C_mn(t)-C_mn(∞), describing the decay of the correlation function with time. Let the Bogoliubov angle of the initial and quenched Hamiltonian be η_k and η̃_k respectively and the difference between the angels is given as α_k=η_k-η̃_k. The relaxation of DC takes the form <cit.>

    δ C_mn(t) = 
       k∈gapped∫sin 2η̃_ksin 2α_k cos(ϵ_k+ϵ_-k)tcos[k(n-m)]= 
        
       k∉gapless∫sin 2η̃_ksin 2α_k cos(ϵ_k+ϵ_-k)tcos[k(n-m)],

where the initial state belongs to the gapped  and gapless regions in Eqs. (<ref>) and (<ref>) respectively. In our case, we have chosen m-n=1, but our result holds for any arbitrary m  and n.

When δ C_mn(t) ∼ t^-χ, χ indicates the dynamical relaxation exponent characterized by the distinctive values of the system parameters.

In the case of the nearest-neighbor XY model, it was shown that the exponent depends on the phase being commensurate or incommensurate. Specifically, in the commensurate phase, the derivative of the dispersion relation with respect to the momentum vanishes at the extreme ends of the Brillouin zone, while in the incommensurate phase, the derivative vanishes inside the Brillouin zone. If the initial state is a product state, i.e., the ground state of the model when the magnetic field goes to infinity and is then quenched with the Hamiltonian corresponding to the commensurate phase, we find χ = 3/2 while χ = 1/2 when quenched with the incommensurate phase <cit.>. 

In the NN XY model with DM interaction, if the initial state belongs to the gapless phase, the exponent changes to χ = 1 if the quenching Hamiltonian is in the commensurate phase, which indicates the presence of DM interaction. In the case of quenching by a Hamiltonian in the incommensurate phase, the exponent is either χ = 1/2 or χ = 1 depending on the parameters of the initial state <cit.>. On the other hand, considering the LR Kitaev chain, it was found that the scaling law behaves similarly to the SR ones when the initial and post-quench Hamiltonian are non-critical although the exponent changes with the parameters chosen from an equilibrium phase transition <cit.>. 

The model considered in this work possesses both LR DM interactions and LR interactions in the xy-plane and hence the interplay between α and D as mentioned in the static scenario can influence the scaling of δ C_mn(t), resulting to different χ. For investigations, possible three situations arise in the non-local regime (α < 1) - (ni) both pre- and post-quench Hamiltonian are in the chiral phase; (nii) only pre-quench Hamiltonian belongs to the chiral phase; (niii) only post-quench Hamiltonian is chosen from the chiral phase. All three cases can also be considered when α is chosen from a quasi-local regime, i.e., when α > 1 which we refer to as (qi) - (qiii). Eg., we find that χ∼ 1.5 for the case of (niii) and when (qii) χ∼ 0.3 (see Fig. <ref>). 

This shows that the exponent in DC can distinguish between the gapped and the gapless phases  of the post- and pre-quench Hamiltonians  (see Fig. <ref>). In addition, the robustness of correlation can be defined as a quantity 1/χ such that if the value of χ is small, the system does not reach the steady state correlation value for a significantly long time. This reinforces the above result that the state from the gapless phase contains a strong correlation both in terms of space and time. 



 §.§ Slow decay of total correlation and two-point correlation with LR DM interaction



We now focus on six scenarios ((ni) - (niii) and (qi) - (qiii)) to study the decay pattern of total correlation and classical correlation with R in a steady state limit, represented by ℐ_R(∞) and 𝒞^xx_R(∞) respectively. Even though the evolution is unitary, as we are studying the properties of a subsystem, a steady state can be reached in the dynamics. When the quenching Hamiltonian is in the gapped phase ((nii)), the steady state maintains both classical and quantum correlations between distant spins, and  persists even at large distance R (see Figs. <ref>(b), (c), (f) and (g)). Conversely, a gapless quenching Hamiltonian ((ni) and (niii)) results in a steady state where classical correlations are present only between nearby spins as depicted in Figs. <ref> (a), (d), (e) and (h). Notice that the decaying natures of ℐ_ℛ(∞) and C^xx_R(∞) are independent of the initial state. Remarkably, the behavior of total and classical correlations remains similar when the distance between the spins is small. 
















 §.§ Effects of long-range in dynamics of EE



In a final attempt to distinguish the gapless and gapped phases under dynamics, we investigate the
growth of entropy with time. The initial states are prepared as the ground states of gapped as well as from the gapless regions and EE, S_l(ρ(t)) ≡ S_t for a fixed l is computed after quenching the system to the gapped or gapless regions by changing the parameters α for a given γ, h and D. Let us denote the initial and final fall-off rates be α_i and α_q respectively. Firstly, we notice that the rate of growth is shown to be linear in time in the case of the XY model with NN <cit.> and long-range interacting model <cit.>, with the initial state being a product state. The trends of S_t with time depend on both α_i and α_q, thereby indicating its dependence on pre- and post-quench Hamiltonian. 

  Secondly, the rate of increment of EE with time is faster if the initial state is in a gapless region and the EE value gets saturated around time t ∼ l/2, as observed in the case of the nearest-neighbor Ising model quenched from a non-critical regime to a critical one <cit.>. However, if the initial and final states are chosen from a gapless region, the increment rate is slow and the saturation is reached faster around t=l/2 compared to a gapped case (see Fig. <ref>(d) and (e)). Also, the scaling of entropy increment until it reaches saturation can be approximated as

    S_t^±∝ a_1t/α_i ±α_q+a_2,

where a_1, and a_2 are constants (see Fig. <ref>).

When both the initial and the final Hamiltonian are gapped or one of the pre- or post-quench Hamiltonian is gapped, the saturation of EE requires much larger time than that obtained with  the Hamiltonians corresponding to the initial and final states being gapless. 








§ CONCLUSION



Quantum long-range systems are intriguing due to their non-local properties, often displaying counter-intuitive phenomena compared to a short-range Hamiltonian. These models are pervasive in physically realizable systems such as trapped ions and atomic, molecular, and optical setups.   Therefore, comprehending their universal features is essential for the advancement of quantum technologies and condensed matter physics.





We found that the long-range extended Dzyaloshinskii-Moriya (DM) interactions with the extended XY model have unique consequences on the critical behavior of the system with a variation of the fall-off rate, α, in which the range of interactions between the sites decays. Specifically,  even with a comparable or more DM interactions than the anisotropy parameter, we proved that the system remains gapped depending on α, which  varies with the direction of the transverse magnetic field although the gapless phase still possesses chiral order.  
By analyzing classical and total correlations between two arbitrary sites of the ground state in the presence of DM interactions, we observed that both  of them decay with the distance between the sites and the decay exponent is independent of α in the gapless region while it depends on the system parameters in the gapped regime and  decays faster in this case than the gapless zone.  Additionally, we exhibited   the departures from the universal trend in the block entanglement entropy in presence of long-range DM interactions which could successfully capture the interplay between long-range and DM interactions.





This study investigated how dynamical  correlations change in time when the energy spectra of the initial and the evolving Hamiltonian are gapped and gapless, driven by  α for a fixed strengths of DM interactions, and magnetic fields. 

The scaling of both classical correlations and mutual information in the  steady states reveals that the sharing of correlations is favorable when the evolving Hamiltonian is gapped.  Moreover, we found that the growth of entanglement entropy with time  is faster in the gapless regime than in the gapped phase and it saturates quickly when both the pre- and post-quench Hamiltonians are gapless.   On the one hand, these findings contribute to the understanding of fundamental aspects of quantum phase transitions, while on the other, sharability of correlations and entanglement  of the ground state as well as their dynamical patterns can be vital for establishing quantum links amongst quantum computers.



We acknowledge the support from the Interdisciplinary Cyber Physical Systems (ICPS) program of the Department of Science and Technology (DST), India, Grant No.: DST/ICPS/QuST/Theme- 1/2019/23. We  acknowledge the use of https://github.com/titaschanda/QIClibQIClib – a modern C++ library for general purpose quantum information processing and quantum computing (<https://titaschanda.github.io/QIClib>), and the cluster computing facility at the Harish-Chandra Research Institute. This research was supported in part by the INFOSYS scholarship for senior students.







 

§ CORRELATION FUNCTIONS




The two-point correlation functions are the fundamental constituents of our study. In the context of the ground state or the thermal state, the two-site correlation function between sites separated by a distance of R can usually be expressed as a determinant of an R × R Toeplitz matrix <cit.>.
However, as the Hamiltonian includes asymmetric interaction, the bipartite reduced state obtained from the N-party ground  state after tracing out N-2 parties includes the correlators like C^xy≡⟨σ^x_i σ^y_i+R⟩ = tr (σ^x ⊗σ^y ρ_R) and C^yx along with  C^ii (i=x,y,z).  Thus for the corresponding density matrix of the two distant parties of the spin chain and the time-evolved state, we need to deal directly with the Pfaffians in order to evaluate the correlations as a function of time as well as distance since there is a formation of chiral phase due to DM interaction. 

Using the Pfaffian formalism, we first write the spin correlation functions as 








    
    .
    [ C^lm_i,i+R = ⟨σ_i^l σ_i+R^m ⟩= c(l,m)pf  |  . I^lm_1,2                                                      …                                             I^lm_1,R-1                                                 J^lm_1                                                 F^lm_1                                               G^lm_1,2                                                    c .                                                      …                                               G^lm_1,r;                                                                                                             …                                                      …                                                      …                                                      …                                                      …                                                    c .                                                      …                                                      …;                                                                                                                                                         I^lm_R-2,R-1                                               J^lm_R-2                                               F^lm_R-2                                             G^lm_R-2,2                                                    c .                                                      …                                             G^lm_R-2,R;                                                                                                                                                                                                                    J^lm_R-1                                               F^lm_R-1                                             G^lm_R-1,2                                                    c .                                                      …                                             G^lm_R-1,R;                                                                                                                                                                                                                                                                               E^lm                                                 D^lm_2                                                    c .                                                      …                                                 D^lm_R;                                                                                                                                                                                                                                                                                                                                    K^lm_2                                                    c .                                                      …                                                 K^lm_R;                                                                                                                                                                                                                                                                                                                                                                                         H^lm_2,3                                                      …                                               H^lm_2,R;                                                                                                                                                                                                                                                                                                                                                                                                                                                       …                                                      …;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     H^lm_R-1,R ] |,






where c(x,x)=c(y,y)=(-1)^R(R+1)/2, 

    I^xx_μ,ν      =  ⟨ A_l+μ(t) A_l+ν(t)⟩, 
    
    J^xx_μ,        =I^xx_μ,R, 
     
    H^xx_μ,ν      = ⟨ B_l+μ-1(t) B_l+ν-1(t)⟩, 
    
    K^xx_ν      = H^xx_1,ν, 
     
    G^xx_μ,ν      = ⟨ A_l+μ(t) B_l+ν-1(t)⟩, 
     
    F^xx_μ      =G^xx_μ,1,  
    
    E^xx      =G^xx_R,1,  
    
    D^xx_ν      =G^xx_R,ν,



    I^yy_μ,ν      =  ⟨ A_l+μ-1(t) A_l+ν-1(t)⟩, 
    
    J^yy_μ      =I^yy_μ,R, 
     
    H^yy_μ,ν      = ⟨ B_l+μ(t) B_l+ν(t)⟩, 
    
    K^yy_ν      = H^yy_1,ν, 
     
    G^yy_μ,ν      = ⟨ A_l+μ-1(t) B_l+ν(t)⟩, 
     
    F^yy_μ      =G^yy_μ,1,  
    
    E^yy      =G^yy_R,1,  
    
    D^yy_ν      =G^yy_R,ν,

with s(x,y)=s(y,x)= -i (-1)^R(R-1)/2, we have

    I^xy_μ,ν      =  ⟨ A_l+μ(t) A_l+ν(t)⟩, 
    
    G^xy_μ,ν      = ⟨ A_l+μ(t) B_l+ν(t)⟩, 
    
    J^xy_μ      = G^xy_μ,0, 
     
    F^xy_μ      =G^xy_μ,1,  
    
    H^xy_μ,ν      = ⟨ B_l+μ(t) B_l+ν(t)⟩, 
    
    E^xy      =H^xy_0,1,  
    
    D^xy_ν      = H^xy_0,ν, 
      
    K^xy_ν      =H^xy_1,ν,

and

    I^yx_μ,ν      =  ⟨ A_l+μ-1(t) A_l+ν-1(t)⟩, 
    
    G^yx_μ,ν      = ⟨ A_l+μ-1(t) B_l+ν-1(t)⟩, 
    
    J^yx_μ      = I^yx_μ,R, 
     
    F^yx_μ      =I^yx_μ,R+1,  
    
    E^yx      =I^yx_r,r+1,  
    
    D^yx_ν      = G^yx_R,ν,  
    
    K^yx_ν      =G^yx_R+1,ν, 
     
    H^yx_μ,ν      = ⟨ B_l+μ-1(t) B_l+ν-1(t)⟩.


Each of the elements in the Pfaffian can be constructed from the expectation values of one of the operators, A_l A_l+R, B_l B_l+R and A_l B_l+R with

A_i=c_i^†+c_i,   B_i=c_i^†-c_i.





When recasted in the same Fourier basis as taken while diagonalization, the operators for the k^th momentum have the matrix form

    (A_l A_l+R)_k =  [[          cos(kR)        -isin(kR)                0                0;        -isin(kR)          cos(kR)                0                0;                0                0 cos(kR)-isin(kR)                0;                0                0                0 cos(kR)+isin(kR) ]],


    (B_l B_l+R)_k =   [[          -cos(kR)           sin(kR)                 0                 0;          -sin(kR)          -cos(kR)                 0                 0;                 0                 0 -cos(kR)+isin(kR)                 0;                 0                 0                 0 -cos(kR)-isin(kR) ]],



    (A_l B_l+R)_k =   [[   cos(kR)  isin(kR)         0         0; -isin(kR)  -cos(kR)         0         0;         0         0         0         0;         0         0         0         0 ]].
 
The single-site transverse magnetization in the same Fourier basis is given by

    σ_z^k = [[ -1  0  0  0;  0  1  0  0;  0  0  0  0;  0  0  0  0 ]].






The corresponding expectation value of the operators for each value of k, say O^k, with respect to the time-evolved  state is computed as ⟨ O ⟩ = ∑_k=1^N/2Tr(ρ_β^k(t) O^k).

Once we find all the single-site magnetizations m_j^α ∀ α={x,y,z} at j=i and j=i+R and all possible two-site correlation functions C_i,i+R^l,m ∀  l,m={x,y,z}  from the Pfaffians described above, we can construct the two-site reduced density matrix between sites i and I+R as we know any two party density matrix for the given Hamiltonian is given as


    ρ_ij=1/4 (𝕀_4 + m^z(σ^z_j+σ^z_i) + ∑_k,l C^kl_ijσ^k_i ⊗σ^l_j),

where k,l∈{x,y}.








 Paper Title: The dynamical alpha-Rényi entropies of local Hamiltonians grow at most linearly in time.








[]d.toniolo@ucl.ac.uk
[]danielet@alumni.ntnu.no
Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United
Kingdom


Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United
Kingdom













§ ABSTRACT 
We consider a generic one dimensional spin system of length L, arbitrarily large, with strictly local interactions, for example nearest neighbor, and prove that the dynamical α-Rényi entropies, 0 < α < 1, of an initial product state grow at most linearly in time. We extend our bound on the dynamical generation of entropy to systems with exponential decay of interactions, for values of α close enough to 1. We provide a non rigorous argument to extend our results to initial low-entangled, meaning O(log L) states. This class of states includes many examples of spin systems ground states, and also critical states. This implies that low entanglement states have an efficient MPS representation that persists at least up to times of order log L. 
The main technical tools are the Lieb-Robinson bounds, to locally approximate the dynamics of the spin chain, a strict upper bound of Audenaert on α-Rényi entropies and a bound on their concavity. Such a bound, that we provide in an appendix, can be of independent interest.



The dynamical α-Rényi entropies of local Hamiltonians grow at most linearly in time
    Sougato Bose 0000-0001-8726-0566
    August 5, 2024
===================================================================================



















§ INTRODUCTION






α-Rényi entropies are generalized quantum entropies that are good quantifiers of entanglement <cit.>. Their properties are listed, for example, in <cit.>.


Several experimental breakthroughs <cit.> have shown how quantum entropies can be detected and measured making them not only crucial quantities to characterized the behavior of many body systems from a theoretical point of view but also susceptible of experimental tests.  


The scaling of α-Rényi entropies, 0 < α < 1, of the reduced density matrix of a state, of a one-dimensional system, have been shown to determine whether such a state could be efficiently represented by a matrix product state (MPS) <cit.>. In higher dimensional systems, instead, a non volume law for the Rényi entropy does not imply an efficient MPS representation <cit.>. 


The variation of the von Neumann entropy of the reduced density of a state evolved in time according to the unitary evolution of a system has attracted a lot of attention starting from the SIE (small incremental entangling) and SIM (small incremental mixing) conjectures attributed by Bravyi in <cit.> to Kitaev. See <cit.> also for early references on this topic. A recent review covering these themes is for example <cit.>. These conjectures roughly say that the rate of increase in time of the von Neumann entropy and the entropy of mixing are upper bounded by a constant independent from the overall system's size. Audenaert <cit.>, Van Acoleyen et al. <cit.> and Marien et al. <cit.> have given proofs of such conjectures. In particular as a consequence of the (at most) linear growth in time of the entanglement entropy the authors of <cit.> were able to prove, using the formalism of the quasi-adiabatic continuation <cit.>, that states within the same phase of matter have the same scaling of the entanglement entropy, implying, in particular, the stability of the area law in dimension larger than one. A phase of matter in this context is identified by the set of gapped eigenstates of local Hamiltonians that can be smoothly connected. Recently a concept of quantum phase has emerged in the more general context of Lindbladian evolution <cit.>. The area law for one-dimensional gapped state has already been proven directly <cit.>. A proof for higher dimensions is still lacking, nevertheless under additional mild conditions Masanes has proven a low entropy law <cit.>.


In our work we are able to upper bound, as a function of time and α, the variation of the α-Rényi entropy, 0 < α < 1,  of the partial trace of the time evolution of a generic product state, when the time evolution is induced by a nearest neighbor Hamiltonian H = ∑_j=-L^L-1 H_j,j+1 in one dimension. With r the local Hilbert space dimension, the bound reads:

    Δ S_α(t) ≤ K   r^1/α-1 max_j{H_j,j+1}  t  log r + K'

with both K and K' constants of order 1.
This is the first rigorous result, as far as we know, on the dynamical evolution of α-Rényi entropy, with 0<α < 1. The upper bound on the von Neumann entropy (entanglement entropy) of the same reduced state follows as the  α→ 1 limit.


We are also able to generalize (<ref>), despite only with a non-rigorous argument, see section <ref>, to the evolution of an initial pure state with α-Rényi entropy (of bipartition) up to O ( log L ), in this case the constant K' becomes of the order of the initial entropy of the state.


The linear dependence on t of the upper bound on  Δ S_α(t) that we have established is the best possible for large t. In fact in <cit.> a lower bound on the dynamical generation, starting from a product state, of the entanglement entropy for the Ising model was proven: Δ S(t) ≥4/3π t -1/2ln t -1.
The Ising model falls into the class of models, 1-dimensional nearest-neighbor, that we are considering here. Δ S(t) denotes the variation of the von Neumann entropy.  α-Rényi entropies, α < 1, are upper bounds of the von Neumann entropy, therefore they cannot growth slower than t for large t.


The bound (<ref>) is easy to generalize to the case of a k-neighbors Hamiltonian. We explicitly consider the case of interactions decreasing exponentially in section <ref>, proving a bound as (<ref>) but only for values of α sufficiently close, from below, to 1. 




Independently from the existence of an energy gap, the bound (<ref>) shows that a state of a one dimensional local system with interactions decaying fast enough, that has an efficient MPS representation, continues upon time-evolution to have an efficient MPS representation up to times at least of the order of log L.


Our work is organised as follows: in the section <ref> we present the physical setting and discuss how the unitary dynamics is approximated making use of Lieb-Robinson bounds. In here we follow the approach  considered in <cit.>. A short proof of the Lieb-Robinson bound, following <cit.>, is in the appendix <ref>. In the section <ref> we present the upper bound on α-Rényi entropies, 0 < α < 1, proven by Audenaert in <cit.> and discuss how it reduces to the more familiar Fannes-Audenaert-Petz bound <cit.> on the von Neumann entropy in the limit α→ 1. In the section <ref> we set up and sum up the series that gives rise to the upper bound on the Rényi entropies (<ref>), discussing how the faster than exponential decrease of the spatial part of the Lieb-Robinson bound of a strictly local Hamiltonian enables the summation of the series for all 0 <  α≤ 1, for arbitrarily large one-dimensional systems. We eventually state the final bound on the dynamical variation of the Rényi entropies (<ref>). In the section <ref> we extend our bound to systems with exponential decay of interactions for values of 0 <  α≤ 1 close enough to 1. We then show in <ref> the generalization of the theory to pure states with entanglement of order log L. In the section <ref> we discuss  possible extensions of our work and relations with other results in the literature. A set of appendices close the paper collecting proofs and technicalities.









§ PHYSICAL SETTING
 





The physical setting is that of a one-dimensional spin chain with sites in the interval [-L,L]. The local Hilbert space is C^r. The Hamiltonian is a sum of nearest neighbors terms:

    H=∑_j=-L^L-1 H_j,j+1

H_j,j+1 is a short hand for 1_r ⊗ ... ⊗ H_j,j+1⊗ ...  ⊗1_r, to make clear  that the support of H_j,j+1 is on the sites {j,j+1}.  We define J := max{  H_j,j+1} < ∞, with · denoting the operatorial norm, that is the maximum singular value.
Our final goal is the evaluation of the variation, associated to time evolution, of the α-Rényi entropies, with 0 < α≤ 1, of a reduced density matrix. Namely we want to upper bound as a function of time:

    Δ S_α (t) :=  | S_α( _[1,L]ρ(t) ) - S_α( _[1,L]ρ) |
 
_[1,L] denotes partial tracing on the Hilbert space ⊗_j=1^L ℋ_j, while ρ(t) is the evolution, at time t of the state ρ, ρ(t) := e^-itHρ e^itH.
We decompose the operator of unitary evolution e^-itH, following the approach of <cit.>, to put in evidence that the term H_0,1 of the Hamiltonian (<ref>) is responsible for the “spread” of entanglement among the two halves of the system. We define

    V(t):= e^it (H_[-L,0]+H_[1,L]) e^-itH = e^it (H-H_[0,1]) e^-itH

Moreover: H_[-L,0] := ∑_j=-L^-1 H_j,j+1 and H_[1,L] := ∑_j=1^L-1 H_j,j+1, that collect the terms in the Hamiltonian (<ref>) with support contained respectively in the intervals [-L,0] and [1,L]. The term H_0,1 has instead support on both these intervals.  
It is easy to verify that replacing e^-itH with V(t) leaves Δ S_α (t) in equation (<ref>) invariant.
A crucial idea from <cit.> is to write V(t) as a recursive product:

    V(t)= V_Λ_L   V^*_Λ_L-1 V_Λ_L-1  ... V^*_Λ_l  V_Λ_l =  ( ∏_j=l+1^L V_Λ_j V^*_Λ_j-1) V_Λ_l

The notation used in (<ref>) is as follows. Λ_j := [-j,j], V_Λ_j is the V-evolution associated to the Hamiltonian  H_[-j,j]:=∑_k=-j^j-1 H_k,k+1, namely:

    V_Λ_j(t):= e^it (H_[-j,0]+H_[1,j])  e^-itH_[-j,j] = e^it (H_[-j,j]-H_[0,1]) e^-itH_[-j,j]

According to (<ref>) it holds: V_Λ_L(t)=V(t). The value of l in the RHS of (<ref>) is a variational parameter, that we will tune, see appendix <ref>, to optimize  the upper bound on the dynamical variation of Rényi entropy. In particular since the LHS of (<ref>) is  l-independent it means that at different instants of time t the value of l that optimizes the bound can be different, therefore the value of l will be in general time-dependent.

Intuitively each factor V_Λ_jV^*_Λ_j-1 in (<ref>) is close to the identity. To compare such  dynamics we will make use of an equivalent formulation of the Lieb-Robinson bounds through the quantity Δ_j(t) that is defined, following <cit.>, and discussed in the appendix <ref>. 










§ AUDENAERT BOUND ON Α-RÉNYI ENTROPIES, 0 < Α < 1 
 





In the paper <cit.> Audenaert presents a sharp form of the Fannes bound on the difference among the von Neumann entropy of a pair of states (density matrices) ρ and σ. The same bound also appears in the book of Petz <cit.>, theorem 3.8, where the proof is attributed to Csiszár.


As a byproduct, Audenaert was able to obtain in <cit.> a sharp bound on the difference among the α-Rényi entropies, with 0 < α < 1, of ρ and σ. The α-Rényi entropies, with α≠ 1, are defined as:

    S_α (ρ) = 1/1-αlogρ^α

It holds S_α (·) ≤ S_β (·) with α≥β. This means that our attention is focused on quantum entropies that upper bound the von Neumann entropy, that is the α→ 1 limit of the α-Rényi entropies. 

    S (ρ) =  -ρlogρ

With  T := 1/2ρ-σ_1 the trace distance among the two states ρ and σ, and d the dimension of the Hilbert space where the states are acting upon, the bound of Audenaert reads as equation A3 of <cit.>: 

    |S_α(ρ)-S_α(σ)| ≤1/1-αlog[ ( 1-T )^α + (d-1)^1-αT^α]

This inequality is sharp, in the sense that for every value of T, 0 ≤ T ≤ 1, there is a pair of states that saturates the bound. These are given by:

    ρ=Diag(1,0, ... , 0), σ=Diag( 1-T,T/d-1, ... ,T/d-1)

The inverse is also true, namely that these are the only states that saturates the bound <cit.>.
We see that with α = 0 the upper bound (<ref>) becomes T independent and therefore trivial, being always equal to log d, that is the maximal value of the α-Rényi entropy with α=0, called Hartley, or max, entropy <cit.>. In what follows we explicitly assume 0 < α≤ 1, with  α = 1 taken as a limit.  
The states (<ref>) play the same role also in the Audenaert-Fannes-Petz inequality on the difference of von Neumann entropy:

    |S(ρ)-S(σ)| ≤ T log (d-1) + H_2(T,1-T)

H_2(T,1-T) denotes the binary Shannon entropy: H_2(T,1-T) := -Tlog T -(1-T) log (1-T).
It can be easily shown that the bound (<ref>) is the limit for α→ 1 of the bound  (<ref>). The Rényi entropies, with the von Neumann entropy being the continuation at α = 1, are decreasing in α for all positive α. This can be shown, for example, using the concavity of the logarithm and the Jensen inequality. The upper bound (<ref>) is also decreasing with 0 ≤α < 1, we explicitly show that in appendix <ref>. 
Both the bounds (<ref>) and (<ref>) are increasing in T only with T ∈ [0,1-1/d]. With T = 1- 1/d the maximum value is reached, that for both entropies is equal to log d. As a consequence, since it is usually only possible to obtain upper bounds on the trace  distance the following slight modification of the bound (<ref>), that is given in Theorem 3.1 of <cit.> is useful. With T ≤ R ≤ 1 it is: 



    |S_α(ρ)-S_α(σ)| ≤1/1-αlog ( (1-R)^α + (d-1)^1-α R^α)    with  R ≤ 1-1/d
    log d    with  R ≥ 1 - 1/d

In (<ref>) Δ S_α is increasing in R. 
For the analogous resetting of the bound given in (<ref>), we refer to Lemma 1 of <cit.> and <cit.>. Recently a new upper bound on the variation of von Neumann entropy that makes use of the trace distance and the norm distance among states has appeared <cit.>.


A further upper bound to (<ref>) and (<ref>), that is not tight but increasing in R, for all 0≤ R ≤ 1, is: 

    |S_α(ρ)-S_α(σ)| ≤1/1-αlog ( 1- α R + (d-1)^1-α R^α)

This follows from (1-u)^α≤ (1-α u) for u> -1 and 0 < α < 1, and the fact that the logarithm is an increasing function.


An important remark about the bounds (<ref>) and (<ref>) is that their derivatives in the trace distance, when evaluated at T=0, diverge. It means that increasing T, from T=0 where the two states coincide, the difference in entropy increases at a diverging rate. For the von Neumann entropy and the α-Rényi entropy, with α > 1, it has been established <cit.> that, for local Hamiltonians systems, the dynamically generated entropy actually grows at a finite rate from t=0. 


In this work we will be interested in the dynamical generation of entropy over any time interval, proving that with strictly local Hamiltonians is always independent form the system's size for all α∈ (0,1) , while when interactions decay exponentially there always exist, for any finite system, a set of α < 1 close to 1 such that the scaling is system-size independent.









§ UPPER BOUNDING Δ S_Α (T) FOR A NEAREST NEIGHBOR HAMILTONIAN
 






We have already defined in (<ref>), Δ S_α (t) :=  | S_α( _[1,L]ρ(t) ) - S_α( _[1,L]ρ) | as the object of our investigation. We now set an upper bound to  Δ S_α (t) in the form of a series making use of the decomposition of V(t) given in equation (<ref>). 


We write  Δ S_α (t) first as a telescopic sum, as shown below, and then use the triangular inequality:

    Δ S_α (t)     =  | S_α( _[1,L] V(t) ρ V^*(t) ) - S_α( _[1,L]ρ) | 
        = | S_α( _[1,L] V_Λ_L(t) ρ V_Λ_L^*(t) ) - S_α( _[1,L]   V_Λ_L-1(t) ρ V_Λ_L-1^*(t) ) + 
        + S_α( _[1,L]   V_Λ_L-1(t) ρ V_Λ_L-1^*(t) ) -  ... + S_α( _[1,L]   V_Λ_l(t) ρ V_Λ_l^*(t) ) - S_α( _[1,L]ρ)  | 
       ≤∑_k=l+1^L | S_α( _[1,L]   V_Λ_k(t) ρ V_Λ_k^*(t) ) - S_α( _[1,L]   V_Λ_k-1(t) ρ V_Λ_k-1^*(t) )| + 
     
        + | S_α( _[1,L]  V_Λ_l(t) ρ V_Λ_l^*(t) ) - S_α( _[1,L]ρ)  |

At this point a crucial observation  follows from the assumption of ρ being a product state

    ρ = ⊗_j=-L^L ρ_j

with ρ_j : C^r →C^r. This  implies that each _[1,L]   V_Λ_k(t) ρ V_Λ_k^*(t) factorizes: the unitary V_Λ_k is supported on the interval [-k,k], with a slight abuse of notation we write V_Λ_k = V_Λ_k⊗1_[-k,k]^c, where [-k,k]^c denotes the complement of [-k,k] in [-L,L]. Moreover with 

    ρ_[-k,k]:= ⊗_j=-k^k ρ_j

we have that:

    _[1,L]( V_Λ_kρ V^*_Λ_k) = _[1,k]_[k+1,L]( V_Λ_kρ_[-k,k] V^*_Λ_k⊗ρ_[-k,k]^c) = _[1,k]( V_Λ_kρ_[-k,k] V^*_Λ_k) ⊗_[k+1,L]ρ_[-k,k]^c

It follows that:

    | S_α( _[1,L]   V_Λ_k(t) ρ V_Λ_k^*(t) ) - S_α( _[1,L]   V_Λ_k-1(t) ρ V_Λ_k-1^*(t) )| 
        = 
     | S_α(_[1,k]( V_Λ_kρ_[-k,k] V^*_Λ_k) ⊗_[k+1,L]ρ_[-k,k]^c) -  S_α( _[1,k]( V_Λ_k-1ρ_[-k,k] V^*_Λ_k-1) ⊗_[k+1,L]ρ_[-k,k]^c) | 
        = | S_α( _[1,k]( V_Λ_kρ_[-k,k] V^*_Λ_k) ) -  S_α( _[1,k]( V_Λ_k-1ρ_[-k,k] V^*_Λ_k-1) ) |

In (<ref>) we have embedded V_Λ_k-1 in the Hilbert space over the interval [-k,k] simply with a tensorization with the identity, that means, with a slight abuse of notation: V_Λ_k-1 = 1_r⊗ V_Λ_k-1⊗1_r. In equation (<ref>) we have used the additivity of Rényi entropies: S_α(ρ⊗σ) = S_α(ρ) + S_α(σ). We then realize that the density matrices in equation (<ref>) are supported on the Hilbert space over the interval [-k,0] that has dimensionality r^k+1. 
We define:

    T_k(t):= 1/2|̱_[1,k]( V_Λ_kρ_[-k,k] V^*_Λ_k) - _[1,k]( V_Λ_k-1ρ_[-k,k] V^*_Λ_k-1)   |̱_1

Using the upper bound (<ref>) for each term of the sum (<ref>) and taking into account (<ref>) we have that:

    Δ S_α (t) ≤1/1-α∑_k=l+1^Llog[ 1-α T_k(t) + (r^k+1-1)^1-αT_k(t)^α] + (l + 1)log r

We notice that we have upper bounded the last term in (<ref>), | S_α( _[1,L]ρ_l(t) ) - S_α( _[1,L]ρ)  |, with the trivial bound for density matrices supported on a Hilbert space of dimension r^l+1, that is (l + 1)log r. This choice will be justified a posteriori after the minimization of (<ref>) with respect to l.


The upper bound on the trace distance that is developed in the appendix <ref>, being J := sup{H_j,j+1} and the rescaled time t':= 4   J   t, reads:

    T_k(t) ≤1/2  V_Λ_kρ_[-k,k] V^*_Λ_k  -V_Λ_k-1ρ_[-k,k] V^*_Λ_k-1_1 ≤1/4(4 J t)^k/k! = 1/4t'^k/k!

We have used the fact that for a generic matrix A defined on a bipartite Hilbert space ℋ_1 ⊗ℋ_2, it holds _ℋ_j A _1 ≤  A _1, with j ∈{1,2}, as a reference see, for example, equations 7 and 17 of <cit.>. This also generalizes to all CPTP maps 𝒩, namely 𝒩(ρ) -𝒩(σ) _1 ≤ρ - σ_1. 


The faster than exponential decrease in k of the upper bound (<ref>) on T_k(t), ensures that, when coupled in (<ref>) with  (r^k+1-1)^1-α,  the sum will converge. We will pick up the L →∞ limit.
Inserting (<ref>) into (<ref>), and defining u_k(t'):= 1/4( e t'/k)^k > 1/4t'^k/k!, see the brief discussion in <ref> on u_k(t'), we get:

    Δ S_α (t) ≤1/1-α∑_k=l+1^Llog[ 1- α u_k(t') + r^(k+1)(1-α) u_k(t')^α] + (l + 1)log r



In the appendix <ref> is proven that (<ref>) has a unique minimum with respect to l.  
A positive integer l that minimizes (<ref>) satisfies the following pair of conditions. Denoting f(l,t') the RHS of (<ref>), the minimum is reached for that value of l such that f(l,t') is smaller of both f(l+1,t') and f(l-1,t').

    f(l,t') ≤ f(l+1,t') 
       f(l,t')  ≤ f(l-1,t')

These equations must be thought at a fixed α and at a fixed time t', with l determined as a function of α and t'. Then the equations for the minimization of Δ S_α(t) are:

    1/1-α log[ 1- α u_l+1(t') + r^(l+2)(1-α) u_l+1(t')^α] ≤log r  
       1/1-α log[ 1- α u_l(t') + r^(l+1)(1-α) u_l(t')^α] ≥log r

 This implies that despite we are not able to solve explicitly the system (<ref>), that can be done numerically, the approximate solution for u_l+1(t') that we are going to provide simply gives rise to a further upper bound to (<ref>). We prefer to provide a loser upper bound but with a transparent physical meaning, than a tighter one. Nevertheless, as discussed after (<ref>), for a generic local (nearest neighbor in this case) Hamiltonian the linear dependence in t' of the upper bound to Δ S_α is the best possible.  



We choose to provide an approximate solution to (<ref>) solving, again approximately, with α far enough from 1, the following equation

    1/1-α log[ r^(l+1)(1-α) u_l(t')^α] ≈log r

That leads to:

    u_l(t) ≈ r^-l1-α/α
       t'^l/4(e/l)^l≈ r^-l1-α/α
       t'/4^1/l(e/l) ≈ r^-1-α/α
        e   r^1-α/α  t' ≈ l   4^1/l

It is: 1 < 4^1/l≤ 4 with l ≥ 1. As the approximate solution of (<ref>) we pick: 

    l_α(t') := c   r^1-α/α  t'

with c > e. The reason for c > e in our approach is to ensure exponential decrease in t of the sum in  (<ref>), as discussed in appendix <ref>.
We stress again that this despite being an approximate solution of (<ref>), when l is replaced with l_α(t') in (<ref>) still provides an upper bound to Δ S_α(t) because of the uniqueness of the minimum in l of (<ref>).



We now plug (<ref>) into the sum (<ref>). We expect that for all t' it will be at most of order 1 because of the fast decrease of the u_k(t'), at a fixed t', with k ≥ l_α(t') +1. The upper bound to (<ref>)  is performed, consistently with the limit α→ 1, in appendix <ref>. Recollecting that t':=4Jt, this leads to 

    Δ S_α(t) ≤ K   r^1/α-1  J   t  log r + K'

with K=4c and K' ≤(1+e/4c(e-1))log r, see appendix <ref>, both of order 1.

It is interesting to compare our upper bound with the entanglement entropy rate obtained in <cit.> in the case of von Neumann entropy, α = 1. Their equation (133), being an upper bound on the rate for all times, implies, in their notations:

    Δ S_1(t) ≤ 2^ν +1 c (log d) A ∑_r r^2νh(r)  t

For comparison we write (<ref>) in the case of a one dimensional system, ν=1, for a strictly local Hamiltonian, in our notations: Δ S_1(t) ≤ 8 (log r) J   t. Looking at the coefficients of proportionality with t in (<ref>), with α=1, that is 4cJ log r, it appears that we do slightly worse than them. 












§ UPPER BOUNDING  Δ S_Α (T) FOR A SYSTEM WITH EXPONENTIAL DECAY OF INTERACTIONS
 






We will extend the results about the dynamical generation of α-Rényi entropy of a strictly local Hamiltonian to the case of an Hamiltonian with exponentially decaying interactions. 


We are able to generalize the approach that we set up in section <ref> for nearest-neighbour Hamiltonians to Hamiltonians that are sum of terms that are supported on the whole system but with decaying interactions. 
The scheme that we develop works for interactions with decay-rate at least that is exponential, f denotes such decay, and potentially could be extended to the case of sub-exponential decay.
Following <cit.>, a function f is defined to have sub-exponential decay if, for any γ <1, |f(t)| ≤ C_γexp(-t^γ), for some C_γ which depends on γ.


Denoting H = ∑_r H_r the Hamiltonian, each term H_r is supported on the whole lattice, we will refer to H_r as “centered” in r. H_r is an interaction decaying according to the function f , meaning that there exist constants J and ξ, that are independent from r, such that with r inside the region X, that for simplicity is assumed connected, it holds:

    1/2^|X^c|( _X^c H_r ) ⊗1_X^c - H_r ≤  J    f ( dist(r,X^c)/ξ)

For simplicity in (<ref>) we are considering the case of qubits, namely, the local Hilbert space dimension is equal to 2.
The physical interpretation of (<ref>) is straightforward: the approximation, in norm, of H_r with an operator that is supported on the region X containing r, improves enlarging the distance of r from the complement of X. The definition (<ref>) is employed, for example, in  <cit.>. We observe that if H_r was strictly local with support contained in X, then the LHS of (<ref>) would be exactly vanishing. Also taking f with compact support the RHS of (<ref>)
vanishes with X large enough, this corresponds to Hamiltonians with k nearest neighbors, a generalization of the case studied in section <ref>.


A Hamiltonian with decaying interactions does not provide an immediate way to identify the terms H_r responsible for the spread of entanglement across the two halves of the system, that was given for a generic nearest-neighbor Hamiltonian by H_I :=H_[0,1]. We introduce a distance b, that will be determined by minimization of the upper bound to the dynamical Rényi entropy, such that, the terms of H mostly responsible for the connection among the left and right halves of the system are those centered within the distance b from the origin: ∑_|r| < b H_r. The physical intuition is that b will be  in relation with the decaying length ξ of definition (<ref>).


With this in mind we set up a different telescopic sum than (<ref>). The first step is to rewrite:

    e^-itH  = e^-it ( H_[-L,-b] + H_[b,L]) e^it ( H_[-L,-b] + H_[b,L]) e^-itH

Where we have introduced, with 0 < b ≤ j ≤ L:

    H_[b,j] := 1/2^|([1,2j] ∩ [1,L])^c|( _([1,2j] ∩ [1,L])^c∑_r=b^j H_r ) ⊗1_([1,2j] ∩ [1,L])^c

H_[b,j] is supported on [1,2j]  ∩ [1,L]. We also define: 

    H_[-j,j] := 1/2^|([-2j,2j] ∩ [-L,L])^c|( _([-2j,2j]∩ [-L,L])^c∑_r=-j^j H_r ) ⊗1_([-2j,2j]∩ [-L,L])^c




We notice that H_[-L,L] = H. 
In equation (<ref>) we have put in evidence at the exponent the sum of the terms centered on the left half and right half of the system outside the interval (-b,b) that, as we said, represents the region that we assume to “connect” the left and right halves and be responsible of the entanglement spread.  

The first factor of (<ref>) is not supported across the two halves of the system therefore it does not contribute to the entropy. Looking at the second and third factor in (<ref>) we realize that, imagining the values of b and -b independent and setting them respectively equal to b=1 and -b=0, if the Hamiltonian H was a nearest neighbor Hamiltonian, then it would coincide with the V operator that we introduced in (<ref>), following <cit.>. 

We define 

    V_Λ_j(t) := e^it ( H_[-j,-b] + H_[b,j])   e^-it H_[-j,j]

that is supported on [-2j,2j]  ∩  [-L,L]. Then:

    Δ S_α (t)     =  | S_α( _[1,L] e^-itHρ e^itH) - S_α( _[1,L]ρ) | 
        = | S_α( _[1,L]V_Λ_L(t)  ρV_Λ_L^*(t)  ) - S_α( _[1,L]ρ)  |  
        = | S_α( _[1,L]V_Λ_L(t)  ρV_Λ_L^*(t)   ) -  S_α( _[1,L]V_Λ_L-1(t)   ρV_Λ_L-1^*(t) ) + 
        +  ... + S_α( _[1,L]V_Λ_l(t)  ρV_Λ_l^*(t) ) - S_α( _[1,L]ρ) | 
       ≤∑_k=l^L-1  | S_α( _[1,L]V_Λ_k+1(t)  ρV_Λ_k+1^*(t)   ) -  S_α( _[1,L]V_Λ_k(t)   ρV_Λ_k^*(t) )| + 
        + | S_α( _[1,L]V_Λ_l(t)  ρV_Λ_l^*(t) ) - S_α( _[1,L]ρ) |

At this point we again assume that ρ is a product state, therefore according to (<ref>) and the  following equations, we have:

    Δ S_α (t)    ≤∑_k=l^L-1  | S_α( _[1,2(k+1)]V_Λ_k+1(t)  ρ_[-2(k+1),2(k+1)]V_Λ_k+1^*(t)   ) -  S_α( _[1,2(k+1)]V_Λ_k(t)   ρ_[-2(k+1),2(k+1)]V_Λ_k^*(t) )| + 
        + | S_α( _[1,2l]V_Λ_l(t)  ρ_[-2l,2l]V_Λ_l^*(t) ) - S_α( _[1,L]ρ_[-2l,2l]) |

In the above we have used the fact that  V_Λ_k(t) is supported on [-2k,2k]. The dimensional reduction associated with product states allows the application of the Audenaert upper bound (<ref>). We then need to provide an upper bound to the trace distance that according to appendix <ref>, equations (<ref>) and (<ref>), reads:

    R_k(t)     := 1/2_[1,2(k+1)]( V_Λ_k+1(t)  ρ_[-2(k+1),2(k+1)]V_Λ_k+1^*(t) ) - _[1,2(k+1)](  V_Λ_k(t)  ρ_[-2(k+1),2(k+1)]V_Λ_k^*(t) ) _1 
       ≤1/2V_Λ_k+1(t)  ρ_[-2(k+1),2(k+1)]V_Λ_k+1^*(t) - V_Λ_k(t)  ρ_[-2(k+1),2(k+1)]V_Λ_k^*(t) _1 ≤   t   J   O (  g(b,J,t)   f ( k/4 ξ) )

With g(b,J,t):= b  t   J((e^vt-1) + a) + c, with a and c of O(1), and with v the Lieb-Robinson velocity of the Hamiltonian. We can now apply to (<ref>) the Audenaert bound (<ref>), that reads:

    Δ S_α (t) ≤1/1-α∑_k=l+1^Llog[ (1-R_k(t) )^α + (2^2(k+1)+1-1)^1-αR_k(t)^α] + 2l +1

We now consider explicitly the case of an Hamiltonian that is a sum of terms with exponential tails f(k/ξ) = e^-k/ξ. In this case the upper bound to  Δ S_α (t) is proven in (<ref>) of appendix  <ref>. The final result, for 1/1+1/8ξln2 < α≤ 1, is:

    Δ S_α (t) ≤β t + A(t)

with β > v/1/4ξ-(2ln2)1-α/α and A(t) decreasing exponentially fast for large t and A(0)=0. This implies that 

    lim_t→∞Δ S_α (t)/t <  2v/1/4ξ- (2ln2)1-α/α

The limit α→ 1, that gives the von Neumann entropy, is:

    lim_t→∞Δ S_1 (t)/t < 8vξ














§ DYNAMICAL Α-RÉNYI ENTROPY FOR INITIAL LOW ENTANGLED STATES
 






In this section we extend the theory that have developed in section <ref> for product states to initial low entangled states. This will allow us to show that, for one dimensional systems, states with an efficient MPS representation continue to have such efficient representation, upon time evolution, at least up to times of the order of log L. 


The main technical tools employed in this section are: an important result about the existence of an efficient MPS representation for low entangled states <cit.> in one dimension and an upper bound on the concavity of α-Rényi entropies, 0 <α <1, proven using the theory of majorization, in appendix <ref>, this result can be of independent interest.  


The results of this section will be, contrarily to all the previous results, based on an unproven assumption about the price paid, in terms of entropy, by replacing a product state with its time evolution, we clearly state this in equation (<ref>).  


Let us consider a low entangled state vector |v⟩∈ (C^2)^⊗ (2L+1), namely we assume that S_α(_[1,L] |v⟩⟨ v| ) = O(log L). The results of <cit.> ensure that a low entangled state |v⟩, together with an additional assumption on the distribution of the tails of the distribution of the Schmidt coefficients of |v⟩⟨ v|, see after equation (4) of <cit.>, has an efficient representation as a matrix product state (MPS). This means that the bond dimension (D) of the MPS is of order poly(L). In fact the rank of the state is upper bounded by D^2 <cit.>.  See, for example, <cit.> for reviews on MPS and related topics. We express this fact saying that |v⟩⟨ v|  has rank of O(poly(L)) or equivalently that is the linear combination of O(poly(L)) terms:

    |v⟩⟨ v| := k_-L,...,k_Lj_-L,...,j_L∑ c_(j_-L,...,j_L)c_(k_-L,...,k_L)   |j_-L⟩⟨ k_-L|  ⊗ ... ⊗ |j_L⟩⟨ k_L|

The states { |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L| } in (<ref>) are all orthonormal. The number of elements of this basis is 2^2L+1. 



In section <ref> we upper bounded Δ S_α(t) for initial product states. Restricting our attention to pure product states, it is  S_α(0)=0. As show in appendix <ref>, equation (<ref>), a convex combination of O(poly(L)) pure product states has dynamical α-Rényi entropy upper bounded by K_α t + K' + O(log L). 

    S_α( _[0,L] U(t) ( ∑_j q_j | j ⟩⟨ j | ) U^*(t) ) ≤ K_α t + K' + O(log L)

(<ref>) implies that, for times t ≤ O(log L), the state U(t) ( ∑_j q_j | j ⟩⟨ j | ) U^*(t) has an efficient MPO (matrix product operator) representation.


Let us consider now a state vector | v ⟩  with low entanglement, as above. We claim that such a state, that has as efficient MPS representation continues to have an efficient representation under time evolution for time O(log L). This claims follows from the following assumption that we think is intuitively justified, but nevertheless unproven:


  for times t ≤ O(log L) we assume:

    S_α( _[0,L]j_-L,...,j_L∑( U(t)  |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L| U^*(t) ) U(t)  | v ⟩⟨ v | U^*(t) ( U(t)  |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L| U^*(t) ) ) ≃
        S_α( _[0,L]j_-L,...,j_L∑(  |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L|  ) U(t)  | v ⟩⟨ v | U^*(t) ( |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L| ) ) + O(poly(log L))


Let us discuss the meaning of <ref>. In equation (<ref>) we have replaced U(t)  |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L| U^*(t) with |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L|, considering this is a product state we know, because of our bound (<ref>), that its entropy increases at most linearly in time, so our reasoning is that overall within a time scale O(log L) the error in entropy in the LHS of (<ref>) would be of the same order. The upper bound of Audenaert, (<ref>), that we have employed to prove (<ref>), turns out to be inefficient in this context because the trace distance that we need to employ that bound is not small enough. At the same time since Audenaert's bound is also designed for two completely generic states while here our states are related by unitary evolution. From assumption <ref>, it follows that:


    S_α( _[0,L] U(t)  | v ⟩⟨ v |  U^*(t) ) ≤
        S_α( _[0,L]j_-L,...,j_L∑(  |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L|  ) U(t)  | v ⟩⟨ v | U^*(t) ( |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L| ) ) 
       ≃  S_α( _[0,L]j_-L,...,j_L∑  U(t) (  |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L| )  | v ⟩⟨ v | ( |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L| ) U^*(t)  ) + O(poly(log L)) 
        = S_α( _[0,L]j_-L,...,j_L∑ |c_(j_-L,...,j_L)|^2  U(t) (  |j_-L⟩⟨ j_-L|  ⊗ ... ⊗ |j_L⟩⟨ j_L| ) U^*(t)  ) + O(poly(log L)) 
       ≤ K_α t + K' + O(poly(log L))

The inequality in (<ref>) follows from pinching, see for example section II.5 of <cit.>, and the fact that pinching with respect to a product basis and partial tracing commute. In (<ref>) we made use of our assumption <ref>, and in (<ref>) we employed the definition (<ref>), where following by the assumption  S_α(_[1,L] |v⟩⟨ v| ) = O(log L), the number non vanishing coefficients c_(j_-L,...,j_L) in (<ref>) is of order O(poly(L)).



The conclusion from (<ref>), and lemma 2 of <cit.>, is that a state |v⟩ with an efficient MPS representation will continue to have, under time evolution, such efficient representation for times t ≤ O(log L).  














§ OUTLOOK
 






In this work we have made the first step towards a general theory that allows to obtain from the Lieb-Robinson bounds of local Hamiltonians an upper bound to the dynamical generation of α-Rényi entropies. In here we were mainly concerned with systems with linear lightcones, in <cit.> we present an application to systems with logarithm lightcones, mostly in relation with the many-body-localization phenomenology. 


A natural way to extend our theory is to look into higher dimensions and to consider explicitly the Hamiltonians with interactions decreasing slower than exponentially. This would allow to address the stability of the area law for α-Rényi entropies in local systems using the formalism of the quasi-adiabatic continuation, in analogy to what done for the von Neumann entropy by the authors of <cit.>.


An intriguing relation of our upper bound on dynamical generation of entropy with the topology of one dimensional spin systems, as quantified by the GNVW index <cit.>, looks possible: on one hand the authors of <cit.> have established that the dynamics of local Hamiltonians has a vanishing GNVW index, on the other the authors of <cit.> have put this index in relation with the Rényi entropy of such evolution via the Choi-Jamiołkowski mapping. We could expect that a tight bound on Δ S_α (t) should imply the vanishing of the GNVW index for the unitary evolution of local Hamiltonians. 










§ ACKNOWLEDGMENTS

 Daniele Toniolo and Sougato Bose acknowledge support from UKRI grant EP/R029075/1. D. T. is glad to acknowledge discussions about topics related to this work with Álvaro Alhambra, Ángela Capel, Angelo Lucia, Dylan Lewis, Emilio Onorati and Lluís Masanes.  






§ APPENDICES









§ DECREASE IN Α OF THE AUDENAERT UPPER BOUND ON Δ S_Α
 






α-Rényi entropies converge to the von Neumann entropy both in the limit α↑ 1 and α↓ 1. Therefore they are continued for  α = 1 into the von Neumann entropy.
A simple application of the Jensen inequality in relation with the concavity of the  log shows that with 0 <  α < 1, S_α is decreasing in α. In the same spirit we show that the Audenaert upper bound on Δ S_α of equation (<ref>), and therefore (<ref>), is decreasing in α.


    d/d  α[ 1/1-αlog( (1-T)^α + (d-1)^1-α T^α) ] = 
        = 1/(1-α)^2log( (1-T)^α + (d-1)^1-α T^α) + 
        + 1/1-α (log(1-T)) (1-T)^α - (log(d-1)) (d-1)^1-α T^α + (d-1)^1-α  (log T) T^α/(1-T)^α + (d-1)^1-α T^α
        = 1/(1-α)^2log( (1-T)^α + (d-1)^1-α T^α) + 
        + 1/(1-α)^2 (log(1-T)^1-α) (1-T)^α  +  log((d-1)^α-1 T^1-α) (d-1)^1-α T^α/(1-T)^α + (d-1)^1-α T^α

Defining 

    p_1 := (1-T)^α/(1-T)^α + (d-1)^1-α T^α≥ 0 
        p_2 := (d-1)^1-α T^α/(1-T)^α + (d-1)^1-α T^α≥ 0 
        x_1 := (1-T)^1-α
        x_2 := (d-1)^α-1 T^1-α

We have  p_1 + p_2 = 1, therefore since the logarithm is a concave function we can apply Jensen inequality:

    ∑_j p_j log x_j ≤log( ∑_j p_j  x_j  )

to the second term in (<ref>). Noticing that:

    ∑_j p_j  x_j = 1/(1-T)^α + (d-1)^1-α T^α

this results in:

    d/d  α[ 1/1-αlog( (1-T)^α + (d-1)^1-α T^α) ] ≤ 0












§ LIEB-ROBINSON BOUNDS FOR STRICTLY LOCAL HAMILTONIANS
 







The results of this section partially appeared in <cit.>, we include it because the faster than exponential decrease of the spatial part of the Lieb-Robinson bounds in (<ref>) is the crucial ingredient for the summation of the series (<ref>). H is as in (<ref>), A is any operator supported in [0,1]. The idea that motivates the definition of Δ_j(t) as below is to quantify how much the evolution in time of an operator is affected by terms in the Hamiltonian that are far away from the support of such operator.



    Δ_j(t)    :=  e^iH_Λ_j+1tAe^-iH_Λ_j+1t - e^iH_Λ_jtAe^-iH_Λ_jt
        = ∫_0^t ds  d/ds[ e^iH_Λ_j+1(t-s)(  e^iH_Λ_jsAe^-iH_Λ_js) e^-iH_Λ_j+1(t-s)] 
        = ∫_0^t ds    e^iH_Λ_j+1(t-s)[  H_Λ_j+1 - H_Λ_j , e^iH_Λ_jsAe^-iH_Λ_js] e^-iH_Λ_j+1(t-s)

We notice that H_Λ_j+1 - H_Λ_j = H_j,j+1 + H_-j-1,-j and H_Λ_j-1 have disjoint supports, therefore they commute, therefore we can insert e^iH_Λ_j-1sAe^-iH_Λ_j-1s in the RHS of the commutator in (<ref>) provided [0,1] that is the support of A does not overlap with [j,j+1] and [-j-1,-j], the smallest j that ensures this to be true is j=2.

    Δ_j(t)    = ∫_0^t ds    e^iH_Λ_j+1(t-s)[  H_j,j+1 + H_-j-1,-j, e^iH_Λ_jsAe^-iH_Λ_js - e^iH_Λ_j-1sAe^-iH_Λ_j-1s] e^-iH_Λ_j+1(t-s)
       ≤ 4 max{  H_j,j+1  ,   H_-j-1,-j}∫_0^t ds   e^iH_Λ_jsAe^-iH_Λ_js - e^iH_Λ_j-1sAe^-iH_Λ_j-1s

We recognize that the integrand in (<ref>) is Δ_j-1(t), meaning that we have set up a recursive equation.
Defining J := max_j { H_j,j+1}, (<ref>) is rewritten as 

    Δ_j(t)    ≤ 4 J ∫_0^t ds Δ_j-1(s) 
       ≤ (4 J)^j-1∫_0^t dt_1 ∫_0^t_1 dt_2 ... ∫_0^t_j-2 dt_j-1Δ_1(t_j-1)

At this point we observe that, based on equations (<ref>)-(<ref>)

    Δ_1(t) ≤ 4 J A  t

In conclusion:

    Δ_j(t) ≤A  (4 J)^j∫_0^t dt_1 ∫_0^t_1 dt_2 ... ∫_0^t_j-2 t_j-1 dt_j-1≤A  (4 J)^jt^j/j!

that renormalizing time as t':= 4 J t, reads:

    Δ_j(t) ≤A t'^j/j!



It turns out that an estimate on Δ_j(t) is equivalent to the usual form of the Lieb-Robinson bounds. Let us sketch this equivalence.


Given A and B operators defined for simplicity on a unidimensional lattice system Λ_L:=[-L,L], with A supported on [0,1] and the support of B on [l+1,l+1+|b|], implying ((A),(B) ) = l, we get:

    [A(t),B]    = [A(t)- e^iH_Λ_ltAe^-iH_Λ_lt + e^iH_Λ_ltAe^-iH_Λ_lt ,B] = [A(t)- e^iH_Λ_ltAe^-iH_Λ_lt ,B]
       ≤  2BA(t)- e^iH_Λ_ltAe^-iH_Λ_lt

With A(t) = e^iH_Λ_LtAe^-iH_Λ_Lt, we get:

    A(t)- e^iH_Λ_ltAe^-iH_Λ_lt =     e^iH_Λ_LtAe^-iH_Λ_Lt - e^iH_Λ_L-1tAe^-iH_Λ_L-1t + 
        + e^iH_Λ_L-1tAe^-iH_Λ_L-1t - e^iH_Λ_L-2tAe^-iH_Λ_L-2t  + 
        + ...  + 
        + e^iH_Λ_l+1tAe^-iH_Λ_l+1t - e^iH_Λ_ltAe^-iH_Λ_lt

Then:

    A(t)- e^iH_Λ_ltAe^-iH_Λ_lt≤A∑_j=l^∞t'^j/j!


    ∑_j=l^∞t'^j/j!    = t'^l  ∑_j=l^∞t'^j-l/j! = t'^l  ∑_k=0^∞t'^k/(k+l)! =  t'^l/l!∑_k=0^∞t'^k/k+l kk!≤ t'^l/l!∑_k=0^∞( t'/l+1)^k

In the last step of (<ref>) we have used k+l k≥(l+1)^k/k!. With t' < l+1 it follows:

    ∑_j=l^∞t'^j/j!≤ t'^l/l!1/1-t'/l+1

With t' ≤l+1/2, it is 1/1-t'/l+1≤ 2, then overall, given that l:= ((A),(B)), it holds:

    [A(t),B]≤ 4 AB t'^l/l!

We stress that the condition  t' ≤l+1/2 is not restrictive, in the sense that the trivial bound to [A(t),B], for generic A and B, is 2 AB, then let us determined the time t' such that t'^l/l! = 1/2, that with l large enough is

    t'= ( l!/2)^1/l≤ (el)^1/ll/e≤l+1/2

For a more general approach that applies also to any dimension, see the derivation of Lieb-Robinson bound in appendix C of <cit.>. 








We conclude this section with two remarks. If the operator A is positive, A ≥ 0, then the rescale time becomes t'=2 J t, in fact we can use A - A/21 = A/2, from <cit.>, for a better upper bound. With B is a generic operator, we have:

    [A,B]  =  [A - A/21,B] ≤ 2A - A/21  B  =  A    B



A more general relation is actually true, namely, if A ≥ 0, denoting λ_max and λ_min the largest and smallest eigenvalues of A, then A - λ_min1 - λ_max-λ_min/21 = λ_max-λ_min/2. With λ_min=0 we recover the previous one.


The second remark regards the extension of (<ref>) to any Schatten norm, that, with |A|:=√(A^*A), is defined as:

    A_p := ( |A|^p )^1/p

Using Hölder's inequality, we have A   B_p ≤A _p  B. This implies:

    e^iH_Λ_j+1tAe^-iH_Λ_j+1t - e^iH_Λ_jtAe^-iH_Λ_jt_p ≤A _p  t'^j/j!

This is relevant when considering quantities like the so called out-of-time-correlators (OTOC): ( σ  A(t)BA(t)B ). Denoting σ=1/d the maximally mixed state, with A=A^*, B=B^*, A^2=B^2=1, and with j the distance among the supports of A and B we get:

    1/d( [A(t),B]^* [A(t),B] )=: 1/d [A(t),B] _2^2 = 2 - 2/d(  A(t)BA(t)B ) ≤1/d(4 A _2  t'^j/j!)^2 =  (  4 t'^j/j!)^2
       1/d(  A(t)BA(t)B ) ≥ 1 - 8 (   t'^j/j!)^2 ≥ 1 -  (  8( e t'/j)^j )^2

At t=0, A and B commutes, moreover, under the assumption A^2=B^2=1, it holds: 1/d(  ABAB ) = 1/d(  A^2B^2 ) = 1, then the bound (<ref>) is saturated. Also:

    1/d | (  A(t)BA(t)B ) | ≤1/d |  A(t)BA(t)B | ≤1/d B A(t)BA(t) _1  ≤1/d B A^2(t)B _1  ≤1/d B ^2 A^2(t) _1 = 1

In (<ref>) we have used the fact that for two matrices C and D such that the product CD is Hermitean then: CD _1  ≤DC_1, that is theorem 8.1 of <cit.>.


The interpretation of (<ref>) is straightforward: the decrease of (  A(t)BA(t)B ) signals the increase of the overlap among the supports of A(t) and B starting from a time scale of the order of the distance among the supports of A and B. 
We stress that with A ≥ 0 and B ≥ 0 both positive then (  A(t)BA(t)B ) is positive as well, is fact a product of positive operators is positive.



 














§ EVALUATING THE TRACE DISTANCE (<REF>)
 





We now upper bound

    T_k(t) :     = 1/2_[1,k]  V_Λ_k(t) ρ_[-k,k]  V^*_Λ_k(t) - _[1,k] V_Λ_k-1(t) ρ_[-k,k]  V^*_Λ_k-1(t) _1 
        = 1/2_[1,k]  V_Λ_k(t) ρ  V^*_Λ_k(t) - _[1,k] V_Λ_k-1(t) ρ  V^*_Λ_k-1(t) _1

To do so we employ the following upper bound where U(t) is a generic time-dependent unitary operator with 
U(0)= 1_ℋ and σ a generic density matrix.

    U(t) σ U^*(t) - σ_1    = ∫_0^t ds  d/ds( U(s)σ U(s)^* ) _1 =  ∫_0^t ds  [ ( d/ds  U(s) ) σ U(s)^* + U(s) σ( d/ds  U^*(s) ) ] _1 
       = ∫_0^t ds  [ ( d/ds  U(s) ) σ U(s)^* - U(s) σ U(s)^* ( d/ds  U(s) ) U^*(s) ] _1 
       = ∫_0^t ds   U(s) [ U^*(s) ( d/ds  U(s) ) σ  -  σ U(s)^* ( d/ds  U(s) ) ] U^*(s)  _1 
       ≤∫_0^|t| ds [ U^*(s)  d/ds  U(s)  , σ] _1 
     ≤ 2 ∫_0^|t| ds   U^*(s)  d/ds  U(s)   
       ≤ 2|t| sup_s∈[0,|t|] U^*(s)  d/ds U(s)



We now apply (<ref>) to (<ref>).



    V_Λ_k+1(t) ρ  V^*_Λ_k+1(t) - V_Λ_k(t) ρ  V^*_Λ_k(t) _1 =    V_Λ_k^*(t)  V_Λ_k+1(t) ρ  V^*_Λ_k+1(t) V_Λ_k(t)  - ρ_1 
       ≤ 2 ∫_s=0^|t| ds  V^*_Λ_k+1(s) V_Λ_k(s) d/ds(  V_Λ_k^*(s)  V_Λ_k+1(s) ) 
        = 2 ∫_s=0^|t| ds  V^*_Λ_k+1(s) V_Λ_k(s) d/ds(  V_Λ_k^*(s) ) V_Λ_k+1(s) + V^*_Λ_k+1(s) d/ds  V_Λ_k+1(s)

Recalling that V_Λ_k(s):= e^is (H_Λ_k -H_I)  e^-is H_Λ_k, where we have defined H_I := H_[0,1], we have that:

    i d/ds  V_Λ_k(s) = e^is (H_Λ_k -H_I) H_I e^-is (H_Λ_k -H_I)V_Λ_k(s)

Then:

    T_k+1(t) ≤∫_s=0^|t| ds  e^is (H_Λ_k -H_I) H_I e^-is (H_Λ_k -H_I)  -  e^is (H_Λ_k+1 -H_I) H_I e^-is (H_Λ_k+1 -H_I)  =  ∫_s=0^|t| ds Δ_k(s) 
       ≤∫_s=0^|t| ds H_I(4 J s)^k/k!≤1/4t'^k+1/(k+1)!

We remark that the dynamics of Δ_k(t) is here generated by H_Λ_k - H_I, but all the steps that lead to (<ref>) are equally valid. To obtain (<ref>) we have used (<ref>), the definition of J := sup{H_j,j+1}, that implies H_I ≤ J, and finally the definition of t':= 4Jt.


 

 
 
 
 
 


§ BRIEF DISCUSSION OF U_K(T)
 






We have defined u_k(t) := 1/4(e   t/k)^k ≥1/4t^k/k!. u_k(t) upper bounds the trace distance. With a fixed k, a sufficient condition for u_k(t) ≤ 1 is given by t ≤k/e. 

It is easy to see that, with a fixed t, u_k(t) is a decreasing function of k, in fact:

    u_k+1(t)/u_k(t) = (e   t/k+1)^k+1/(e   t/k)^k=e   t/k+1( k/k+1)^k < e   t/k+1

The condition t ≤k/e, that implies u_k(t) < 1, also ensures that u_k(t) is a decreasing function of k.




 
 
 
   
   
 



§ ALGORITHM FOR THE MINIMIZATION OF THE SUM THAT UPPER BOUNDS Δ S_Α(T)
 





In this section we discuss the existence and uniqueness of the solution l_α(t) that satisfies the two equations (<ref>). We consider α∈ (0,1), the value α=1 can be dealt with analogously. 



  * We fix t  and consider a value l̅ of l such that both the LHSs of the equations in (<ref>) are larger equal than 1 such that u_l(t) ∈ [0,1-1/2^l̅ +2].  We will discuss the minimal t that allows the existence of such an l̅ at the end of this section.


  * We increase l from l̅ to l̅ +1. We remark the crucial fact that the value taken from the LHS of the second equation is now the value that the LHS of the first equation in (<ref>) had at the previous step. Therefore they cannot jump in one step from being both larger than one to be both smaller than one. Since u_l(t) < 1 is a decreasing function of l and both the LHSs of (<ref>) are increasing in u with u_l(t) ∈ [0,1-1/2^l̅ +2] they  will both decrease. 


  * We will repeat increasing l till the LHS of the first equation will become smaller than 1. This will certainly happen at a certain point, in fact with u of the order of 2^-(l+2)1-α/α  the LHS of (<ref>) will be of order 1. This is the value of l that minimizes (<ref>). 


What can almost certainly be improved of our approach is the constant of proportionality of t in the upper bound (<ref>). That is left for future research.   







 


§ ESTIMATE OF THE SERIES IN (<REF>)
 






We want to estimate the series in (<ref>), that is copied below (<ref>),  using  l_α(t') := c   r^1-α/α  t', with c > e as in (<ref>), and u_k(t'):= 1/4( e t'/k)^k. The result will be that for every t'>0 and α∈ (0,1] (<ref>) is upper bounded by a constant of order 1. 



    1/1-α∑_k=l_α+1^Llog[ 1 - α u_k(t') + r^(k+1)(1-α) u_k(t')^α]



Let us discuss the fact that the argument of the log in (<ref>) is an increasing function of 0 ≤ u ≤ 1.

    0 = ∂/∂ u[ 1- α u + r^(k+1)(1-α) u^α] = -α + α r^(k+1)(1-α) u^α-1

That has the solution, corresponding to a maximum, as it is seen by the always negative second derivative, u = r^k+1. r^k+1 has minimal value, reached for k=1 (corresponding to t=0), equal to r^2. The minimal value of r is 2, therefore, in the interval 0 ≤ u ≤ 1, the RHS of (<ref>) is an increasing function of u. It means that replacing u with an upper bound (smaller than r^2) we still obtain an upper bound, despite the negative term - α u in the argument of the log in (<ref>). 


We observe that:

    r^(k+1)(1-α) u_k(t')^α = r^(k+1)(1-α)/4^α( et'/k)^k α = r^(1-α)/4^α r^kα1-α/α( et'/k)^k α = r^(1-α)/4^α( r^1-α/αet'/k)^k α = r^(1-α)/4^α( e/c l_α/k)^k α

We then upper bound (<ref>):

    1/1-α∑_k=l_α +1 ^Llog[ 1- α u_k(t') + r^(k+1)(1-α) u_k(t')^α] 
        = 1/1-α∑_k=l_α +1 ^Llog[ 1- α1/4( e/c r^1-α/αl_α/k)^k + r^(1-α)/4^α( e/c l_α/k)^k α] 
       ≤1/1-α∑_n=1 ^∞[ - α1/4(e/c r^1-α/α)^l_α + n(  l_α/l_α + n)^l_α + n + r^(1-α)/4^α(e/c)^α(l_α + n)(  l_α/l_α + n)^α(l_α + n)]



In the last step we have changed the variable of summation from k to n, with k=l_α +n,  moved the upper limit of the sum to ∞, and used the inequality log(1+x) ≤ x with x ≥ -1, as discussed above.


It is well known that the function (x/x+n)^x+n = (1-n/x+n)^x+n is increasing in x, with lim_x→∞(1-n/x+n)^x+n = e^-n. With x=l_α this corresponds to the t →∞ limit. Then going back to  (<ref>) we can further upper bound it with:

    ≤1/1-α[ - α1/4(e/c r^1-α/α)^l_α +1∑_n=1 ^∞ e^-n + r^(1-α)/4^α(e/c)^α(l_α + 1)∑_n=1 ^∞  e^-α n] 
        = 1/1-α[ - α1/4(e/c r^1-α/α)^l_α +11/e-1 + r^(1-α)/4^α(e/c)^α(l_α + 1)1/e^α-1] 
        = 1/1-α[ - α1/4(e/c r^1-α/α)^cr^1-α/αt' +11/e-1 + r^(1-α)/4^α(e/c)^α(cr^1-α/αt' + 1)1/e^α-1]

It is easy to see that for all t > 0, in the limit α→ 0 the upper bound (<ref>) tends to zero. To achieve this, it has been crucial the introduction of the  parameter c > e. Moreover for all t ≥ 0, the limit α→ 1 of (<ref>) gives:

    lim_α→ 11/1-α   [ - α1/4(e/c r^1-α/α)^l_α +11/e-1 + r^(1-α)/4^α(e/c)^α(l_α + 1)1/e^α-1] 
        = log r/4(e-1)(e/c)^et'+1≤log r/4(e-1)(e/c)

For qubits, r=2, and c/e=1.2, 1/4(e-1)e/c≃ 0.12. For all t > 0 the upper bound (<ref>) is exponentially decreasing in t, this is also the case in the limit α→ 1 as seen in (<ref>). 




Denoting K'-log r the upper bound in (<ref>), in agreement with (<ref>), it is:

    Δ S_α(t) ≤ l_α(t')  log r + K'

Then:

    Δ S_α(t) ≤  c   r^1/α-1  t'  log r  + K'

Recollecting that t':= 4   t max_j {H_j,j+1} =: 4   t J, we obtain, with K := 4    c

    Δ S_α(t) ≤ K   r^1/α-1  J   t  log r + K'

as in (<ref>).











§ ACCOUNTING FOR TAILS OF INTERACTIONS IN THE HAMILTONIAN FOR AN UPPER BOUND OF Α-RÉNYI ENTROPY
 








In this section we provide the details to obtain the upper bound (<ref>).
Applying (<ref>) we obtain:

    V_Λ_k+1(t) ρV^*_Λ_k+1(t) -  V_Λ_k(t) ρV^*_Λ_k(t) _1 ≤ 2 t sup_s∈[0,t]V^*_Λ_k+1(s) V_Λ_k(s) d/ds(  V_Λ_k^*(s) ) V_Λ_k+1(s) + V^*_Λ_k+1(s) d/dsV_Λ_k+1(s)

Recalling that 

    V_Λ_k(t) := e^it ( H_[-k,-b] + H_[b,k])   e^-it H_[-k,k]

it holds:

    i d/dsV_Λ_k(s) = e^it ( H_[-k,-b] + H_[b,k])( H_[-k,k] - H_[-k,-b] - H_[b,k]) e^-it ( H_[-k,-b] + H_[b,k])V_Λ_k(s)

Comparing this equation with (<ref>), where the generator of V_Λ_k was a unitary conjugation of the same Hermitean operator, H_I, for all k, we see that in (<ref>) instead we have the operator H_[-k,k] - H_[-k,-b] - H_[b,k] that, despite decaying outside of the interval [-b,b], is supported on [-2k,2k], therefore it is k-dependent. All together we have:  

    V_Λ_k+1(t) ρV^*_Λ_k+1(t) -  V_Λ_k(t) ρV^*_Λ_k(t) _1 
       ≤ 2 t sup_s∈[0,t]  e^is ( H_[-k,-b] + H_[b,k])( H_[-k,k] - H_[-k,-b] - H_[b,k]) e^-is ( H_[-k,-b] + H_[b,k]) +  
        - e^is ( H_[-k-1,-b] + H_[b,k+1])( H_[-k-1,k+1] - H_[-k-1,-b] - H_[b,k+1]) e^-is ( H_[-k-1,-b] + H_[b,k+1])
       ≤ 2 t sup_s∈[0,t]  e^is ( H_[-k,-b] + H_[b,k])( H_[-k,k] - H_[-k,-b] - H_[b,k]) e^-is ( H_[-k,-b] + H_[b,k]) +  
        - e^is ( H_[-k-1,-b] + H_[b,k+1])( H_[-k,k] - H_[-k,-b] - H_[b,k]) e^-is ( H_[-k-1,-b] + H_[b,k+1]) +  
        + 2 t  H_[-k-1,k+1] - H_[-k-1,-b] - H_[b,k+1]  -  ( H_[-k,k] - H_[-k,-b] - H_[b,k])

We see that the first operator norm of (<ref>) has almost the structure of a Δ_k(s) as in appendix <ref>, the difference with (<ref>) is that, in the case of a strictly local Hamiltonian the operator whose Heisenberg evolutions are computed had a fixed support, while in (<ref>) it depends, despite only with tails, on k.
We recollect that with r inside the region X, that for simplicity is assumed connected, it holds:

    1/2^|X^c|( _X^c H_r ) ⊗1_X^c - H_r ≤  J    f ( dist(r,X^c)/ξ)



Let us show that the second operator norm of (<ref>) is bounded by O(J C_ξ f(k/ξ)), with f denoting the rate of decrease of interactions, as defined in (<ref>). We start considering H_[b,k+1] - H_[b,k]. To shorten the equations we define: _[m,n]^c(A) := 1/2^|[m,n]^c|_[m,n]^c (A) ⊗1_[m,n]^c. Let us assume for now 2k < L, the case 2k ≥ L is discussed in (<ref>). 

    H_[b,k+1] - H_[b,k] :=   _([1,2(k+1)]∩[-L,L])^c∑_r=b^k+1 H_r  -   _([1,2k]∩[-L,L])^c∑_r=b^k H_r  
        =  _([1,2(k+1)]∩[-L,L])^c( 1 -  _ ([1,2k]∩[-L,L])^c ∖ ([1,2(k+1)]∩[-L,L])^c ) ∑_r=b^k H_r   +  _([1,2(k+1)]∩[-L,L])^c   H_k+1

The norm of the first term in (<ref>) is small, of order J f( k/ξ), see (<ref>), in fact being the complements of the sets evaluated with respect to [-L,L]:

    ([1,2k]∩[-L,L])^c ∖ ([1,2(k+1)]∩[-L,L])^c = [2k+1,2(k+1)]∩[-L,L]

 the Hamiltonian term of ∑_r=b^k H_r with the closest centre to [2k+1,2(k+1)]∩[-L,L] is H_k. We also observe that the normalized trace defined above is a projection, see <cit.>, namely: _[m,n]^c( _[m,n]^c(A) ) = _[m,n]^c(A), therefore, as a superoperator, it has norm equal to 1. This implies that:

    _([1,2(k+1)]∩[-L,L])^c( 1 -  _([2k+1,2(k+1)]∩[-L,L])) ∑_r=b^k H_r ≤( 1 -  _ ([2k+1,2(k+1)]∩[-L,L]) ) ∑_r=b^k H_r  
       ≤   J ∑_r=b^k f(2k+1-r/ξ)  ≤   J C_ξ( f(k/ξ) - f(2k-b/ξ) )

with C_ξ a function of ξ. If, for example, the interactions have exponential tails, it is C_ξ = ξ, in fact:

    J ∑_r=b^k f(2k+1-r/ξ)     = J ∑_r=b^k e^-2k+1-r/ξ = J ∑_l=k+1^2k+1-b e^-l/ξ = J ∑_l=0^2k+1-b e^-l/ξ - J ∑_l=0^k e^-l/ξ
        = J ( 1-e^-2k+2-b/ξ/1-e^-1/ξ -  1-e^-k+1/ξ/1-e^-1/ξ) 
     ≤ J ξ( e^-k/ξ - e^-2k-b/ξ)

In the last line we have used e^1/ξ-1 ≥1/ξ 


It is important to discuss what happens with 2k ≥ L, in this case we must recollect that the support, for example, of H_[b,k] is [1,2k] ∩ [-L,L], therefore in (<ref>) we have: 

    ( [1,2k] ∩ [-L,L] )^c ∖( [1,2(k+1)]  ∩ [-L,L] )^c  = [-L,0] ∖ [-L,0] =  ∅

The partial trace associated with the empty set is the identity, therefore the term among parenthesis in (<ref>) is vanishing. 


We mention the fact that within this formalism we are able to recover the results for nearest neighbor Hamiltonians that we obtained in the first part of this paper, and actually to generalize those to the case of a k-neighbor Hamiltonian. In fact with f compactly supported and ξ the size of its support, the RHS of (<ref>) is vanishing. More in detail, considering

    f(j/ξ) = δ( ⌊ j - 1/2/ξ⌋ , 0 )

⌊ a ⌋ denotes the largest integer such that ⌊ a ⌋≤ a. δ(·,·) is the Kronecker delta. We take ξ integer, and ξ≥ 1. From the assumption r ∈ X in (<ref>), follows that (r,X^c) ≥ 1, then j ≥ 1 and integer. According to (<ref>), with  j ≤ξ, it is f(j/ξ) = 1, with j > ξ, f(j/ξ) = 0. This also shows that the value of b, that we have introduced in (<ref>), for the case of a strictly local Hamiltonian that is the sum of ξ-neighbors terms  equals ξ.


We are now ready to upper bound the operator norm of the second term in equation (<ref>). To ease the notation all the sets are meant to be the intersection with [-L,L].

    H_[-k-1,k+1] - H_[-k-1,-b] - H_[b,k+1]  -  ( H_[-k,k] - H_[-k,-b] - H_[b,k]) 
        = ( _[-2(k+1),2(k+1)]^c( 1 -  _ [-2k,2k]^c ∖ [-2(k+1),2(k+1)]^c ) ∑_r=-k^k H_r )  +  _[-2(k+1),2(k+1)]^c( H_-(k+1) + H_k+1) 
        -( _[-2(k+1),-1]^c( 1 -  _ [-2k,-1]^c ∖ [-2(k+1),-1]^c ) ∑_r=-k^-b H_r )  -  _[-2(k+1),-1]^c   H_-(k+1) 
        -( _[1,2(k+1)]^c( 1 -  _ [1,2k]^c ∖ [1,2(k+1)]^c ) ∑_r=b^k H_r )  -  _[1,2(k+1)]^c   H_k+1

Combining together in (<ref>) the partial traces on H_k+1 and H_-(k+1), using the same procedure as for (<ref>), and employing (<ref>), we obtain that (<ref>) is upper bounded by O( J C_ξ f(k/ξ)).


More precisely from (<ref>) we have five terms to keep into account. Term I:

    ( _[-2(k+1),2(k+1)]^c( 1 -  _ [-2k,2k]^c ∖ [-2(k+1),2(k+1)]^c ) ∑_r=-k^k H_r ) ≤( 1 -  _ [-2k-2,-2k-1] ∪ [2k+1,2k+2] ) ∑_r=-k^k H_r  
       ≤( 1 -  _ [-2k-2,-2k-1] ∪ [2k+1,2k+2] ) ∑_r=0^k H_r   + ( 1 -  _ [-2k-2,-2k-1] ∪ [2k+1,2k+2] ) ∑_r=-k^0 H_r  
       ≤ 2 J ∑_r=0^k f(2k+1-r/ξ) ≤ 2 J C_ξ( f(k/ξ) - f(2k/ξ) )



Term II:

    ( _[-2(k+1),-1)]^c( 1 -  _ [-2k,-1]^c ∖ [-2(k+1),-1]^c ) ∑_r=-k^-b H_r ) ≤( 1 -  _ [-2k-2,-2k-1]  ) ∑_r=-k^-b H_r  
       ≤  J ∑_r=b^k f(2k+1-r/ξ) ≤  J C_ξ( f(k/ξ) - f(2k-b/ξ) )



Term III:

    ( _[1,2(k+1))]^c( 1 -  _ [1,2k]^c ∖ [1,2(k+1)]^c ) ∑_r=b^k H_r ) ≤   J C_ξ( f(k/ξ) - f(2k-b/ξ) )



Term IV:

    _[-2(k+1),2(k+1)]^c H_k+1 -  _ [1,2(k+1)]^c  H_k+1 = _[-2(k+1),2(k+1)]^c( 1 -  _ [1,2(k+1)]^c ∖ [-2(k+1),2(k+1)]^c )  H_k+1
       ≤( 1 -  _ [-2(k+1),0]) H_k+1≤ J f(k+1/ξ)



Term V:

    _[-2(k+1),2(k+1)]^c H_-(k+1) -  _ [-2(k+1),-1]^c  H_-(k+1)≤ J f(k+1/ξ)

Overall: 

    Term I + ... + Term V≤ 4 J C_ξ( f(k/ξ) - f(2k/ξ) ) + 2 J f(k+1/ξ)

  
  
So far with (<ref>) we have upper bounded the second term in equation (<ref>), as mentioned above the first term is in relation with the object that in appendix <ref> is called Δ_k(t). In fact, from the first term in (<ref>):

    e^is ( H_[-k,-b] + H_[b,k])( H_[-k,k] - H_[-k,-b] - H_[b,k]) e^-is ( H_[-k,-b] + H_[b,k]) +  
        - e^is ( H_[-k-1,-b] + H_[b,k+1])( H_[-k,k] - H_[-k,-b] - H_[b,k]) e^-is ( H_[-k-1,-b] + H_[b,k+1])
    
      =∫_0^s du    d/du[  e^i(s-u) ( H_[-k-1,-b] + H_[b,k+1]) 
     e^iu ( H_[-k,-b] + H_[b,k])( H_[-k,k] - H_[-k,-b] - H_[b,k]) 
        e^-iu ( H_[-k,-b] + H_[b,k])  
      e^-i(s-u) ( H_[-k-1,-b] + H_[b,k+1])] 
    ≤∫_0^s du    [ H_[-k-1,-b] + H_[b,k+1] - 
      H_[-k,-b] - H_[b,k] ,  
        e^iu ( H_[-k,-b] + H_[b,k])( H_[-k,k] - H_[-k,-b] - H_[b,k])
      e^-iu ( H_[-k,-b] + H_[b,k])]

H_[-k-1,-b] + H_[b,k+1] - H_[-k,-b] - H_[b,k] is “mostly” supported around ± (k+1) but with tails on [-2k,2k], whereas H_[-k,k] - H_[-k,-b] - H_[b,k] is “mostly” supported around [-b,b] with tails on [-2k,2k].
The commutator in (<ref>) has the structure of a Lieb-Robinson bound, once we have shown that these two operators  can be approximated with operators that have supports disjoint and enough far apart to ensure a small upper bound. We now perform such approximations.


Analogously to what done in (<ref>) we have that:

    H_[-k-1,-b] + H_[b,k+1] -   H_[-k,-b] - H_[b,k]≤ 2J C_ξ(f(k/ξ)-f(2k-b/ξ) ) +  H_k+1 + H_-(k+1)
       ≤ 2J C_ξ(f(k/ξ)-f(2k-b/ξ) ) +  2J ≤ O(2J)

(<ref>) bounds the norm of the first entry in the commutator of (<ref>).


To apply the theory of Lieb-Robinson bounds for local Hamiltonians with tails we need to ensure that the two operators that enter the commutator are far apart, since they are actually terms of the Hamiltonian, namely they have tails, we need to restrict their supports, provided the error in doing so is small. Let us consider H_k+1 and H_-(k+1), that have supports [1,2(k+1)] and [-2(k+1),-1]. We define:

    H_k+1 := _[ ⌊3(k+1)/4⌋, ⌊5(k+1)/4⌋ ]^c H_k+1
       H_-(k+1) := _[ ⌊-5(k+1)/4,-3(k+1)/4⌋ ]^c H_-(k+1)

⌊ a ⌋ denotes the largest integer such that ⌊ a ⌋≤ a.  With this choice the size of the support is approximately |k|/2. It is easy to see that the error made is upper bounded by  J f(k/4 ξ), in fact:

    H_k+1 -  _ [ ⌊3(k+1)/4⌋, ⌊5(k+1)/4⌋ ]^c   H_k+1 = ( _[1,2(k+1)]^c( 1 - _ [ ⌊3(k+1)/4⌋, ⌊5(k+1)/4⌋ ]^c ∖ [1,2(k+1)]^c )  H_k+1)

is such that: [ ⌊3(k+1)/4⌋, ⌊5(k+1)/4⌋ ]^c ∖ [1,2(k+1)]^c = [1,⌊3(k+1)/4⌋ -1 ]  ⋃  [ ⌊5(k+1)/4⌋ , 2(k+1) ], therefore the distance of k+1 from this set is approximately ⌊k/4⌋.


On the other hand:

    H_[-k,k] - H_[-k,-b] - H_[b,k]
        =  _[-2k,2k]^c(∑_r=-k^- ⌊k/4⌋-1 H_r  + ∑_r=-⌊k/4⌋^⌊k/4⌋ H_r   + ∑_r=⌊k/4⌋+1^k H_r  )   
       - _[-2k,-1]^c( ∑_r=-k^-⌊k/4⌋-1 H_r + ∑_r=-⌊k/4⌋^-b H_r  )   - _[1,2k]^c( ∑_r=b^⌊k/4⌋ H_r + ∑_r=⌊k/4⌋+1^k H_r )

We pair terms from (<ref>) as follows:

    ( _[-2k,2k]^c - _[-2k,-1]^c) ∑_r=-k^-⌊k/4⌋-1 H_r ≤ J C_ξ(  f ( k/4 ξ) - f ( k/ξ) ) 
       ( _[-2k,2k]^c - _[1,2k]^c) ∑_r=⌊k/4⌋ +1^k H_r ≤ J C_ξ( f ( k/4 ξ) - f ( k/ξ) )

It is left _[-2k,2k]^c∑_r=-⌊k/4⌋^⌊k/4⌋ H_r - _[-2k,-1]^c∑_r=-⌊k/4⌋^-b H_r - _[1,2k]^c∑_r=b^⌊k/4⌋ H_r. We replace this operator with the same one but with support restricted to [-⌊k/2⌋, ⌊k/2⌋], we denote this operator as:

    H_[-b,b] : = _[-⌊k/2⌋,⌊k/2⌋]^c∑_r=- ⌊k/4⌋^⌊k/4⌋ H_r - _[-⌊k/2⌋,-1]^c∑_r=-⌊k/4⌋^-b H_r - _[1,⌊k/2⌋]^c∑_r=b^⌊k/4⌋ H_r

As shown in (<ref>), it follows that: 

    _[-2k,2k]^c∑_r=-⌊k/4⌋^⌊k/4⌋ H_r - _[-2k,-1]^c∑_r=- ⌊k/4⌋^-b H_r - _[1,2k]^c∑_r=b^⌊k/4⌋ H_r -H_[-b,b]
        =  _[-2k,2k]^c( 1 - _[- ⌊k/2⌋,⌊k/2⌋]^c ∖ [-2k,2k]^c ) ∑_r=-⌊k/4⌋^⌊k/4⌋ H_r 
        - _[-2k,-1]^c( 1 - _[-⌊k/2⌋,-1]^c ∖ [-2k,-1]^c ) ∑_r=-⌊k/4⌋^-b H_r 
       - _[1,2k]^c( 1 - _[1,⌊k/2⌋]^c ∖  [1,2k]^c ) ∑_r=b^⌊k/4⌋ H_r 
       ≤ J C_ξ( 4f ( k/4 ξ) - 2f ( k/2 ξ) - 2f ( k/2 - b /ξ) )



To upper bound the norm of the commutator (Lieb-Robinson bound) in (<ref>), we also need to upper bound the norm of each entry of the commutator. The norm of the first entry is provided by (<ref>).


The norm of the second entry of the commutator in (<ref>) is bounded as follows:

    H_[-k,k] - H_[-k,-b] - H_[b,k] := 
     _[-2k,2k]^c∑_r=-k^k H_r - _[-2k,-1]^c∑_r=-k^-b H_r
     - _[1,2k]^c∑_r=b^k H_r 
       ≤ | _[-2k,2k]^c( 1 - _ [-2k,-1]^c ∖ [-2k,2k]^c ) ∑_r=-k^-b H_r  + _[-2k,2k]^c( 1 - _ [1,2k]^c ∖ [-2k,2k]^c ) ∑_r=b^k H_r  +  H_[-b+1,b-1]
       ≤ |  ( 1 - _ [-2k,-1]^c ∖ [-2k,2k]^c ) ∑_r=-k^-b H_r  + ( 1 - _ [1,2k]^c ∖ [-2k,2k]^c ) ∑_r=b^k H_r  +  (2b-1)J 
       ≤ 2J ∑_r=b^k f( r/ξ) +  (2b-1)J ≤ 2JC_ξ(f( b/ξ) - f( k/ξ) ) +  (2b-1)J ≤ O(2bJ)



Overall the norm of the commutator in (<ref>) is bounded as follows:

    [ H_[-k-1,-b] + H_[b,k+1] - 
      H_[-k,-b] - H_[b,k] ,  e^iu ( H_[-k,-b] + H_[b,k])( H_[-k,k] - H_[-k,-b] - H_[b,k])
      e^-iu ( H_[-k,-b] + H_[b,k])] 
       ≤[ H_k+1 + H_-(k+1) ,  e^iu ( H_[-k,-b] + H_[b,k])( H_[-k,k] - H_[-k,-b] - H_[b,k])
      e^-iu ( H_[-k,-b] + H_[b,k])]  + 
        + 2 ( 2J C_ξ(f(k/ξ)-f(2k-b/ξ) ) + 2 J  f(k/4ξ) ) ( 2JC_ξ(f( b/ξ) - f( k/ξ) ) +  (2b-1)J  )  
       ≤[ H_k+1 + H_-(k+1) ,  e^iu ( H_[-k,-b] + H_[b,k])H_[-b,b] 
      e^-iu ( H_[-k,-b] + H_[b,k])]  + 
        + 8 O(bJ^2) ( 2 C_ξ(f(k/ξ)-f(2k-b/ξ) ) + 2  f(k/4ξ) )  + 4 O(C_ξ J^2)   ( 6f ( k/4 ξ) - 4f ( k/2 ξ) - 2f ( k/2 - b /ξ) )  
       ≤ O(bJ^2)(e^vt-1) f( k/2ξ) + J^2 D_ξ

In the last line, (<ref>), we have introduced D_ξ≤ O (f( k/4ξ) ) to shorten the notation. We see that in comparison to the Lieb-Robinson bound of strictly local operators there is a term, D_ξ that vanishes for large distances, of O(k), among the supports of H_k+1 and H_[-b,b], as f( k/2ξ). We stress that when f has a compact support, as discussed before in (<ref>), we recover the case of strictly local Hamiltonians and D_ξ is identically vanishing. 
The norm of the commutator in (<ref>) is upper bounded by a Lieb-Robinson bound for Hamiltonians with rapidly decaying interactions <cit.>, we denote v the corresponding velocity. We notice that the supports of the operators involved in the Lieb-Robinson bound are at a distance of the order of k/2. 


We are now ready to go back to (<ref>), that we copy below, to obtain the bound on the trace distance that we are looking for:

    1/2V_Λ_k+1(t) ρV^*_Λ_k+1(t) -  V_Λ_k(t) ρV^*_Λ_k(t) _1  
       ≤  t sup_s∈[0,t]  e^is ( H_[-k,-b] + H_[b,k])( H_[-k,k] - H_[-k,-b] - H_[b,k]) e^-is ( H_[-k,-b] + H_[b,k]) +  
        - e^is ( H_[-k-1,-b] + H_[b,k+1])( H_[-k,k] - H_[-k,-b] - H_[b,k]) e^-is ( H_[-k-1,-b] + H_[b,k+1]) +  
        +  t  H_[-k-1,k+1] - H_[-k-1,-b] - H_[b,k+1]  -  ( H_[-k,k] - H_[-k,-b] - H_[b,k]) 
       ≤  t sup_s∈ [0,t]∫_0^s du  [ H_[-k-1,-b] + H_[b,k+1] - 
      H_[-k,-b] - H_[b,k] ,  
        e^iu ( H_[-k,-b] + H_[b,k])( H_[-k,k] - H_[-k,-b] - H_[b,k])
      e^-iu ( H_[-k,-b] + H_[b,k])]  + 
        + 4   t    J   C_ξ( f(k/ξ) - f(2k/ξ) ) + t  J   f(k+1/ξ)  
       ≤  t^2 O(bJ^2) (e^vt-1) f( k/2ξ) + t^2J^2 D_ξ   +  t    J   D'_ξ
       ≤ t   J   O (  g(b,J,t)   f ( k/4 ξ) )

In (<ref>) we have introduced 

    D'_ξ:= 4    C_ξ( f(k/ξ) - f(2k/ξ) ) + f(k+1/ξ) ≤ O (f( k/ξ) )

When f has a compact support, see equation (<ref>), D'_ξ, as D_ξ, vanishes.
In (<ref>), it is: 

    g(b,J,t) ≤ b  t  J  ( (e^vt-1) + O(1) ) + O(1)

The minimization with respect to b of the trace distance (<ref>) is straightforward, in fact being g proportional to b in (<ref>) up to corrections of O(1), the minimum is reached for the smallest b that makes the theory consistent. Since we have seen for the strictly local case, see (<ref>), that b=ξ, then we pick b=ξ also in the general case.


We are now ready to upper bound the sum in (<ref>), that we copy below (<ref>), to obtain Δ S_α(t) of equation (<ref>) in the case of an Hamiltonian with exponential decrease of interactions .
As done in the case of strictly local Hamiltonians we need to find out the value of l that minimizes the sum. This will turn out to be dependent on α, v ad ξ. We adopt a different, but equivalent, approach than the one of appendix <ref> where we had an approximate value of l from the solution of the minimization condition. Here we will pick up l=β t with a β to be determined in such a way to hold a linear increase in t for large t of Δ S_α(t), we know in fact that a “slower” increase has been already ruled out by <cit.>. 
With, for large enough t, R_k(t) ≤ξ (t J)^2 e^vte^-k/4ξ and 

    Δ S_α (t)    ≤1/1-α∑_k=l+1^Llog[ (1-R_k(t) )^α + (2^2(k+1)+1-1)^1-αR_k(t)^α] + 2l 
       ≤1/1-α∑_k=l+1^∞( -α R_k(t)  + 2^(2k +3)(1-α)R_k(t)^α) + 2β t 
       ≤1/1-α∑_n=1^∞( -α R_l+n(t)  + 2^(2(l+n) +3)(1-α)R_l+n(t)^α) + 2β t 
       ≤1/1-α∑_n=1^∞( -αξ (t J)^2 e^vte^-β t+n/4ξ  + e^ln 2(2(β t+n) +3)(1-α)(ξ (t J)^2 )^α e^α vte^-αβ t+n/4ξ) + 2β t 
       ≤1/1-α∑_n=1^∞( -αξ (t J)^2 e^vte^-β t+n/4ξ  + e^3ln 2(1-α)(ξ (t J)^2 )^α e^α vte^-λ(β t+n)) + 2β t
 
In the last step we have introduced λ:= α/4ξ - 2ln2(1-α). We now carry out two important steps. The first is to ensure that the sum over n is finite, this will give a condition on set of allowed α. The second step would be to impose that the sum of the series decreases exponentially in time, this will provide a restriction on the value of β and therefore the large t behaviour of Δ S_α (t).
From the second sum in (<ref>) we get a restriction on the value of α as follows:

    λ > 0 ⇒α/4 ξ - 2ln2(1-α) > 0 ⇒α > 1/1+1/8ξln2

With ξ≪ 1, that corresponds to the Hamiltonians' term becoming almost on site, (<ref>) reduces to α≳ 0. In the limit of extendend interactions instead ξ≫ 1, being by assumption 0 < α≤ 1, (<ref>) gives α≈ 1.


On the other hand the restriction on β to ensure that the sum in (<ref>) decreases exponentially in t provides:

    λβ - α v > 0 ⇒β > α v/λ = v/1/4ξ- (2ln2)1-α/α

It can be checked that the condition for the exponential decrease in time of the first term of the sum in (<ref>) is less restrictive than (<ref>).


Overall, making use of the conditions (<ref>) and (<ref>) we have that:

    Δ S_α (t)  ≤ -α/1-αξ (t J)^2 e^vte^-β t/4ξ1/e^1/4ξ-1 + 1/1-α e^(3ln 2)(1-α)(ξ (t J)^2 )^α e^α vte^-λβ t1/e^λ -1  + 2β t 
       lim_t→∞Δ S_α (t)/t <  2v/1/4ξ- (2ln2)1-α/α

The limit α→ 1, that gives the von Neumann entropy, is:

    Δ S_1 (t)  ≤ (3ln 2) ξ (t J)^2 e^vte^-β t/4ξ1/e^1/4ξ-1  + 2c t 
       lim_t→∞Δ S_1 (t)/t < 8vξ
















§ AN UPPER BOUND ON THE CONCAVITY OF THE Α-RÉNYI ENTROPIES, 0 < Α < 1
   

 
 
 
 
 
 
Given a convex combination of states ρ := ∑_i=1^N p_i ρ_i, with ρ_i: C^d →C^d, denoting ρ_i= ∑_j=1^n_iλ_j^i |e_j^i ⟩⟨ e_j^i | the spectral  decomposition of each ρ_i, for 0 < α < 1 it holds:

    ∑_i=1^N p_i S_α (ρ_i) ≤ S_α (ρ) ≤ H_α{p_i λ_j^i }

where H_α{p_i λ_j^i } := 1/1-αlog∑_i,j ( p_i λ_j^i)^α is the α-Rényi entropy of the probability distribution {p_i λ_j^i }. Moreover:
 
    H_α{p_i λ_j^i }≤ H_α{p_i} + max_i { S_α(ρ_i) }

Therefore if {ρ_i } are pure states, it is S_α (ρ) ≤ H_α{p_i }, with the upper bound depending only on the probability distribution { p_i }.




We now sketch a proof, for more details consult lemma 1.24 (page 18) and example 11.12 (page 178) of <cit.>.
The first inequality in (<ref>) is the concavity of the Rényi entropy with 0 < α <1, <cit.>.
 The proof of the second inequality in (<ref>) is a simple application of the theory of majorization, see chapter II of <cit.>. The spectral decomposition of ρ reads: ρ = ∑_k=1^m a_k |v_k ⟩⟨ v_k |. Since {|v_k ⟩} is an orthonormal set, it follows that m ≤ d. The set { |e_j^i ⟩}, that collects the eigenvectors of all {ρ_i }, is in general not orthonormal. In fact, for example, two states ρ_i can share an eigenvector.  It follows that, completing the matrix V with entries V_j,k^i := √(p_i λ_j^i/a_k)⟨ v_k |e_j^i ⟩ to a unitary matrix U, it holds
 
    |e_j^i ⟩ = ∑_k U_j,k^i |v_k ⟩

It follows from the definition of V, and the unitarity of U, that 

    p_i λ_j^i = ∑_k |U_j,k^i|^2 a_k

The matrix with entries |U_j,k^i|^2 is a double stochastic matrix, meaning that ∑_k |U_j,k^i|^2 = ∑_i,j |U_j,k^i|^2 = 1, this implies, see theorem II.1.10 (page 33) of <cit.>,   that the set {p_i λ_j^i } is majorized by the set  { a_k }. Each set is supposed to be ordered in a non increasing fashion. It then follows from theorem II.3.1 (page 40) of <cit.>, and the concavity of the function x^α, with 0< α <1, that:

    ∑_k a_k^α≤∑_i,j (p_i λ_j^i)^α

That implies, being the logarithm an increasing function: S_α (ρ) := H_α{a_k }≤ H_α{p_i λ_j^i }. 


We now prove equation (<ref>).
 
    H_α{p_i λ_j^i } := 1/1-αlog∑_i,j ( p_i λ_j^i)^α =
     1/1-αlog∑_i=1^N  p_i^α∑_j (λ_j^i)^α
       = 1/1-αlog∑_i=1^N  p_i^αexp( (1-α) 1/1-αlog∑_j (λ_j^i)^α) = 1/1-αlog∑_i=1^N  p_i^αexp( (1-α) S_α (ρ_i) ) 
       ≤1/1-αlog∑_i=1^N   p_i^αexp( (1-α) max_i { S_α(ρ_i) }) =  H_α{p_i} + max_i { S_α(ρ_i) }




Equation (<ref>) holds also for the von Neumann entropy, with the Rényi entropy H_α replaced by the Shannon entropy H. In fact the same proof applies with the function x^α replaced by -x log x. See equation 2.3 of <cit.>. See also Theorem 11.10 (page 518) of <cit.>, or theorem 3.7 (page 35) of <cit.> for alternative proofs.  
For the von Neumann entropy we can see that the upper bound takes a more explicit form:
 
    S(ρ) ≤ H{p_i} + ∑_i p_i S(ρ_i)

In fact:

    H {p_i λ_j^i }:= -∑_i,j p_i λ_j^i log ( p_i λ_j^i ) = -∑_i,j p_i λ_j^i ( log p_i + logλ_j^i ) 
        = -∑_i p_i log p_i  -∑_i,j p_i λ_j^i logλ_j^i = H{p_i} + ∑_i p_i S(ρ_i)

This implies that the difference S(ρ) - ∑_i p_i S(ρ_i) ≤  H{p_i} is upper bounded by a quantity only depending on the probability distribution { p_i }, but not on other quantities like the states {ρ_i } or the dimension of the Hilbert space. The bound (<ref>) has been improved in theorem 14 of <cit.>, where also appears an upper bound on the difference of Holevo quantities for different ensembles that is independent from the Hilbert space dimension.


In general this is not the case for the Rényi entropy. Let us consider the state ρ of equation (<ref>) and the maximally mixed state of C^d, 1/d1_d. It is easy to see that S_α( p ρ + (1-p) 1/d1_d ), with fixed α, p and d ≫ 1 goes like log d. This also follows from (<ref>). We see that contrarily to (<ref>), for α-Rényi entropy it is: S_α( p ρ + (1-p) 1/d1_d ) -(1-p) S_α( 1/d1_d ) ≈ p log d. 


Within the setting of the physical system described in <ref>, and the notations of Lemma <ref>, let us consider the case where max{ S_α(ρ_i) }≤ O( log L ), and N ≤ O(L), then according to equation (<ref>), we have:

    S_α(ρ) ≤1/1-αlog∑_i=1^N p_i^α + O(log L ) ≤ O(log L)

This shows that when convex combinations of few, O(L), low entangled states, S_α(ρ_i) ≤ O(log L), are considered then ρ = ∑_i p_i ρ_i is still low entangled: S_α(ρ) ≤ O(log L).  
















































 














 Paper Title: Post-Quantum Cryptography (PQC) Network Instrument: Measuring PQC Adoption Rates and Identifying Migration Pathways.










Post-Quantum Cryptography (PQC) Network Instrument:
 Measuring PQC Adoption Rates and 
 Identifying Migration Pathways
    Jakub Sowa 0009-0009-0115-4822^1, Bach Hoang 0000-0003-0827-2564^1, Advaith Yeluru^1, Steven Qie^2,  
 Anita Nikolich^2, Ravishankar Iyer^2, Phuong Cao*Corresponding author: Phuong Cao; Data: https://pmcao.github.io/pqchttps://pmcao.github.io/pqc 0000-0001-6028-0583^1,2,*




^1National Center for Supercomputing Applications,
^2University of Illinois at Urbana-Champaign


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



§ ABSTRACT 
The problem of adopting quantum-resistant cryptographic network protocols or post-quantum cryptography (PQC) is critically important to democratizing quantum computing. The problem is urgent because practical quantum computers will break classical encryption in the next few decades. Past encrypted data has already been collected and can be decrypted in the near future. The main challenges of adopting post-quantum cryptography lie in algorithmic complexity and hardware/software/network implementation. The grand question of how existing cyberinfrastructure will support post-quantum cryptography remains unanswered.

This paper describes: i) the design of a novel Post-Quantum Cryptography (PQC) network instrument placed at the National Center for Supercomputing Applications (NCSA) at the University of Illinois at Urbana-Champaign and a part of the FABRIC testbed; ii) the latest results on PQC adoption rate across a wide spectrum of network protocols (Secure Shell – SSH, Transport Layer Security – TLS, etc.); iii) the current state of PQC implementation in key scientific applications (e.g., OpenSSH or SciTokens); iv) the challenges of being quantum-resistant; and v) discussion of potential novel attacks. 

This is the first large-scale measurement of PQC adoption at national-scale supercomputing centers and FABRIC testbeds. Our results show that only OpenSSH and Google Chrome have successfully implemented PQC and achieved an initial adoption rate of 0.029% (6,044 out of 20,556,816) for OpenSSH connections at NCSA coming from major Internet Service Providers or Autonomous Systems (ASes) such as OARNET, GTT, Google Fiber Webpass (U.S.) and Uppsala Lans Landsting (Sweden), with an overall increasing adoption rate year-over-year for 2023-2024. Our analyses identify pathways to migrate current applications to be quantum-resistant.





§ INTRODUCTION


The problem of adopting quantum-resistant cryptographic network protocols or post-quantum cryptography (PQC) is critically important to democratizing quantum computing. The problem is urgent because practical quantum computers <cit.>, will break classical encryption in the next few decades. Major applications such as financial services, high-performance computing (HPC) or supercomputing, and health analytics must be migrated to be quantum-resistant. The main challenges of adopting post-quantum cryptography lie in both algorithmic complexity and hardware/software/network implementation. The grand question of how existing cyberinfrastructure will support post-quantum cryptography remains unanswered.

This paper describes: i) the design of a Post-Quantum Cryptography (PQC) network instrument placed in a national-scale supercomputing center, ii) the latest results on PQC adoption rate across a wide spectrum of popular network protocols (Secure Shell – SSH, Transport Layer Security – TLS, etc.), iii) the current state of PQC implementation in key scientific applications (e.g., OpenSSH, SciTokens), iv) the challenges of being quantum-resistant and v) discussion of potential novel attacks. Our result is critically important regarding adopting National Institute of Standards and Technology (NIST)'s draft algorithms, such as CRYSTALS-Kyber for encryption, FALCON and SPINCS+ for digital signature, and KEMTALS for key exchange <cit.> and protocol-specific PQC adaptation such as Hybrid Streamlined NTRU (Ring-Based Public Key Cryptosystem) Prime sntrup761 and x25519 with SHA-512 (sntrup761x25519-sha512) in OpenSSH <cit.>.




Motivation.
The cryptographic algorithms we use to secure and verify all this data today are far more complex than in centuries past, yet still heavily reliant on the "hardness" of certain mathematical problems. These "hard" problems, such as integer prime factorization and the discrete logarithm problem, form the basis of modern cryptosystems such as RSA and Ecliptic Curve Cryptography (ECC), respectively. As long as no computer can efficiently solve these problems, which are assumed to be hard, the security of the cryptosystem should be maintained. For example, it would take an exascale supercomputer such as Frontier (≈ 1.20 exaflops <cit.>) longer than the age of the universe, on average, for an attack to reveal the 256-bit private key in the curve25519 ECC cryptographic scheme.

With new developments in quantum computers and quantum algorithms on the rise, many of these problems previously assumed to be hard appear much easier. The 1994 Shor's algorithm for solving both discrete logs and prime factorization efficiently on quantum computers shows the need for this Post-Quantum Cryptography (PQC), which is made even more urgent when you consider an adversary who is storing classically encrypted traffic now to be decrypted with a quantum computer in the near future when they are powerful and reliable enough. The grand question of how existing cyberinfrastructure will support post-quantum cryptography remains unanswered and will be addressed in this paper.




Key contributions:

    
  * A novel network instrument to measure PQC adoption in real-time and provides historical trends of the adoption rates across seven layers of computer network protocols. 
    
    
    
    
  * The first statistical study of PQC's readiness across seven layers of computer network and HPC applications with a unique vantage point: a petascale supercomputing center with nation-scale visibility through the FABRIC TeraCore network.
    
    
  * Release of an open sample dataset containing metadata of cryptographic suites across major network protocols (SSH, TLS, RDP, etc.  <cit.>), available online at https://pmcao.github.io/pqc
    
    
  * Discussion of potential novel PQC attacks and characterization of major challenges blocking the adoption of PQC in HPC applications. 
    
    
    
    
    


Major results.
Our results highlight the difficulty of ensuring all systems are updated and using the most secure connection options available, which is important to reach widespread PQC adoptions as follows:



  * OpenSSH and Google Chrome have successfully implemented PQC and achieved an initial adoption rate of 0.029% (6,044 out of 20,556,816) for OpenSSH connections at NCSA.


  * The adoption rate for OpenSSH is increasing year-over-year for 2023-2024. 


  * Top U.S. large networks (Autonomous Systems – ASes) such as OARNET, GTT, Google Fiber Webpass, and Comcast; and Uppsala Lans Landsting (Sweden) are hosting clients that have already adopted PQC in OpenSSH. 


  * Over 83% of Server-side SSH protocols were from 2019 and earlier, 3 years before sntrup761x25519 was even introduced. Only about 65% of connections used TLS version 1.3, the most recent and most secure option for this use case The rest used TLS version 1.2, which still supports many of the weak cipher suites we found. 



Our analyses identify pathways to migrate current applications to be quantum-resistant. Through experimental deployment at different network vantage points, our approach may help identify empirical novel attacks against PQC implementations and give feedback to the National Institute of Standards and Technology (NIST) as well as developers to improve their drafts and successfully migrate to PQC <cit.>.



[width=, colback=white!95!black, boxrule=0.5pt, left=2pt,right=2pt,top=1pt,bottom=1pt]
remark
Remark remark:

Perspective w.r.t the state of the art.
To the best of our knowledge, this is the first network instrument that exists to monitor PQC adoption publicly. There are a few related work <cit.>, e.g., Cloudflare released a snapshot of TLS adoption in Feb 2024 through their content delivery network <cit.>. On the other hand, our network instrument is the first of such to monitor the adoption of both OpenSSH, TLS, and other protocols at large-scale, high-speed (gigabit to terabit) interconnected networks with a wide spectrum of not only traditional cloud but also scientific workloads. 








§ APPROACH OVERVIEW

This section describes the network topology, architecture of the network optical tap, placement of our network instrument, data collected, and statistical methods used.









Figure <ref> describes our network instrument. Using FABRIC testbed (A), we deployed our instrument at the NCSA site (B) to ingest Zeek connection metadata and parse session and application layer cipher suite information (C, D, E). An example of PQC key exchange and statistics that we parsed is shown for Secure Shell (SSH) protocol (F, G). Our instrument is embedded in a nation-scale network for monitoring a wide spectrum of scientific workloads as follows.

FABRIC testbed. FABRIC <cit.> is a nation-scale research infrastructure consisting of a TeraCore network between universities, national labs, and supercomputing facilities such as NSF Cloud testbeds CloudLab, NCSA, and Chameleon (Figure <ref>A). FABRIC powers exploratory networking research at scale in a variety of applications, including cybersecurity, distributed computing, high-speed storage, artificial intelligence, and HPC workloads. The unique approach to FABRIC is its Application Programming Interface (API), allowing experiments to programmatically compose extensible, high-speed interconnected optical link networks, compute, and storage. 

NCSA. The National Center for Supercomputing Applications (NCSA) offers cutting-edge cyberinfrastructure through the National Petascale Computing Facility (NPCF) that houses major NCSA infrastructure, including the Blue Waters supercomputer, to support diverse research needs. Major computing resources include: 1) Delta – NSF-funded GPU system ideal for GPU-accelerated applications (A100) and gateway-based workflows with nearline storage, Infiniband interconnect; 2) HAL Cluster supporting deep learning system with IBM POWER9 CPUs; and testbeds for ScienceDMZ, honeypot experiments and FPGA boards. All these resources are interconnected with long-term tape archives (Petabytes) and high-speed external connectivity (+400Gbps) to major research networks such as the Energy Sciences Network (ESnet) (Figure <ref>B).



Zeek network security monitor.
Zeek (formerly Bro <cit.>) is a flexible, open-source platform with decades of development history in network security monitoring. Defenders use Zeek's network analyzers to perform deep packet inspections and provide a broad view of many network protocols. Zeek enables high-performance, high-level, stateful semantic analysis at the application layer and is used operationally at various large sites such as Berkeley Lab and NCSA. The collection and analysis of these data provide a comprehensive view of PQC adoption, as described below.

Ethical considerations.
Our approach works on connection metadata produced by Zeek parsers. We did not use man-in-the-middle or traditional SSL termination equipment. Thus, we only see the encrypted data and not the original content. However, we do have access to the timing information of connection establishment to derive the trend and seasonality of the PQC adoption rate. The connection metadata generally includes standard information such as IP address, certificate, and negotiation of cipher suites. We do not see Personally Identifiable Information (PII).

 

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remark
Remark remark:

Our Zeek-integrated network instrument to measure PQC adoption has three main components: 

    
  * Parallel log parser to ingest large-volume of Zeek logs (metadata) from a +400Gbps network.
    
  * Historical analyses scripts that perform regression, trend, and statistics on the parsed metadata of cipher suites.
    
  * Real-time snapshot and comparison of our results vs. others (if any) to detect novel attacks and give feedback to the National Institute of Standards and Technology (NIST).
















§ DATASETS


This section presents the dataset that our network instrument has collected, which is built upon our prior work on the security testbed at NCSA <cit.>. We have collected a heterogeneous dataset containing metadata of cryptographic network protocols powering high-speed scientific workloads at NCSA between 2023-2024. In total, approximately 13TB of metadata has been generated and collected using the Zeek network observability framework from a 400Gbps network border link on the NCSA network and partner facilities, which supports a wide range of scientific applications <cit.> such as Globus, SciTokens, and Kubernetes. 

A summary of our dataset is specified in Table <ref>. Our dataset contains network connections in encrypted application-layer protocols (above layer 3 TCP/IP), such as Secure Shell (SSH), Remote Desktop Protocol (RDP), Hypertext Transfer Protocol Secure (HTTPS), etc. With metadata including network connections from various sources and destinations, we recorded a wide range of cryptographic algorithms (e.g., ECDH, RSA, SHA-256, etc.) and elliptic curves (e.g., x25519, secp256r1, etc.) that are used for encryptions tasks such as key exchange or key encapsulation mechanism.  Since this dataset is collected 24/7 for more than 16 months (January 2023 - present), it provides the orthogonal view of PQC adoption in scientific network traffic. 

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remark
Remark remark:

Our dataset is unique in its placement at a strategic network vantage point to observe metadata of a wide spectrum of applications across different protocols, validating our PQC measurement for both historical and real-time adoption rates.


























 



§ METHOD & RESULTS
 PQC NETWORK INSTRUMENT ON OPENSSH





This section presents both: i) longitudinal studies of PQC adoption on the Secure Shell (SSH) protocol over 16 months (Jan 2023 - Apr 2024) and ii) a detailed analysis of cipher suites used in OpenSSH on a sample working day. 

The first and most interesting protocol we studied, owing to its wide usage across most operating systems, was the Secure Shell (SSH) protocol which is well understood through our prior studies <cit.>. SSH metadata included information on four types of cryptographic algorithms used to secure the connections, the data for each is shown in Table 1. The most immediately apparent detail in the data is that the vast majority of the connections seem to be using some version of OpenSSH. Additionally, there are a few deprecated and insecure cryptographic algorithms in use, which would be a cause for more concern if not for their low usage numbers.

The data, stored as logs using the Zeek network analysis framework, is focused solely on connection metadata. The data was parsed and visualized in Python using its library, Matplotlib, and are shown in Figure 
<ref>,
<ref>,
<ref>. We first provide a background on PQC implementation in the latest version of OpenSSH (version ≥ 9.0), then describe our results based on SSH data.



 §.§ Background on PQC adoption in OpenSSH

This section presents details regarding the implementation
of PQ authentication into the OpenSSH library. The first version that OpenSSH supports PQC is 9.0 <cit.>. 
This developing standard PQC protocol has three main components:

    
  * sntrup761 is a key-encapsulation mechanism (KEM) in a family of ring-based public key cryptosystems called Streamlined NTRU Primes  <cit.> <cit.>, in which 761 is a prime number serving as a parameter. Cryptosystems in this family are parameterized by 3 positive integers (p, q, w), in which p, q are prime numbers, x^p - x - 1 is irreducible in the polynomial ring ℤ/q[x]. The parameters of sntrup761  KEM are a triplet of numbers p = 761, q = 4591, w = 286 <cit.>.
    
    
  * x25519 is the widely implemented Elliptic-Curve-Diffie-Hellman (ECDH) key exchange protocol, using Curve25519 as a underlying curve. Curve25519 <cit.> is an Montgomery elliptic curve in the form of y^2 = x^3 +  Ax^2 + x over the field ℤ/p, in which p = 2^255 - 19 is a prime number, and A = 486662 is an integer that A^2 - 4 is not a square modulo p.
    
    
  * SHA-512 is a secure cryptographic hash algorithm to ensure data integrity <cit.>. The input for SHA-512 is a message with a size up to 2^128 - 1 bits and the output is a word with a consistent length of 512 bits. 


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remark
Remark remark:

The sntrup761 KEM is a lattice-based post-quantum algorithm. Lattice-based algorithms are relatively new in the cryptography world and are thus not as strongly tested against even pre-quantum adversaries as some more traditional schemes. Many PQC implementations such as this one thus combine a more established cryptosystem with the intention of ensuring immediate security if just one of the involved algorithms are proven weak <cit.>.



To visualize this PQC protocol, we show key exchange with a client and server in Figure <ref>.



    
  * Ephemeral Key Generation:  Client and server generate temporary (ephemeral) key pairs using both sntrup761 and x25519 algorithms.

    
  * Public Key Exchange: Client and server exchange their public keys for both sntrup761 and x25519.

    
  * Shared Secret Calculation: Each side uses its private key and the other's public key to calculate a shared secret for both the sntrup761 and x25519 algorithms.

    
  * Combining and Hashing: The two shared secrets are combined and then hashed using the SHA-512 algorithm.

    
  * Final Key: The result of this process is the final, strong key used to encrypt and decrypt the SSH communication.


As SSH is widely adopted and frequently updated against vulnerabilities, it is more likely that SSH clients and servers will be the first to adopt PQC schemes. The described protocol is secure because it combines: 1) a hybrid approach, using both a post-quantum resistant algorithm (sntrup761) and a traditional algorithm with strong security (x25519) to provide fault tolerance; and 2) SHA-512, a hashing function securing the final key such that no information about the individual shared secrets can be easily derived.










 §.§ Results on PQC protocols adoption in SSH

First, we compare the adoption rate of SSH with the publicly available adoption rate from Cloudflare. Figure <ref> shows cross-protocol and cross-site comparison of adoption rate between SSH protocol at NCSA (our analysis) and TLS 1.3 protocol at Cloudflare  <cit.>. NCSA records 0.029% (6044 out of 20,556,816) monthly adoption rate for SSH, while Cloudflare recorded ≈ 1.78 percent adoption rate for TLS 1.3; more than 99% adoption came from Chrome <cit.>.

Figure <ref> shows a histogram of autonomous systems adopting PQC in SSH showing that the top 5 ASes (OARNE, GTT, Google Fiber, Comcast, Advanced Communications Technology, etc.)  from U.S. and Uppsala Lans Landsting (Sweden) accounted for the majority of PQC in the head of the distribution. A long list of ASes is shown in the long tail.

Figure <ref> shows the increasing adoption rate of PQC in SSH key exchange (sntrup761x25519-sha512@openssh.com), starting from January 2023 with only 37 exchanges, up to 1,585 exchanges in April 2024.






More interestingly, it is clear that there are no PQ algorithms in use except for one, the key exchange algorithm sntrup761x25519 using the SHA512 hash. This algorithm is an implementation of the post-quantum NTRU Prime cryptographic scheme, specifically a hybrid Streamlined NTRU Prime paired with the x25519 ECDH key exchange method to preserve security against classical adversaries in case a vulnerability in NTRU is found. It was introduced in the OpenSSH version 7.9 and added to the list of defaults in version 9.0 in 2022. It was developed by OpenSSH before the third round selections by NIST took place later that year, which actually ended up removing NTRU after the second round in the process. Unfortunately, we could only identify an insignificant amount of connections using this algorithm for key exchange, 0.08% for a sample day. The overall average adoption rate is  0.029% (6044 out of 20,556,816) for the period in which we compare SSH adoption rate with Cloudflare's TLS adoption rate (Figure <ref>).

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remark
Remark remark:

SSH is among the first protocols that implement PQC and actually have clients adopt their PQC protocol widely, as shown in our results.




 §.§ Results on classical protocols in SSH

Here, we provide further detailed traffic analysis for a regular working day in June 2023 for the OpenSSH protocol at NCSA. Network metadata was collected from a 400 Gbps network border link on the NCSA network. 

Importantly, we see the presence of various deprecated algorithms used by some connections as shown in Table  <ref>. This poses a security risk outside the scope of PQC and warrants another look at the versions of SSH protocols in use. 

Through further inspection of metadata logs, we determined that over 83% of Server-side SSH protocols were from 2019 and earlier, three years before sntrup761x25519 was even introduced. This highlights a common problem in the technology space—the difficulty of ensuring all systems are updated and using the most secure connection options available. This is important if we are to ever reach full PQC adoption.






§ METHOD AND RESULTS 
 PQC NETWORK INSTRUMENT ON OTHER PROTOCOLS

This section describes our analysis extended to other major network protocols such as Remote Desktop Protocol (RDP), Domain Name System (DNS), and Transport Layer Security (TLS). Note that this analysis is focused on a sample working day to illustrate the adoption. The objective is to measure the readiness level of applications using these protocols with regard to Post-Quantum Cryptography.



 §.§ RDP

The next network protocol we looked at was the Remote Desktop Protocol (RDP) for remote desktop visualization of Windows machines. Although the protocol today has no simple solution for using PQC, it is still worth analyzing the cryptography available given the protocol's prevalence in the IT space and its history as a tool for attackers <cit.>. RDP plays a crucial role for many organizations as a way to manage servers, troubleshooting, and remote work. With the rise in popularity of remote access solutions, ensuring RDP meets future quantum-safe standards is necessary to not only protect the integrity of data transmitted via RDP but also ensure that RDP sessions remain secure in the face of quantum computing advancements.  

Even though for RDP, there was a minimal data set of 26 connections, we were able to identify an interesting issue. Generally, RDP can be configured to use two types of cryptographic security, Enhanced Encryption and Network-Layer Authentication (NLA). Enhanced Encryption for RDP offers encryption over TLS of everything after the Connection Initiation stage and also allows authentication of the server from the client side. NLA requires the client to be authenticated before establishing a session with the server. However, out of all 26 connections, only 2 used the HYBRID-EX security protocol setting, which uses both Enhanced Encryption and NLA. As with SSH, this calls for more careful management of individual machines and making sure they are all configured to use both Enhanced Encryption and NLA for optimal security.

A final thing of note regarding RDP is that on connections where both server and client are using Windows 11 machines, it can be configured to run over TLS 1.3 to add additional levels of security.







 §.§ DNS


We then briefly examined the Domain Name System (DNS) protocol used for deriving IP addresses from domain names. Historically with DNS, packets are not encrypted - this is also true for the NCSA network. However, there are various ways of encrypting DNS traffic, namely through the HTTPS protocol, or over a TLS connection.

HTTPS can be used with DNS, called DoH, to encrypt DNS queries with HTTPS over port 443. This type of DNS can be enabled on some browsers such as Firefox and Chrome. As a side effect, DNS-over-HTTPS makes distinguishing it from general web traffic in logs difficult. 

Another form of secure DNS is DNS-over-TLS (DoT). DoT creates encrypted TLS channels for DNS queries with a DNS-over-TLS domain name resolver instead of the typical unencrypted DNS resolver.

With both these methods of securing DNS and many others not discussed here, it should be noted that any security properties of DNS are transferred from other network protocols, hence creating an opportune security dependence chain discussed further in later sections.










 §.§ TLS


The final network protocol investigated in our study was the Transport Layer Security (TLS) protocol. As with RDP and DNS, there has not been tremendous work finished in the community regarding PQC implementation in this protocol. Some companies and organizations have begun developing specifically tailored PQC implementations for their own use, but little has been developed for the common internet user. Microsoft, for example, remains in the process of collaborating with OpenSSL to integrate PQC into their open-source TLS libraries, but that is still in the works <cit.>. With that in mind, it remains to be said that no PQC was found in the TLS network traffic data at NCSA. As with RDP though, it is again worth looking at the security and PQC readiness of TLS here anyway.

Primarily, the security of TLS lies in its various cipher suites, or groupings of cryptographic algorithms, used for securing internet traffic. This data was collected in Table 2, where we show the top ten cipher suites appearing in the connections. While the most commonly found cipher suites in the data are today considered to be secure against classical adversaries, such as the top cipher suite TLS-AES-128-GCM-SHA256, a sizable portion of connections (over 8%) used weak cipher suites that have a concerning vulnerability.

Even though Chrome browser 116 and above offers TLS 1.3 hybridized Kyber support for PQC, the client does not turn it on by default. In addition, this problem is in part due to the other data point we examined, the TLS versions in use on the server end of the connections: only about 65% of connections used TLS version 1.3, the most recent and most secure option for this use case. The rest used TLS version 1.2, which still supports many of the weak cipher suites we found. Continuing this message of heightened security maintenance would greatly improve the resilience of network traffic to adversaries at NCSA if all machines were to adopt support for TLS v1.3. 

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remark
Remark remark:

This difficulty in adopting TLS v1.3, though, is still prevalent throughout the internet and not just here at NCSA, and it brings to light the unfortunate truth that if widely adopting the newest TLS has taken this many years, PQC adoption may show the same challenge.
























§ CURRENT PQC ADOPTION OF NETWORK PROTOCOLS AND THEIR APPLICATIONS AND LIBRARIES


This section describes the current adoption of critical network protocols regarding post-quantum cryptography (PQC). The problem of migrating applications to be quantum resistant is critical to ensure the security of key exchanges (KEX) and the integrity of digital signatures. As there is an increasing concern about future attacks made by quantum algorithms and computers, knowing the current adoption rate in critical protocols is crucial to ensure that these protocols are all secure against all these quantum computers. To the best of our knowledge, very few, if any, studies have done a systematic survey of PQC adoption across network protocol layers and the wide spectrum of workloads. 



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remark
Remark remark:

The key takeaway from our study is that only SSH and SSL have had PQC implementation and some real-world adoptions. The rest of the protocols have to roll their own crypto or encapsulate their data in TLS 1.3 (hybrid quantum key exchange). 



Table <ref> described the current PQC adoptions of some of the most important protocols in the world. Some protocols are one of many protocols lying in the applications layer of Internet Protocol Suite such as HTTP, DNS, DHCP, SSH, SSL, etc. In addition, there are widely used communications and authentication protocols such as Kerberos, Modbus, MySQL, SIP, and RDP. As described in Table <ref>, most critical protocols are not equipped with quantum-resistant algorithms, except for Secure Shell (SSH) and Secure Socket Layer (SSL). SSH protocol, OpenSSH library has implemented PQC for encryption in sntrup761x25519-sha512@openssh.com key exchange method after the release of its 9.0 version. SSL protocol, OpenSSL has another library called oqsprovider <cit.>, which is a provider to standard OpenSSL to implement quantum-safe cryptography for KEM key establishment in TLS1.3. Some quantum-safe algorithms that oqsprovider has implemented include KEM algorithms (BIKE, CRYSTALS-Kyber) and signature algorithms (Falcon, CRYSTALS-Dilithium).   




§ CASE STUDY: CHALLENGES OF MIGRATING 
 SCITOKENS TO PQC


SciTokens enables a federated ecosystem for authorization on distributed scientific computing infrastructures, enabling researchers to authenticate themselves as valid users to access scoped scientific computing resources<cit.>. Major supercomputing and scientific instruments (e.g., Laser Interferometer Gravitational-Wave Observatory (LIGO), Open Science Grid, Extreme Science and Engineering Discovery Environment (XSEDE)) rely on SciTokens. It is critical to safeguard SciTokens with quantum-resistant cryptography to avoid the forgery of digital signatures used in tokens (Figure <ref>).

Background.
The SciTokens protocol uses JSON web tokens (JWTs) between various parties, which will validate any access to resources. The process starts off by including information within the JWT. The information included most commonly consists of who issued the token, who the token is assigned to, when the token will expire, and what permissions and access the end user of such token will have. The tokens are then signed cryptographically to be authentic and certified when called upon to use. The benefit of this approach is that each organization or institution can issue its own tokens, allowing for easier and quicker access to resources without relying on one central authority to control the distribution of such tokens. 

Making SciTokens Quantum Resistant.
To be quantum-resistant, existing software must migrate encrypting in three main steps: signing, key exchange, and encryption. Key exchange and encryption are currently effective, but the looming threat of quantum computing could compromise these safeguards, potentially exposing sensitive data. For SciTokens, the main critical component that must be quantum secure is the signing step. When a digital signature is forged, the root of trust about the integrity of the signed token is broken. This undermines trust in historical transaction records, as the digital signatures securing them could be compromised by a quantum computer.




Existing work.
The NIST (National Institute of Standards and Technology) has determined four post-quantum resistant cryptographic algorithms. While there are four determined, one of the algorithms applies to just general encryption, while the other three refer to the digital signing aspect of quantum-resistant cryptography. The general cryptography algorithm is CRYSTALS-Kyber, which is beneficial due to its small encryption keys <cit.>. The three digital signing algorithms are CRYSTALS-Dilithium, FALCON, and SPHINCS+, which are distinct from each other due to the high efficiency that CRYSTALS-Dilithium and FALCON provide but also due to the variance in terms of mathematical approach SPHINCS+ has <cit.>.

Challenges of SciTokens migration to PQC.

1. As these four algorithms that NIST recommended are relatively new, the mitigation documents are scarce. There are protocol descriptions such as CRYSTALS-Dilithium and Falcon algorithms along with SPHINCS+ <cit.> <cit.> <cit.>, but implementation guidelines are very few if there are any.

2. Simply adapting SciTokens using existing cipher suites such as RSA-512 will create a new integration problem. For example, extending the length of the RSA digital signature from 256 bits to 512 bits will exceed the length of the HTTP header as defined in RFC 2616, breaking JWT when transmitted through intermediate proxies and interpreted by standard HTTP servers.

3. Emerging side-channel attacks targeting PQC implementations, such as Kyber and Dilithium, show that it is challenging to have a correct implementation, despite that the PQC protocols are correct <cit.>.


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remark
Remark remark:
To the developer community, it is unclear what is the process of determining which post-quantum secure algorithm to use and how to implement those correctly. In the future, we will work with the SciTokens community to correctly implement its quantum-resistant digital signature algorithm.






§ LESSONS LEARNED & 
 ACTION ITEMS TO PREEMPT NOVEL PQC ATTACKS

End-to-end PQC requires both client-server support.
To completely support PQC, we have found that both client and server have to support the same cipher suite, and widespread adoption hinges upon effective protocol negotiation when establishing the initial handshake, followed by key exchanges and digital signatures. For effective post-quantum security to be possible, it is thus necessary that as many machines as possible adopt post-quantum versions of network protocols.


Internet-wide scan for PQC implementations.
Another direction is to create a web spider that measures Post-Quantum Cryptography at NCSA and other organizations, constituting a "Network of PQC telescopes" that actively scans IPv4 and IPv6 space for PQC adoption across web services.
In addition, each organization can adopt its own internal PQC compliance checking tool as an HTTPS service so users can visit and perform self-assessments.

TLS v2.0 supporting PQC by default.
A more ambitious, yet significant direction for PQC adoption would be the introduction of PQC directly into the TLS standard as TLS v2.0. While some organizations are already working on integrating PQC as additions onto their TLS versions 1.2 and 1.3 <cit.>, distinguishing a fully post-quantum implementation with a new version number could prove to be essential in marketing the dire urgency in TLS adoption of PQC internet-wide.

This urgency is highlighted by the incredible potential of securing most network traffic, against both quantum and classical adversaries under TLS. We have shown in earlier sections that there are various methods to secure protocols such as DNS and RDP using just TLS as a wrapper. In addition to those methods, TLS Termination Proxies can also be used as wrappers around current legacy infrastructures to secure traffic seamlessly <cit.>. With many TLS-based solutions for cryptographically lacking protocols already out there, it should not be unrealistic to extend this security to the post-quantum world under a new version number, implementation, and standard.

Preempting PQC attacks.
Despite numerous attack attempts  <cit.> and analyses against current PQC drafts, none of the attacks have been publicly confirmed. We assert that if there were any successful attacks, they would highly likely leave traces in the network metadata, which would be measured by our network instrument or a honeypot <cit.>, given that we can achieve a widespread deployment on the Internet scale. Therefore, it is important to continue this line of network instruments for PQC measurement to preempt attacks <cit.> and failures <cit.>, particularly in HPC environments.

PQC downgrade attack. The security of a PQC scheme depends on both the protocol itself, the implementation, and the effective negotiation of cipher suites (e.g., hybrid) during the migration period. We hypothesize that future attacks may not attempt to exploit the PQC scheme directly but rather downgrade a PQC protocol to a classical protocol, which is less secure, similar to the SSL downgrade attack shown in  <cit.>.



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remark
Remark remark:
In the future, we will release an open dataset containing metadata of cryptographic suites across major network protocols (SSH, TLS, RDP, etc.) upon the paper's publication. With our network instrument in place, we will track the adoption of PQC algorithms, give feedback to NIST, expect to discover potential novel PQC attacks in the wild and help migrate HPC applications to be quantum-safe. Finally, we will expand our network instrument to a set of networked vantage points (PQC telescopes) to continuously measure PQC adoption at the Internet scale.








§ CONCLUSION AND FUTURE WORK

We have successfully implemented and deployed a novel network instrument to measure PQC adoption in real-time and provide historical trends of the adoption rates across seven layers of computer network protocols at the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign. The instrument is placed at the network border router of nation-scale high-performance supercomputing centers and is a testbed for measurements. Using only metadata of network connections, without compromising user's privacy, we present the first statistical study of PQC's readiness across seven layers of computer network and HPC applications with a unique vantage point: a petascale supercomputing center with nation-scale visibility through the FABRIC TeraCore network.




§ ACKNOWLEDGEMENTS

This work was partly supported by the National Science Foundation under contract CCF #2319190. We acknowledge Dr. Jim Basney for the useful discussions about SciTokens, Dr. Santiago Nunez-Corrles, Dr. Edoardo Giusto, members of NCSA Quantum Task Force, Dependable Classical-Quantum Computing Systems Engineering Working Group, and the Illinois Quantum Information Science and Technology Center (IQUIST) faculty members for insightful feedback. We also want to recognize the following organizations/programs: NCSA Student Pushing Innovation Program, IBM-Illinois Discovery Accelerator Institute, FABRIC, PPoSS, and the National Science Foundation's CyberCorps Scholarship for Service (SFS) Program, NCSA Integrated Cyberinfrastructure/IRST team, and NIST HPC Security Working Group. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of their employers or the
sponsors.






















IEEEtran





 Paper Title: Persistent Topological Negativity in a High-Temperature Mixed-State.






Department of Physics, University of California, Santa Barbara, CA 93106, USA	


Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA, 93106


Department of Physics, University of California, Santa Barbara, CA 93106, USA	

§ ABSTRACT 
We study the entanglement structure of the Greenberger–Horne–Zeilinger (GHZ) state as it thermalizes under a strongly-symmetric quantum channel describing the Metropolis-Hastings dynamics for the d-dimensional classical Ising model at inverse temperature β. This channel outputs the classical Gibbs state when acting on a product state in the computational basis.  When applying this channel to a GHZ state in spatial dimension d>1, the resulting mixed state changes character at the Ising phase transition temperature from being long-range entangled to short-range-entangled as temperature increases. Nevertheless, we show that the topological entanglement negativity of a large region is insensitive to this transition and takes the same value as that of the pure GHZ state at any finite temperature β>0. We establish this result by devising a local operations and classical communication (LOCC) “decoder" that provides matching lower and upper bounds on the negativity in the thermodynamic limit which may be of independent interest. This perspective connects the negativity to an error-correction problem on the (d-1)-dimensional bipartitioning surface and explains the persistent negativity in certain correlated noise models found in previous studies. Numerical results confirm our analysis. 


 Persistent Topological Negativity in a High-Temperature Mixed-State
    Sagar Vijay
    February 2024
===================================================================



Introduction: The nature and fate of quantum entanglement in mixed-states of quantum many-body systems – arising in a description of thermal quantum matter  or after a pure state experiences local decoherence – have been recently investigated in order to understand the universal quantum correlations that can persist in open quantum systems <cit.>. Patterns of long-range-entanglement (LRE) in mixed quantum matter are intricately related to quantum many-body phases which can act as quantum error-correcting codes <cit.>.  Thermal states of quantum many-body systems are also now known to be efficiently classically-preparable at sufficiently high temperatures <cit.>.

The entanglement negativity provides a well-studied diagnostic of bipartite entanglement in a mixed-state, which vanishes if the state can be prepared by from a product state by local operations and classical communication (LOCC) across the entanglement bipartition <cit.>.  While the negativity of a region A grows with the area of its boundary in a generic Gibbs state <cit.>, contributions to the negativity which are independent of coarse-grained details of the bipartitioning surface signal the presence of LRE in a mixed-state and can probe the presence of mixed-state topological quantum order <cit.>.  Diagnostics of non-local mixed-state entanglement, such as the topological entanglement negativity <cit.>, are challenging to study analytically, and the subtle behavior of mixed-state LRE under a finite-depth local quantum channel (FDLC) remains to be fully understood <cit.>.  

In this work, we investigate the fate of mixed-state quantum entanglement under a Metropolis-Hastings <cit.> quantum channel describing the equilibration of a thermal classical system, specifically that of a d-dimensional Ising model at inverse temperature β. When acting on a pure state in the computational basis, this channel simply prepares a classical thermal state of the d-dimensional Ising model. 

The nature of the mixed quantum many-body state produced by the action of this channel on a generic input state is more striking. When the initial state is chosen to be a Greenberger–Horne–Zeilinger (GHZ) state, a symmetric, long-range-entangled ground-state of the ferromagnetic  phase, we show that the resulting quantum many-body mixed state at any finite temperature β > 0 has a negativity which is asymptotically exactly that of the pure GHZ state in the limit that the boundary of the sub-region becomes large. In spatial dimensions d>1, the negativity of a large, contiguous region in this mixed state ρ_β is thus a constant, independent of geometric details of the bipartitioning surface, and insensitive to the thermal phase transition, as summarized in Eq. (<ref>). Nevertheless, the entanglement properties of the mixed-state do change across this transition; we show that ρ_β cannot be expressed as a convex sum of short-range-entangled (SRE) pure states in the ordered phase but is preparable using an FDLC acting on a product initial state in the high-temperature, disordered phase.  This observed behavior is in stark contrast to the negativity of the Gibbs state of a local, quantum many-body Hamiltonian which necessarily vanishes above a finite temperature <cit.>.  

This result is established by using the fact that the entanglement negativity is an entanglement monotone <cit.>, so that the negativity of a subregion A of any state ρ, denoted (ρ), cannot increase under an LOCC operation  across the bipartitioning surface

    ([ρ])≤(ρ).

This property lends a new perspective on mixed-state negativity, which is of possibly broader interest outside of the specific mixed-states that we study. We explicitly construct an LOCC operation that recovers the GHZ state at any finite temperature and for any bipartition in the asymptotic limit that the boundary A becomes large.  We show that for a contiguous bipartition, this channel is the decoder for a (d-1)-dimensional repetition code, which the two parties in the LOCC protocol must use to determine the precise unitary circuit that must be applied to recover the GHZ state.  The well-known success of the (d-1)-dimensional repetition code up to a maximal error strength in dimensions d > 1 is intimately related to the recovery of the GHZ state, and thus the constant entanglement negativity, at any finite temperature. 

We comment on broader utility of this perspective for understanding mixed-state negativity, by demonstrating that this can provide a simple understanding of why the topological entanglement negativity can remain a constant in correlated noise models (e.g. in <cit.>).


Setup: Consider N qubits on a d-dimensional hypercubic lattice, in a state described by the density matrix

    ρ_β≡1/_β∑_σe^-β H|ψ_σ⟩⟨ψ_σ|.

Here, 

    H ≡ -J∑_⟨ i,j⟩Z_iZ_j

is the energy of a classical Ising model on the d-dimensional hypercubic lattice with ferromagnetic (J>0) nearest-neighbor interactions, while _β≡(e^-β H) is the corresponding partition function at inverse temperature β.  Furthermore, the state
|ψ_σ⟩≡ (|σ⟩ + U|σ⟩)/√(2), 
where |σ⟩≡|σ_1,…,σ_N⟩ is a product state in the Pauli Z basis, and U≡∏_jX_j is a unitary transformation which generates the Ising symmetry of Eq. (<ref>).

The mixed-state ρ_β is naturally obtained by starting with a GHZ state |ψ_+⟩ = (|↑⋯⟩ + |↓⋯⟩)/√(2), which is an equal-amplitude superposition of the macroscopically-distinct all-up and all-down qubit configurations, and then applying a channel Φ_β which describes a Metropolis-Hastings algorithm <cit.> in which the energy H is measured, and then a unitary operation X_j is applied on a random qubit with probability min{1, exp(-β Δ E_j)}, where Δ E_j≡ E_new - E_old is the energy difference between the new and old qubit configurations. 

Repeated measurements and feedback, starting from any product state in the Pauli Z basis, will eventually produce a Gibbs ensemble of the d-dimensional classical Ising model at inverse temperature β <cit.>.   We specifically define Φ_β to be  repeated application of the Metropolis-Hastings algorithm so as to produce a thermal steady-state when acting on the all-up state ρ_0≡|↑⋯⟩⟨↑⋯|, i.e.

    Φ_β[ρ_0] = 1/_β∑_σe^-β H|σ⟩⟨σ|.

Each operation that comprises the channel Φ_β manifestly commutes with the Ising symmetry transformation U.  Therefore, U is a strong symmetry of this channel <cit.>, i.e.    UΦ_β[ρ]=Φ_β[Uρ] for any state ρ, so that

    Φ_β[|ψ_+⟩⟨ψ_+|] = 1+U/√(2)Φ_β[ρ_0]1+U/√(2) = ρ_β.


Mixed-State Entanglement:
The state ρ_β witnesses a thermal phase transition with increasing temperature in spatial dimensions d>1 between a ferromagnetically-ordered and a disordered state. At any temperature, the state ρ_β has a strong Ising symmetry Uρ_β = ρ_β.  However, the ferromagnetically-ordered phase is also characterized by long-range-order in local operators which are charged under this symmetry (ρ_βZ_iZ_j) |i-j|→∞0.  These conditions are sufficient to establish  <cit.> that this mixed-state cannot be described as an ensemble of SRE pure-states.  

We may also show that in the disordered phase, ρ_β can be prepared by an FDLC, and is thus an SRE mixed-state [The ability to prepare ρ_β by an FDLC implies that this mixed-state describes an ensemble of SRE pure state.].  In the Supplemental Material <cit.>, we formally show that ρ_β = ℰ_ZZ[|ϕ(β)⟩⟨ϕ(β)|], 
where ℰ_ZZ is a quantum channel that describes maximal dephasing of Z_iZ_j along all nearest-neighbor bonds.  Furthermore, we show <cit.> that the wavefunction |ϕ(β)⟩≡_β^-1/2∑_μe^-β H/2|μ⟩, 
where |μ⟩ denotes a product state in the Pauli X basis, is the exact ground-state of the  Hamiltonian

    H_0(β) = ∑_j[∏_i∈NN(j)e^-β J Z_iZ_j -X_j],

which exhibits a phase transition at a critical value of β=β_c(d) which is in the d-dimensional Ising universality class.  When β<β_c(d),  |ϕ(β)⟩ is the Ising-symmetric ground-state in the disordered phase and can thus be prepared from a product state using a finite-depth local unitary circuit.  Thus in this range of temperatures, ρ_β can be prepared by an FDLC by composing this circuit with the quantum channel ℰ_ZZ.


Conditional Mutual Information:
We now investigate entanglement properties of ρ_β and their relation to the Ising phase transition. The conditional mutual information (CMI) is a measure of tripartite correlations in a state and has been used to study various aspects of entanglement in quantum systems<cit.>, including probing mixed state quantum phases of matter<cit.>. 
Given subsystems A, B, and C, the CMI is defined as

    I(A:C|B)
           =S(AB)+S(BC)-S(B)-S(ABC).

Consider the partitioning shown in Fig. <ref> with r>0, where C is taken to be the complement of AB. Note that for any proper subset A of qubits, ρ_β,A=_A̅(ρ_β)=ρ_β,A^Cl, where ρ_β^Cl=e^-β H/_β is the classical Gibbs state [This in particular shows that for mutual information we have I(A:C)=I^Cl(A:C) provided that r>0, where I^Cl is computed with respect to ρ_β^Cl.]. This shows that the first three terms in Eq. (<ref>) take the same value when computed for ρ_β as for ρ_β^Cl. Moreover, it is straightforward to use  Eq.(<ref>) to show S(ρ_β)=S(ρ_β^Cl)-log(2). Therefore I(A:B|C)=log(2)+I^Cl(A:C|B),
where I^Cl(A:C|B) is the CMI computed for ρ_β^Cl. On the other hand, since B is assumed to be non-empty, there is no interaction term in H with support on both A and C and hence I^Cl(A:C|B)=0 <cit.>, and

    I(A:B|C)=log(2),

independent of r, temperature β and the system size. It is worh mentioning that in contrast, for thermal Gibbs state of local quantum Hamiltonians, CMI decays exponentially with r <cit.>. 


Entanglement Negativity:
We now investigate the entanglement negativity of a subregion A of the mixed-state (<ref>), (ρ_β) ≡ (||ρ_β^T_A||_1 - 1)/2.  We will show that for a region A, 

    (ρ_β) = {[ 1/2 - O(exp(-|∂ A|))                  β>0;                     ;                    0                  β=0 ].

where |∂ A| is the number of qubits on the boundary of A. The second result is easily established. At infinite temperature, ρ_β = 0 = (1 + U)/2^N, so that ρ_β^T_A = ρ_β and (ρ_β=0)=0. 

The first result is determined as follows. The negativity is a convex function <cit.> so that 

    (ρ_β) ≤1/_β∑_σe^β J∑_⟨ i,j⟩σ_iσ_j(|ψ_σ⟩⟨ψ_σ|) = 1/2

for any region A, since |ψ_σ⟩ is related to the GHZ state |ψ_+⟩ by single qubit gates, so that (|ψ_σ⟩⟨ψ_σ|) = (|ψ_+⟩⟨ψ_+|).  The latter is exactly 1/2 for any region A. The negativity is also an entanglement monotone <cit.>, see Eq. (<ref>).  Here, we show that given the state ρ_β for any β > 0 and any region A, there is an LOCC operation  with respect to A and its complement A̅ that recovers the GHZ state with a fidelity that approaches unity exponentially quickly in the size of the boundary of A.  Combining this with (<ref>), allows us to establish Eq. (<ref>).  



We first construct the LOCC channel  for a contiguous bipartition, before generalizing this construction for a non-contiguous subregion A.  The state ρ_β describes the ensemble of pure-states |ψ_σ⟩ drawn with probability e^β J∑_⟨ i,j⟩σ_iσ_j/_β. Given a state from this ensemble, we measure the operators Z_iZ_j along all nearest-neighbor bonds lying exclusively within A or .  The resulting pattern of domain walls – an example of which is shown in the left panel of  Fig. <ref> in two spatial dimensions – is consistent with having drawn the state  |ψ_σ⟩ or the state  ∏_j∈ AX_j|ψ_σ⟩ and thus leaves one classical bit's worth of ambiguity in determining the precise state.  Equivalently, this ambiguity corresponds to the two possible ways that the observed domain walls can connect across the bipartition, as shown in Fig. <ref>.   

The preparation of the GHZ state through LOCC thus maps onto a decoding problem in a classical (d-1)-dimensional repetition code living on the interface between A and A.  The classical bit of information is the relative alignment of any two qubits across the bipartition in the Pauli Z basis (e.g. in Fig. <ref>, the classical bit can be taken to be τ_1≡σ_1σ_1).  The measured “syndromes" of the repetition code (e.g. τ_1τ_2 = (σ_1σ_2)(σ_1σ_2)) and the Boltzmann weights for the d-dimensional Ising model are used to perform maximum-likelihood decoding to guess the correct state.  Specifically, let w_1 and w_2 denote the Boltzmann weights corresponding to states (i) |ψ_σ⟩ and (ii) ∏_j∈ AX_j|ψ_σ⟩, respectively. We will guess that the state we have is (i) with probability p ≡ w_1/(w_1+w_2) and (ii) with probability 1-p.  We will subsequently apply the appropriate single-site unitary gates to turn this into the GHZ state.
The LOCC channel is thus 


  * Measure Z_iZ_j along all nearest-neighbor bonds lying exclusively within A or .

  * Let |σ⟩ be a product state in the Pauli Z basis which is consistent with the measurement outcomes. Apply the unitary U_1 = ∏_jX_j^(1-σ_j)/2 with probability p, or the unitary U_2 = ∏_j∈ AX_j∏_jX_j^(1-σ_j)/2 with probability 1-p. 


At any finite temperature, the relative probability p/(1-p) = w_1/w_2 scales exponentially in the size of the boundary of A, due to the fact that the energy of an Ising domain wall is extensive in its length.  Thus, the decoder for the repetition code should be successful at any finite temperature, and fail exactly at infinite temperature when w_1 = w_2 = 1/2.  This is consistent with the known behavior of the repetition code with single-qubit bit-flip errors with probability q, for which perfect decoding is possible for any q≤ 1/2. 

We may generalize this channel for the case where A is a non-contiguous region.  In this case, step 1 (above) is replaced by measurements of Z_iZ_j across all pairs of sites entirely within A and entirely within A̅.  As before, only one classical bit's worth of ambiguity thus remains after these measurements, in determining the precise state.  The quantum circuit that takes this state onto the GHZ state is then determined by a decoder for a repetition code with |∂ A| sites.  Thus, decoding the correct quantum circuit should again be successful at any β > 0, as |∂ A| becomes large, regardless of the geometry of the bipartition.  

Consistent with these expectations, we show <cit.> that for any bipartition, the fidelity of the recovered state R(ρ_β) with the GHZ state is exactly

    ℱ((ρ_β),|ψ_+⟩⟨ψ_+|) = 1/2[1 - ⟨tanh(β H_∂)⟩]

where H_∂ denotes terms in the Ising Hamiltonian acting exclusively along bonds that cross the bipartition between A and A and the expectation value is taken in the thermal state for the classical Ising model.  At any finite temperature, the energy H_∂ is negative, and extensively large in the boundary of A, so that ⟨tanh(β H_∂)⟩→ 1 as A becomes large.

Numerical Results: 
To numerically confirm this result, we obtain an alternate expression for the entanglement negativity of any bi-partition <cit.> 

    𝒩_A[ρ_β]=1/4|1-e^2β H_∂[σ]|.

One may now use classical Monte Carlo simulations to compute the thermal average in Eq.(<ref>).
However, care must be taken with sampling the spin configurations, since the expression inside the absolute value could be exponentially large for configurations with an exponentially small Boltzmann weight, in such a way that the overall contribution to the expectation value is O(1). 




The aforementioned issue can be avoided by using a technique known as importance sampling. Let q[σ] denote the probability distribution defined as q[σ_A,σ_]=1/2p[σ_A,σ_]+1/2p[σ_A,_], where p[σ]=e^β J∑_⟨ i,j⟩σ_iσ_j/𝒵_β. To sample from q[σ], one can simply sample from p[σ] and then with 1/2 probability, flip the spins in . We may thus write Eq.(<ref>) as,

    𝒩_A[ρ_β]
           =1/4∑_σ q[σ]p[σ]/q[σ]|1-e^2β H_∂[σ]|
       =1/2|tanh(β H_∂[σ])|_q.

where the final expectation value is taken with respect to the distribution q. The advantage of Eq.(<ref>) compared to Eq.(<ref>) is that the expression inside the expectation value is now bounded by unity everywhere, so that contributions from rare configurations are no longer a concern. 

In the following we present the numerical results for a 2D system of qubits placed on a L× L square lattice with periodic boundary conditions. 
Fig. <ref> shows the negativity 𝒩_A(ρ_β) when A is taken to be the L×L/2 cylinder. As is clear from Fig. <ref>, in the thermodynamic limit, the negativity is equal to 1/2 even above the Ising critical temperature (marked by the dashed line in Fig.<ref>), as expected from the expression in (<ref>). The fact that the negativity of subregions with extensive boundary is 1/2 irrespective of temperature, means that if one defines the topological negativity as the constant contribution to the negativity as in Ref. <cit.>, one would find that the resulting mixed state has 𝒩_topo=-1/2 even above the critical temperature, despite the fact that for T>T_c – as shown above – ρ_β is an SRE mixed-state. <cit.> (cf. Ref. <cit.>). 

Although the thermal phase transition at T_c has no effect in the negativity of subregions with extensive boundaries, the negativity of finite subregions (while continuous) becomes singular at T=T_c. Fig. <ref> shows the derivative of the negativity of a single site with respect to temperature which shows a singularity at T=T_c, which is consistent with the singularity of the heat capacity at the d=2 classical Ising phase transition, as to be expected from Eq. (<ref>). 




Discussion and Outlook: Using LOCC to recover states with a known pattern of LRE may provide another perspective that can be used to investigate mixed-state LRE under decoherence. For correlated decoherence, this is apparent in specific examples.  The two-dimensional toric code dephased by local operators Z_j+δX_j at each site j, with the vector δ given in Fig. <ref>, was studied in Ref. <cit.>.  Quantum state trajectories under this channel only contain patterns of ℤ_2 charge and flux which are separated by δ.  Therefore, for any dephasing strength, it is possible to use LOCC across the bipartition in Fig. <ref> to deterministically recover the toric code ground-state. The ℤ_2 charge and flux lying on the bipartitioning surface, which cannot be directly measured in the LOCC protocol, can be perfectly inferred from stabilizer measurements in A and A due to the noise correlations.  The negativity for this bipartition and for any dephasing strength <cit.> is thus exactly that of the toric-code ground-state <cit.>. Another example for which the LOCC perspective presented here could be insightful is the study of entanglement structure in the toric code with single-site Y decoherence <cit.>. In this case, while the noise model itself is not correlated, the LOCC protocol can benefit from the correlations that are present in the syndromes due to the extensively many strong symmetries of this channel <cit.>. The detailed study of this model and other correlated noise models is left to a future work. 

We identify other directions implied by our results.  First, it is interesting to consider the fate of mixed-state entanglement under thermalization by genuinely quantum channels which are strongly-symmetric.
More concretely, we may consider thermalizing the GHZ state via a strongly symmetric implementation of the quantum Metropolis-Hastings dynamics<cit.> for the d-dimensional transverse field quantum Ising model. The resulting mixed state would be the projection of the quantum Gibbs state onto the symmetric sector. A similar LOCC protocol to recover the GHZ state can be used to relate the entanglement negativity to the performance of a (d-1)-dimensional repetition code under a coherent correlated noise model. Energetic considerations can be used to argue about the success of the LOCC protocol in turning the resulting mixed state into the GHZ state. This lower bounds the negativity of the resulting mixed-state and may suggest that it 
would have a non-trivial negativity at high temperatures, though this remains to be confirmed. 

Second, while the persistence of topological negativity at high temperatures in our work can be understood as a consequence of the repetition code possessing maximum error threshold, it is interesting to search for settings where a similar line of argument would lead to a different code with a finite threshold. Such a system, would exhibit a finite temperature phase transition in negativity, which may not necessarily coincide with the standard order-disorder thermal phase transition. Furthermore, while we focused on the Metropolis-Hastings channel for the classical Ising model, a possibly fruitful future direction to pursue is to consider other classical systems with different symmetries and study the fate of entanglement in an initially symmetric state (e.g. continues symmetries, higher-form symmetries or non-Abelian symmetries <cit.>). 


Lastly, given that the topological negativity is unable to detect the phase transition in the complexity of the mixed-state ρ_β (from LRE to SRE), it is interesting to look for other quantities that would be able to capture the complexity of the quantum phase transition in our system as well as in more general settings. 

Acknowledgments:
We thank Yimu Bao, Matthew Fisher, Tarun Grover, Tibor Rakovszky, Subhayan Sahu, Shengqi Sang and Yaodong Li for useful discussions. This research was supported in part by grant NSF PHY-2309135 to the Kavli Institute for Theoretical Physics (KITP). Use was made of computational facilities purchased with funds from the National Science Foundation (CNS-1725797) and administered by the Center for Scientific Computing (CSC). The CSC is supported by the California NanoSystems Institute and the Materials Research Science and Engineering Center (MRSEC; NSF DMR 2308708) at UC Santa Barbara. SV is partly supported as an Alfred P. Sloan Research Fellow.








    Supplemental Material





§ ENTANGLEMENT OF Ρ_Β

Here, we show that the state (<ref>) is short-range-entangled (SRE) throughout the disordered phase at sufficiently high temperatures, and in any spatial dimension.  First, observe that this state may be written as

    ρ_β = ∑_μp_μ|ϕ_μ⟩⟨ϕ_μ|

where |ϕ_μ⟩≡ρ_β^1/2|μ⟩/ √(p_μ) and p_μ≡⟨μ|ρ_β|μ⟩, while |μ⟩ is a product state in the Pauli X basis.  Because of the strong symmetry of this state, (ρ_βU) = 1, where U ≡∏_jX_j.  As a result, each pure state appearing in (<ref>) must satisfy ∏_jX_j|ϕ_μ⟩ = |ϕ_μ⟩.  

Furthermore, observe that

    ρ_β^1/2 = 1/√(_β)∑_σe^-β H/2|ψ_σ⟩⟨ψ_σ|

It is evident from this expression that [Z_iZ_j,ρ_β^1/2]=0 for any i, j.  As a result, any state |ϕ_μ⟩ appearing in (<ref>) – for which U|ϕ_μ⟩ = |ϕ_μ⟩ – can be obtained by applying single-site unitary gates on the reference state |ϕ_+⟩ = ρ_β^1/2|+⋯ +⟩/√(p_+), where |+⋯ +⟩ is a product state in which X_j=+1 for each qubit:

    |ϕ_μ⟩ = ∏_jZ_j^1-μ_j/2|ϕ_+⟩

Finally, we note that p_μ = ⟨μ|ρ_β|μ⟩ = ⟨+⋯ +|∏_jZ_j^1-μ_j/2ρ_β∏_jZ_j^1-μ_j/2|+⋯ +⟩ = ⟨+⋯ +|ρ_β|+⋯ +⟩ = p_+.  As a result, p_μ = 1/2^N-1 where N is the total number of qubits.  To summarize, we may write (<ref>) as

    ρ_β   = 1/2^N-1∑_μ  s.t.  ∏_jμ_j= +1∏_jZ_j^1-μ_j/2|ϕ_+⟩⟨ϕ_+|∏_jZ_j^1-μ_j/2

We now re-cast the summation by introducing Ising degrees of freedom σ_ij along bonds of the hypercubic lattice, and letting μ_j≡ A_j[σ] ≡∏_i∈ NN(j)σ_ij, where the product is taken over sites that are nearest-neighbor to j.  With periodic boundary conditions, ∏_jA_j[σ] = +1, and as a result, the constraint on the summation ∏_jμ_j = +1 is naturally satisfied by this replacement.  With this replacement, we may write

    ρ_β   = 1/2^N_b∑_σ∏_jZ_j^1-A_j[σ]/2|ϕ_+⟩⟨ϕ_+|∏_jZ_j^1-A_j[σ]/2

Here, N_b = d· N is the total number of bonds on the d-dimensional hypercubic lattice.  The pre-factor ensures the normalization of the density matrix[Another way to derive this pre-factor is to observe that for a given configuration of μ_j satisfying the constraint ∏_jμ_j = 1, there are 2^N_b/2^N-1 different configurations of σ_ij such that μ_j = A_j[σ]. ]. Finally, using the fact that Z_j^1-A_j[σ]/2 = ∏_i∈NN(j)Z_j^1-σ_ij/2, we may write ∏_jZ_j^1-A_j[σ]/2 = ∏_⟨ i,j⟩(Z_iZ_j)^1-σ_ij/2, where the product in the second expression is taken over distinct, nearest-neighbor bonds.  As a result,

    ρ_β   = 1/2^N_b∑_σ∏_⟨ i,j⟩(Z_iZ_j)^1-σ_ij/2|ϕ_+⟩⟨ϕ_+|∏_⟨ i,j⟩(Z_iZ_j)^1-σ_ij/2

We conclude that ρ_β may be obtained by dephasing Z_iZ_j on each bond of the lattice, with maximal strength, a manifestly finite-depth, local quantum channel.

Finally, we may argue that the state |ϕ_+⟩ is SRE when the system is in a disordered phase.  We do this by first writing

    |ϕ_+⟩ = √(2^N-1)ρ_β^1/2|+⋯ +⟩ = √(2^N-1/_β)∑_σe^-β H/2|ψ_σ⟩⟨ψ_σ|+⋯ +⟩ = 1/√(_β)∑_σe^-β H/2|σ⟩

We now observe that (<ref>) is the ground-state of the Hamiltonian

    H = -∑_jQ_j

where

    Q_j≡ -X_j + ∏_i∈NN(j)e^-β J Z_iZ_j


 


§ LOCC OPERATIONS AND RECOVERING THE GHZ STATE


Here, we will show the existence of an LOCC protocol   with respect to bi-partition A and , which transforms state ρ_β back to the GHZ state in the thermodynamic limit. 
The LOCC protocol consists of two steps:


    
  * Measure Z_iZ_j on each link that is entirely in A or in  subregions, finding the outcome μ_ij=± 1. This does not include the links across the boundary, ∂ A.

    
  * For any given set of measurement outcomes μ={μ_i,j}, there exist two possible states which have domain wall configuration consistent with μ. 
     We pick one of the two states based on a coin toss weighted with their respective Boltzmann weight in ρ_β (Eq.(<ref>)) and
      apply a product of Pauli X_i operators to eliminate the domain walls in that assumed configuration.

In the following, we will derive Eq.(<ref>) for the fidelity between the GHZ state, |ψ_+⟩, and the output state (ρ_β).


 §.§ Fidelity


Let {M_μ} be the projective measurements that measure whether there is a domain wall or not on the links which are entirely
in region A or A̅,


    M_μ≡∏_<i,j>∈ A, A̅1+μ_i,jZ_iZ_j/2,

where μ_ij is the measurement outcome of measuring Z_iZ_j and the product only includes next nearest sites which are both in A or both in .
Note that,

    M_μ|ψ_σ⟩ = ∏_<i,j>∈ A, A̅δ_μ_i,j,σ_iσ_j|ψ_σ⟩.
 

A set of measurement outcomes μ specifies a certain domain wall configuration in A and , with some domain walls ending on the boundary ∂ A. Let 1∈ A and 1̅∈ be two adjacent sites across the boundary and define τ=± 1 to be a marker for whether or not there is a domain wall across the edge connecting 1 and 1̅, i.e. τ≡σ_1 σ_1̅ (see Fig. <ref>). Note that μ and τ together uniquely specify the global domain wall configuration, because there should be even number of domain walls across each plaquette of the square lattice. In particular if we label the sites along the boundary as shown in Fig.<ref>, we have,

    σ_i σ_ i̅=τ∏_j=1^i-1μ_j, j+1μ_j̅, j̅+1.

Let X_μ,τ be the appropriate product of single qubit Pauli-Xs that corresponds to the domain wall configuration specified by μ and τ. In particular, if M_μ|ψ_σ⟩=|ψ_σ⟩, we have

    X_μ,τ|ψ_σ⟩≡|ψ_+⟩   if σ_1σ_1̅=τ
    ∏_j∈ AX_j|ψ_+⟩   if σ_1σ_1̅≠τ.

Given the measurement outcomes μ, the decoder  applies X_μ,τ with τ=± 1 chosen randomly with probability P_μ,τ. The probability P_μ,τ is chosen to be proportional to the Boltzmann weight of the spin configuration specified by μ and τ and according to Eq.(<ref>) can be written as,

    P_μ,τ = e^β J τ∑_i≥ 1∏_j=1^i-1μ_j, j+1μ_j̅, j̅+1/∑_τ'=± 1 e^β J τ' ∑_i≥ 1∏_j=1^i-1μ_j, j+1μ_j̅, j̅+1.

Putting everything together, we can compute the average “decoding" fidelity,


    F = ψ_+(ρ_β)ψ_+= ∑_σ,μ,τe^-β H[σ]/𝒵_βP_μ,τ⟨ψ_+|X_μ,τM_μ|ψ_σ⟩^2.


Using Eqs.(<ref>) and (<ref>) this simplifies to


    F = ∑_σ,μ,τe^-β H[σ]/𝒵_β P_μ,τ∏_<i,j>∈ A, A̅δ_μ_i,j,σ_iσ_jδ_σ_1σ_ 1̅,τ.

Plugging the expression for P_μ,τ into Eq. (<ref>), we get


    F =   ∑_σ,τe^-β H[σ]/𝒵_βe^β Jτ∑_i≥1σ_1σ_iσ_ 1̅σ_ i̅/∑_τ'=±1 e^β Jτ'∑_i≥1σ_1σ_iσ_ 1̅σ_ i̅δ_σ_1σ_ 1̅,τ
    
         =   ∑_σe^-β H[σ]/𝒵_βe^β J∑_i≥1σ_iσ_ i̅/∑_τ'=±1 e^βτ'σ_1 σ_ 1̅ J∑_i≥1σ_i σ_ i̅
    
         =   1/2∑_σe^-β H[σ]/𝒵_β(1-tanh(β J∑_i≥1σ_i σ_ i̅))
    
         =   1/2[1-⟨tanh(β H_∂)⟩]

where H_∂ is the boundary Hamiltonian between A and A̅. 



 §.§ Mapping to the d-1 repetition code


The problem of choosing which domain wall configuration to assume can be understood as a decoding problem for a (d-1)-dimensional classical repetition code. To see this, note that the decoding task in a classical error correcting code is basically figuring out what error has happened based on a given set of syndromes. In particular the decoding task in the classical repetition code on N bits can be described as follows. Consider an error vector e∈{1,-1}^N such that e_i=-1 if a bit flip has happened on the i'th bit and e_i=1 otherwise. 
The goal of the decoder is to figure out e, knowing only the syndrome values, i.e. e_i e_j for adjacent i and j on the appropriate graph, e.g. the d-dimensional square lattice in the standard d-dimensional repetition code. Moreover, the decoder assumes an underlying noise model, i.e. a random distribution from which the vector e is drawn. For example, in the independent and identically distributed (iid) noise model, it assumes each component e_i is -1 with probability p and 1 with probability 1-p, independent of other components. 

In our setting finding the domain wall configuration is clearly the same task, with e_i defined to be e_i≡σ_i σ_ i̅. Note that we can not know the value of e_i, but we know the value of e_i e_j=μ_i,j μ_i̅,j̅ for adjacent i and j. At low temperatures, the domain walls are rare so most e_is would be 1, and as one increases the temperature more e_i would be -1. The underlying noise model is given by P_μ,τ as described in the previous section. This noise model is not iid, but rather a correlated noise model. However, as we argue in the next section, the repetition code is able to guess e correctly with probability 1 in the thermodynamic limit unless β=0, at which point each e_i is ± 1 compeletly at random. 



§ THRESHOLD FOR THE REPETITION CODE

Here we argue that the repetition code decoding under the noise distribution P_μ,τ is successful with probability 1 for any β>0 in the thermodynamic limit. First we review the argument for maximum error correction threshold in the presence of the iid noise and then we argue that the same should remain true for more generic noise models. We use quantum notation for convenience although the problem we are considering is inherently classical. 

Consider an initial state |Ψ⟩ undergoing some errors, leaving it in the mixed state

    ρ = ∑_e p_e e|Ψ⟩⟨Ψ|e

where e is a product of X bit flips and p_e is the probability of having that error. We rewrite the sum to distinguish between the errors that leave ρ with fewer than N/2 bit flips and errors with higher weights:

    ρ = ∑_e: |e|<N/2p_e e|Ψ⟩⟨Ψ|e + ∑_e:|e|≥ N/2p_e e|Ψ⟩⟨Ψ|e.

The error correction channel, , measures the syndromes, and between two possible error configuration consistent with a syndrome, applies the one with the smaller weight. Therefore it always successfully corrects the errors when there are less than N/2 bit flips, hence,

    (ρ)=   ∑_e: |e|<N/2 p_e (e|Ψ⟩⟨Ψ|e) + ∑_e:|e|≥ N/2p_e  (e|Ψ⟩⟨Ψ|e)
    
        =   ∑_e: |e|<N/2 p_e|Ψ⟩⟨Ψ| + ∑_e:|e|≥ N/2p_e (e|Ψ⟩⟨Ψ|e).


The fidelity of this post-error correction state with the original state is lower bounded as,


    F =   ⟨Ψ|(ρ)|Ψ⟩
    
        =   ∑_e: |e|<N/2 p_e +  ∑_e:|e|≥ N/2p_e ⟨Ψ|(e|Ψ⟩⟨Ψ|e)|Ψ⟩≥∑_e: |e|<N/2 p_e


where the lower bound is because the second term in the expression above is non-negative. Now we will show that this lower bound is equal to one in the thermodynamic limit. First, we rearrange the sum to be over |e|=k, the number of bit flips error, and because  p_k is a binomial distribution,
p_k = Nkp^k(1-p)^N-p.

In the limit of large N, we can approximate the binomial distribution as a Gaussian distribution with mean Np and standard deviation √(Np(1-p)) 
via the central limit theorem. Since N/2 is N/2-Np/√(Np(1-p))=O( √(N)) standard deviations away from the peak of the Gaussian, we see that 

lim_N→∞∑_k=0^N/2-1 p_k = 1

when p<1/2. Since fidelity is upper-bounded by one, the fidelity is indeed equal to one in the thermodynamic limit. Thus, the repetition code will work successfully when p<1/2. 

For the more general case where e is drawn based on the P_μ,τ distribution, the bound in Eq.(<ref>) is still valid. Therefore, to show perfect recovery we only need to show that with probability one, less than half of the edges contain a domain wall. Intuitively, this is true because for β>0, not having a domain wall on an edge is energetically favorable. To make it rigorous, one can define indicator variables I_i=(1-e_i)/2 in terms of which the number of domain walls can be written as k=∑_iI_i. When β >0 it is more probable for I_i to be 0 rather than 1 and hence the expectation value of k/|∂ A| is less than 1/2. Noting that lim_|i-j|→∞Cov(I_i,I_j)=0, the weak law of large numbers for weakly dependent variables<cit.> shows that k/|∂ A| concentrates around its mean, which is less than 1/2 when β>0 in the thermodynamic limit. 



§ NEGATIVITY SPECTRUM

Here we calculate the entanglement negativity between a sub-region A and its complement A̅, in the state ρ_β. 
Let  denote the spin configuration that is related to σ by flipping every spin. Accordingly, the state |ψ_σ⟩ can be written as |ψ_σ⟩=|σ⟩+|⟩/√(√(2)) and Eq. (<ref>) can be rewritten as,


    ρ_β = 1/2𝒵_β∑_σ e^-β H[σ](|σ⟩⟨σ| + |σ̅⟩⟨σ̅| +|σ⟩⟨σ̅| +|σ̅⟩⟨σ|),

where H[σ]=-J ∑_i,jσ_i σ_j is the classical Ising energy of the spin configuration σ. Taking the partial transpose of ρ_β with respect to sub-region A yields,

    ρ_β^T_A   = 1/2𝒵_β∑_σ[ e^-β H[σ](|σ⟩⟨σ| + |σ̅⟩⟨σ̅|) +e^-β H[σ_A,σ_A](|σ_A,σ_A⟩⟨σ_A, σ̅_A| +|σ_A, σ̅_A⟩⟨σ_A,σ_A|)]
       =1/2𝒵_β∑_σ[ e^-β H[σ](|σ⟩⟨σ| + |σ̅⟩⟨σ̅|) +e^-β H[σ_A,_A](|σ⟩⟨σ| +|σ⟩⟨σ|)]


where σ_A and σ_A̅ denote the spin configuration σ restericted to regions A and A̅ respectively. Thus, the eigenvalues and eigenstates of ρ_A^T_A are

    λ_σ;±=e^-β H[σ]± e^-β H[σ_A,σ̅_A]/_β, |ψ_σ^(±)⟩≡|σ⟩±|σ⟩/√(2).



We can now calculate the entanglement negativity  from the sum of all the negative eigenvalues. Firstly,
it is only the λ_σ;- eigenvalues which can be negative. 
We can also see that λ_σ;-=λ_σ̅;- from the ℤ_2 symmetry of the Hamiltonian and they correspond to the same eigenvector, up to some phase. 
Moreover, 
note that λ_σ;-=-λ_σ_A,σ̅_A̅;- so for a given spin configuration, either λ_σ;- or λ_σ_A,σ̅_A̅;- would be negative. Thus, we account for the two incidences of double counting with an additional 1/4 and get


    𝒩_A[ρ_β] = 1/4_β∑_σe^-β H[σ] - e^-β H[σ_A,σ_A].


Factoring out e^-β H[σ] we get


    𝒩_A[ρ_β]    = 1/4_β∑_σ e^-β H[σ]1 - e^-β( H[σ_A,σ_A]- H[σ])
       =1/4⟨1 - e^2β H_∂[σ])⟩.






 Paper Title: Relaxation Fluctuations of Correlation Functions: Spin and Random Matrix Models.






pathak.tanay@yukawa.kyoto-u.ac.jp
Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics,
 Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan


§ ABSTRACT 
Spectral statistics and correlations are the usual way to study the presence or absence of quantum chaos in quantum systems. We present our investigation on the study of the fluctuation average and variance of certain correlation functions as a diagnostic measure of quantum chaos and to possibly characterize quantum systems based on it. These quantities are related to eigenvector distribution and eigenvector correlation. Using the Random Matrix Theory certain analytical expressions of these quantities, for the Gaussian orthogonal ensemble case, were calculated before. So as a first step we study these quantities for the Gaussian unitary ensemble case numerically, and deduce certain analytical results for the same. We then carry out our investigations in physical system, such as the mixed-field Ising model. For this model, we find that although the eigenvalue statistics follow the behaviour of corresponding random matrices, the fluctuation average and variance of these correlation functions deviate from the expected random matrix theory behaviour. We then turn our focus on the Rosenzweig-Porter model of the Gaussian Orthogonal Ensemble and Gaussian Unitary Ensemble types. By using the fluctuation average and variance of these correlations, we identify the three distinct phases of these models: the ergodic, the fractal, and the localized phases. We provide an alternative way to study and distinguish the three phases and firmly establish the use of these correlation fluctuations as an alternative way to characterize quantum chaos.

Relaxation Fluctuations of Correlation Functions:

Spin and Random Matrix Models
    Tanay Pathak  https://orcid.org/0000-0003-0419-2583

    < g r a p h i c s >

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

            Report Number: YITP-24-96







§ INTRODUCTION

The dynamics of classical systems have been classified into a well-known hierarchy, the ergodic hierarchy <cit.> which ranges from integrable systems to Bernoulli systems. However, the situation is not quite clear in quantum systems <cit.> and there is no analogous neat categorization and it is not easy to extend the results of the classical phase space to Hilbert space with ease. The signature of quantum chaos is often attributed to the observation that there is a level repulsion amongst the eigenvalues in quantum Hamiltonians whose classical counterpart is chaotic which is called the Bohigas-Giannoni-Schmit (BGS) conjecture <cit.>. This behaviour of eigenvalues and their correlations are well described by the random matrices <cit.>. On the other hand, we have Berry-Tabor conjecture <cit.>, for the quantum system whose classical counterpart is integrable and the eigenvalues in quantum systems do not have any correlation, show level clustering and in the semi-classical limit behaves statistically similar to a random process. Thus, there is an intricate relationship between random matrices as described using Random Matrix Theory (RMT) and quantum chaos <cit.>. Over the years, these properties of quantum chaotic systems have been studied in great detail using various tools, not necessarily related to RMT as well. Some of these include (a non-exhaustive list) the eigenstate thermalization hypothesis (ETH) <cit.>, eigenvalue correlation measures such as the spectral form factor <cit.> and ⟨r|$⟩-value <cit.>, entanglement measures <cit.> and measures utilizing operator growth such as out of time-ordered correlator (OTOC) <cit.> and recently introduced universal operator growth hypothesis and associated Krylov complexity <cit.>, see <cit.> for a detailed review on recent works.

With the motivation to study quantum chaotic systems and possibly to classify them, similar to their classical counterpart, we utilize the ideas put forth in <cit.> regarding the relaxation fluctuations of the correlation function and its intrinsic connection to chaos. The study was motivated by and studied using models having well-defined classical limits which are known to be chaotic such as in quantum maps <cit.>. By studying the quantum correlation functions which can be related to classical correlations and also to the intricate issues of ergodicity and mixing, an attempt was made to classify quantum dynamics similar to their classical counterpart. Of central interest in this study is the correlation function constructed in such a way that they have a well-defined classical limit. In this paper, we will study these quantities with their original formulation and without necessarily considering the classical limits to illuminate properties of certain quantum spin systems, such as the mixed-field Ising model and other models such as the Gaussian Orthogonal Ensemble (GOE) and Gaussian Unitary Ensemble (GUE) Rosenzweig Porter (RP) models. The mixed field Ising model is a simple model of1/2-spins in a 1D chain with a nearest neighbour interaction and two transverse external fields. The model is known to exhibit both integrable and quantum chaotic limits <cit.>, depending on the strength of the transverse field. The eigenvalues of the model follow the RMT behaviour, yet the model does not have the expected chaotic dynamics, and exhibit weak chaos<cit.>. The RP model is a simple one-parameter model of random matrices that has a rich structure and is well studied in literature <cit.> for its properties. This simple one-parameter model remarkably has a fractal phase (an extended phase) apart from the usually found ergodic (chaotic) and localized (integrable) phases, depending on the value of its parameters. Our findings suggest that even if the eigenvalue statistics follow RMT behaviour it is still possible that the eigenvector correlations do not follow the same, as we will see in the case of the mixed-field Ising model. Such conclusions have also been made for another simple model of quantum chaos, Baker's map <cit.>. Furthermore, for models like the RP model, we show that the average and variance of relaxation fluctuations can used to detect the transition quite well. Our work here also try to advocate the use of eigenvector distribution and their correlations along with usual eigenvalue statistics to quantify quantum chaos.

The outline of the paper is as follows. We start with describing the setup used in Section <ref> and then discuss the results obtained previously for the GOE random matrices. We then briefly use the method to revisit Baker's map in section <ref> and study some of it in some detail. In Section <ref> we then study the mixed field Ising model within the same framework and describe the similarities of its results with the Baker's map. We then study both the GOE and the GUE-RP model in Section (<ref>) and finally end the paper with the conclusion and outlook in Section <ref>.





§ SETUP AND RANDOM MATRIX THEORY

In this section, we first set up the framework. We will mostly review the procedure already outlined in <cit.>. 

We will be interested in the study of Hilbert spaceℋof dimensionD. Let there be two distinct operators on ,P_AandP_B. For our present purpose, we will focus on the case when these operators are projection operatorsP_A + P_B = 𝐈_D,𝐈_Dis theDdimensional identity matrix. The reason for such choice originally was the classical limits of these operators, where classically these correspond to choosing the partitionAandBof relevant phase space𝒫such thatA ∪B= 𝒫 . Consider a unitary evolution operatorUwhich acts on the state in Hilbert space  of dimensionD. The quantity of interest considered in <cit.> is the correlation function 

    C_AB(t)=  (U^t P_A U^-t P_B)/D,

which physically captures the overlap of subspaceAevolved for timetwith the subspaceB. For the case whent →∞,C_AB →(P_A)(P_B)/D^2 = f_Af_B wheref_Aandf_Bdenotes what fraction of total space , sub-spacesAandBare respectively. Using this, we normalize Eq. (<ref>), and hence study the following quantity

    c(t) = C_AB(t)/f_Af_B.

Furthermore, for simplicity, we will choose projectors to be diagonal in the discrete position basis where 

    P_A = ∑_n^f_AD|n⟩⟨n|,

andP_B = 𝐈_D - P_ASinceU(or for that matterH) has a discrete and finite spectrum, the correlation given by Eq. (<ref>) is a sum of purely oscillatory terms, hence giving rise to fluctuation in correlation function even after equilibrium has set in. We will mainly be interested in the study of the average and the variance of these fluctuations denoted byandσ_c^2respectively. We can representc(t)in terms of eigenfunctions of the evolution operatorU, denoted byψ_kwith the following eigenvalue equation 


    U |ψ_k⟩ = e^i E_k|ψ_k⟩   k = 1, ⋯ D,

whereE_kis the eigenenergy. The correlation function can then be written as 

    c(t)   = 1/D f_A(1-f_B)∑_k_1,k_2 ∑_n_1=1 ^D f_A∑_n_2= D f_A+1^D e^i t (E_k_2-E_k_1
       ×⟨n_2|ψ_k_1|⟨%s|%s⟩⟩ψ_k_1|n_2⟨n_1|ψ_k_2|⟨%s|%s⟩⟩ψ_k_2|n_2,

where we have used the fact thatf_A +f_B = 1.
The time average of the above quantity can be simply taken by substitutingk_1 = k_2. Denotingf_Aasfwe get following after simplification

    = 1/D f(1-f)∑_k =0^D ∑_n_1=1 ^D f∑_n_2= D f + 1^D 
        |⟨n_1|ψ_k|⟨%s|%s⟩⟩n_2|ψ_k|^2.

Thus,is a measure of the distribution of eigenvectors in the two sub-spacesAandB. Another feature of the average of correlations is that they capture the amount of spread of eigenstates in the subspaceAandB. If they are equally distributed then the average tends towards the classical time average and is equal to 1. This progression of results towards classically results means the result in the largeNlimit (noticeħ_effis defined to be equal to1/Nfor such systems and largeNimpliesħ_eff →0). To capture this feature it is important to look at the scaling ofwithN.  In <cit.>
it was suggested via various different models that< 1, which implies the reluctance to participate equally in both partitions. However, there is still a possibility of≥1if there are symmetries present in the systems for e.g. the parity symmetry which forces the wavefunctions to be essentially identical in the two subspacesAandB. Although in the absence of any symmetries. 
Similarly the variance ofc(t)can be written as follows after simplification

    σ_c^2 = 2/D^2f^2(1-f)^2∑_k_2>k_1∑_n=1^f D-1|⟨n|ψ_k_1|⟨%s|%s⟩⟩ψ_k_2|n|^4,

which implies that the variance is a measure of correlations between distinct pairs of eigenvectors. 

Using the properties, that the real eigenvectors of GOE matrices are uniformly distributed over aD-dimensional sphere and none of the eigenvectors is preferred, it was analytically derived in <cit.> that the average and variance of the fluctuations are given by 

    ⟨c_avg|_⟩GOE   = N/N+2∼ 1- 2/N, 
    ⟨σ_c^2|_⟩GOE   = 3/N^2,

where⟨·|$⟩ denotes the ensemble average of the quantities.
It is important to note that the above results are independent of the fraction f. We have plotted the above results in Fig. (<ref>) along with the numerical results obtained using the GOE matrices. We have also plotted the results for the GUE matrices. For numerical purposes for both GOE and GUE type random matrices, we have taken f= 1/2 and system sizes N= 128 (2048), 256 (1024), 512 (512), 1024(128), 2048(64) , 4096(32), where the numbers in the bracket denote the number of samples. Using the detailed numerical analysis, as outlined in Appendix <ref>, we deduce the following results for GUE matrices

    ⟨|_⟩GUE   = N/N+1∼ 1- 1/N,
    ⟨σ_c^2|_⟩GUE   = 2/N^2 + 𝒪(1/N^2).

These are also plotted along with the GUE results in Fig. (<ref>) ( the dashed black lines). These behavior of ⟨|$⟩ and⟨σ_c^2|$⟩ are universal, in the sense that they are independent of the partition f used. 
 


§ BAKER'S MAP: A REVISIT

As a brief overview of the setup described in the previous section, we consider a simple quantum model exhibiting chaos, the quantum Baker's map <cit.>. Classically the map is defined via the following recursion corresponding to the phase space points (q_n,p_n)

    (q_n+1, p_n+2)= 
        (2 q_n, p_n/2)              0< q_n < 1/2,
    (2q_n-1, (p_n+1)/2)     1/2 < q_n < 1.

The quantized version of the map corresponds to the following unitary operator U, in position representation, with Hilbert space dimension N

    B = G_N^-1([  G_N / 2        0;        0 G_N / 2, ])

where G_N denotes the discrete Fourier transform and is given as 
(G_N)_l,m = 1/√(N)exp(-2 π i (l+1/2)(m+1/2)).
The classical map can be shown to be chaotic and mixing with the Lyapunov exponent equal to ln(2), which is also equal to the Lyapunov exponent obtained using OTOC in the quantized Baker's map too <cit.>. We remark that we can form various kinds of generalized Baker's map similar to Eq. (<ref>), by taking into account the different partition schemes in the classical version, e.g. Eq. (<ref>) is isomorphic to  (1/2,1/2) Bernoulli process <cit.>. As another example, we can consider a classical map with Bernoulli scheme (2/3,1/3) <cit.> given as follows

    (q_n+1, p_n+2)= 
        ( 3q_n/2, 2 p_n/3)              0< q_n≤2/3,
    (3q_n-2, (p_n+2)/3)   2/3 < q_n < 1.

The corresponding unitary operator is then given by 

    B = G_N^-1([ G_2 N / 3         0;         0  G_N / 3. ])

The (1/2,1/2) scheme also has a parity symmetry with the parity operator given by P = -G_N^2. The eigenvalues of the unitary operator U are given by e^iθ, where θ is the eigenangle, θ∈ [0,2 π] which are uniformly distributed on a circle. The spacing distribution of eigenangles for the chaotic operator follows the GOE distribution. In Fig. (<ref>) we show the eigenangle spacing distribution of Baker's map with Bernoulli scheme (1/2,1/2) after symmetry resolution and unfolding. We observe quite a good agreement with the GOE spacing distribution, which is given by the red solid line in Fig. (<ref>). We now study the c_avg and σ_c^2 of the fluctuations for various schemes of Baker's map. 











The Fig (Bottom). (<ref>) suggest the following trend of  and σ_c^2 with system size N

    = 1- α/N^β, 
    σ_c^2   = α_σ/N^β_σ.

For various schemes, we obtain the following values of the best-fit parameter for =

    (2/3,1/3)    : α = 1.07 , β = .78 , 
    
        (1/4,3/4)    : α = .37 , β = .98 , 
    
        (1/3,2/3)    : α = .94 , β = 1.02.


Similarly, for various schemes, we obtain the following values of the best-fit parameter for σ_c^2

    (2/3,1/3)    : α_c = 2.84 , β_c = 1.97, 
    
        (1/2,1/2)    : α_c = 4.94, β_c = 1.97, 
    
        (1/4,3/4)    : α_c = 2.39 , β_c = 1.95, 
    
        (1/3,2/3)    : α_c = 3.24 , β_c = 1.97.


We observe that for σ_c^2 for the Baker's map corresponding to the (1/2,1/2) scheme, the slope is the same as other schemes, β_σ≈ -2, yet it has an offset from other schemes and stands markedly different than other curves which nearly overlap on top of each other. Similar results reporting deviation of (1/2,1/2) scheme from (2/3,1/3) schemes were also reported in <cit.> and a possible reason suggested for such deviation is the presence of a large number of scarred states in Baker's map for this particular scheme <cit.>. We merely report these results of Baker's map as an exercise to understand the machinery of the method described. However, it is important to further understand the origin of these results in greater detail.
 


§ MIXED-FIELD ISING MODEL

We now consider a simple spin chain model that exhibits quantum chaos depending on the values of its parameters, the mixed field Ising model. The model is given as follows 

    H = - ∑_i=1^N-1σ_i^zσ_i+1^z - ∑_i=1^N-1 ( h_xσ_i^x+ h_zσ_i^z),

where σ_i^x and σ_i^y are the Pauli matrices at site i, defined as 

    σ_k^i= 𝐈_2^⊗(i-1)⊗σ_k^i⊗𝐈_2^⊗(N-i)

for k ∈{x,y,z}. We consider open boundary conditions, with N sites with the Hilbert space of dimension d = 2^N.

The model is known to exhibit both chaotic and integrable regimes <cit.> depending on the strength of longitudinal and transverse fields as follows

    (h_x,h_z)=    
          (-1.05,0.5),   Chaotic
    
         (-1,0).       Integrable

Since the model has parity symmetry hence we look for these properties by focusing on symmetry resolves blocks, so as to eradicate the effect of symmetry on these quantities. Once the symmetries are resolved we then carry out our analysis.



In Fig. (<ref>) we look at the variation of c(t) with time for the two regimes. For practical purposes, and to bypass the need to further resolve the extra symmetry arising for the integrable case, unlike Eq. (<ref>), we take the parameters to be (h_x,h_z) = (-1,0.001). We observe that for the integrable regime, the fluctuations are larger than those of the chaotic regime, and there is a clear distinction between the two. 



The integrable and chaotic regimes are quite evident from the scaling of  and σ_c_+ with N. For the integrable case, the results in Fig. (<ref>), suggest the scaling behaviour of the following form 

    ()_+ = 1- α_+/ N ^β_+,

with fitting parameters: α = 0.058, γ = 0.12. The slope of the best-fit line in Fig (Top).(<ref>) is really small ∼ -0.12. In fact, we practically observe that for larger N the value saturates implying that as N increases  does not approach unity and is constant. This tendency of  to not scale with N is a sign of the integrability of the system. We also explore the variance and observe the following scaling behaviour

    σ^2_+ = κ_+/ N ^δ_+,

with best-fit parameters κ_+ = 0.46, δ_+= 1.54. Unlike the behaviour  for the variance, we see that the scaling of variance with N. It has been observed in <cit.> as well that for integrable systems σ_c^2 can scale with N but the coefficient is non-universal with a strong dependence on partition f. 




For the chaotic regime, the results for the deviation of average relaxation from unity are shown in Fig (Top). (<ref>) for both the parity sectors. We observe the following power-law-like behaviour of average fluctuation with N 

    ()_± = 1- α_±/ N ^β_±,

where ± denotes the +1 and -1 parity sectors respectively with α_±≈ 1, β_±≈7/10, which deviates from the RMT predictions as discussed in Section <ref>, where α= 2 and β =1. This implies that in some sense, the Ising model is less chaotic as compared to other models of quantum chaos which follow the RMT behaviour closely. We have already observed similar features in the case of the quantum Baker's map. These results hint towards a hierarchy amongst systems exhibiting quantum chaos and suggest that not all systems are as chaotic as others. Furthermore, it shows that eigenvalue statistics are not enough to characterize the dynamics of the quantum systems, as it is possible that although the eigenvalue statistics of these systems follow RMT behaviour closely, the correlation of eigenvectors as discussed in the present context is not in agreement with the corresponding RMT results. We would also like to remark that spin 1/2 systems are the most quantum systems we can have and thus they have peculiar features within the context of quantum chaos as has been reported before in various studies using other measures as well <cit.>. 




§ ROSENZWEIG-PORTER MODEL

To elucidate the features of integrable to chaotic transition in more detail we now study a random matrix model, the Rosenzweig-Porter (RP) model <cit.>. The model is given by 

    H = A + 1/N^γ/2 B ,

where A is a diagonal matrix with its elements chosen identically and independently from a standard normal distribution and B is a GOE matrix, whose elements are drawn from a normal distribution with zero mean (for both diagonal and off-diagonal elements) and variance ⟨B_ii|^⟩2= 1 and ⟨B_ij|^⟩2 = 1/2 for i ≠ j respectively. Depending on the value of the free parameter γ, the model has the following three phases

    γ = 
         [0,1]     Ergodic ,
    
        [1,2]      Fractal ,
    
        [2,∞)   Localized .


These phases are characterized using the fractal dimensions which are defined using the inverse participation ratio (IPR). For a particular state ψ_k, we define the inverse participation ratio  in a basis ϕ_i <cit.> as follows

    IPR = ∑_i| ⟨ϕ_i|ψ_k||⟩^4.

The scaling of IPR is given as follows

    IPR∼ N^-D,

where D is called the fractal dimensions. For the ergodic phase when the states are uniformly delocalized over all the available states D= 1, however, for the localized phase, it is D= 0 as the states are localized over a single state. The fractal phase is interesting as D lies between 0 and 1 implying the states are delocalized over the basis, but not fully. 



 Paper Title: Quantum repeater protocol for deterministic distribution of macroscopic entanglement.








pyrkov@icp.ac.ru
Federal Research Center of Problems of Chemical Physics and Medicinal Chemistry RAS, Acad. Semenov av. 1, Chernogolovka, Moscow region, Russia, 142432



Federal Research Center of Problems of Chemical Physics and Medicinal Chemistry RAS, Acad. Semenov av. 1, Chernogolovka, Moscow region, Russia, 142432


tim.byrnes@nyu.edu
New York University Shanghai, 567 West Yangsi Road, Shanghai, 200126, China; NYU-ECNU Institute of Physics at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China; Shanghai Frontiers Science Center of Artificial Intelligence and Deep Learning, NYU Shanghai, 567 West Yangsi Road, Shanghai, 200126, China.
State Key Laboratory of Precision Spectroscopy, School of Physical and Material Sciences, East China Normal University, Shanghai 200062, China
 Center for Quantum and Topological Systems (CQTS), NYUAD Research Institute, New York University Abu Dhabi, UAE.
Department of Physics, New York University, New York, NY 10003, USA



             


§ ABSTRACT 
Distributing long-distance entanglement is a fundamental goal that is necessary for a variety of tasks such as quantum communication, distributed quantum computing, and quantum metrology.  Currently quantum repeater schemes typically  aim to distribute one ebit at a time, the equivalent of one Bell pair's worth of entanglement.  Here we present a method to distribute a macroscopic amount of entanglement across long-distances using a number of operations that scales only linearly with the chain length.  The scheme involves ensembles of qubits and entangling them with an S^z S^z interaction, which can be realized using atomic gas ensembles coupled by a shared optical mode. Using only local measurements on the intermediate ensembles, this leaves the ensembles at the ends of the chain entangled.  We show that there are particular “magic” interaction times that allow for distribution of entanglement with perfect fidelity, with no degradation with chain length.  The scheme is deterministic, such that with suitable local conditional unitary corrections, the same entangled state can always be prepared with good approximation.  





                             

                              
Quantum repeater protocol for deterministic distribution of macroscopic entanglement
    Tim Byrnes
    August 5, 2024
====================================================================================






§ INTRODUCTION


Entanglement distribution in quantum networks is an important task in the field of quantum information, with potential applications such as secure communication, quantum metrology, distributed quantum computation <cit.>. Photons can carry quantum information over long distances with almost negligible decoherence and are compatible with existing telecommunication fiber technology.  
However, while photons can be transferred relatively efficiently between two locations in the same laboratory or city, distances exceeding this begin to be problematic due to photon loss. 
For example, in a typical optic fiber, the photon loss rate become significant beyond distances of ∼ 200 km and the maximum distance is estimated as ∼ 600 km <cit.>. 
In classical signal transmission line this problem can be solved with optical amplifiers but it is not possible to amplify quantum signal due to the famous no-cloning theorem <cit.>. To overcome this difficulty, Briegel, Zoller and colleagues proposed the quantum repeater protocol <cit.>, 
aiming to establish long-distance quantum entanglement. 
In the quantum repeater scheme, the transmission channel is divided into several short-distance links, where entanglement is generated between the nearest neighbor nodes using photon transmission.  Then entanglement between distant nodes is achieved through entanglement swapping <cit.>. 

In the entanglement swapping protocol, the probabilistic nature of generating photonic entanglement (e.g. by parametric down conversion) across the elementary links means that the entanglement may not be present synchronously.  One potential solution to this challenge involves storing the generated entanglement in a quantum memory until it has been successfully prepared in both links <cit.>. It has been shown that many physical systems, including single neutral atoms <cit.>, 
cold atoms <cit.>, 
trapped ions <cit.>, 
and the Nitrogen-vacancy centers in diamonds <cit.>  
can absorb and store incoming photons, 
for instance using techniques such as electromagnetically induced transparency <cit.>.  This allows them to serve as the nodes of a quantum network with an embedded quantum memory <cit.>. 

As a platform to realize a quantum memory, atomic gas ensembles are an attractive candidate due to their ability to perform strong and controllable coupling between the atoms and photons.  The primary advantanges of using an ensemble is that there is a collective enhancement due to the large number of atoms with long coherence times. Duan, Lukin, Cirac, and Zoller (DLCZ) proposed a scheme based on single-photon detection to implement a quantum repeater and realize scalable long-distance quantum communication with atomic ensembles <cit.>. 
In the scheme, probabilistic write-out photons are generated from two remote atomic ensembles, and the correlated photons are sent to a beam splitter generating a heralded Fock-state type entanglement between the atomic ensembles. Kuzmich, van der Wal, and co-workers experimentally realized a quantum memory using the DLCZ protocol with trapped cold Cs atoms <cit.> and Rb atoms <cit.>. 

This was followed by several other works studying entanglement generation between atomic ensembles  <cit.>.  In the continuous variable (CV) framework, Polzik produced entanglement between atomic ensembles using a quantum nondemolition measurement approach <cit.>. Another prominent design for a CV repeater involves creating entangled two-mode quadrature squeezed states of light using spontaneous parametric downconversion (SPDC) sources. This is followed by CV non-Gaussian quantum error correction through the injection and detection of single photons <cit.>. The process also includes storing heralded link-level entanglement in quantum memories, as well as entanglement swapping through coherent detection <cit.> or other non-Gaussian methods <cit.>. 






Such approaches for generating entanglement between ensembles work in either the single excitation  (e.g.  DLCZ) or Holstein-Primakoff regime (for CV), such that the amount of entanglement is of the order of one ebit, corresponding to the entanglement in a single Bell pair.  However, the very large number of atoms in an atomic ensemble provides an opportunity to create entanglement in a different regime, creating a type of macroscopic entanglement.  Such an approach was discussed in works such as Refs. <cit.>, where a scheme for producing macroscopic entanglement using an S^z S^z interaction was analyzed and proposed to be used for quantum computation. Here, we define macroscopic entanglement as von Neumann entropy or logarithmic negativity that is of order of log_2 D, where D ≫ 1 is the dimensionality of the subsystems that participate in the entanglement.  In the case of atomic ensembles, the effective dimensionality is equal to the number of atoms <cit.>, which is in the range of 10^3 to 10^12.   These type of states have been proposed to be used as a resource for several quantum information processing protocols, such as quantum teleportation, remote state preparation, and others <cit.>. Experimentally, 
the creation of many-particle entanglement localized in a single spatial location <cit.> and spatially separate regions <cit.> within one Bose-Einstein condensate (BEC) has been observed. 
Bell correlations have been experimentally confirmed in a BEC <cit.> and a thermal atomic ensemble <cit.>. Recently, an experiment successfully demonstrated the achievement of entanglement between a spatially split BECs <cit.>. 

In this paper, we introduce a new protocol for creating long-distance macroscopic entanglement between atomic ensembles. The basic idea is to use entanglement swapping, in a similar way to that proposed in existing quantum repeaters, but perform this at the macroscopic scale. As illustrated in Fig. <ref>, we consider a chain of qubit ensembles within a quantum network, where nearest neighbors can be entangled with a S^z S^z interaction.  Only local projective measurements are required on the intermediate ensembles leaving the ensembles at the ends of the chain in an entangled state (Fig. <ref>(b)).  Operations at the ensemble level are only performed for the protocol (i.e. no microscopic operations of single qubits), and the resources required for the protocol only depends upon the chain length and not the number of qubits in an ensemble. 
The type of state is considerably more complex than the qubit version due to the many-body entangled state created by the S^z S^z interaction between the atomic ensembles. Despite this, we are able to find an analytic solution at particular entanglement times.  We show that there are “magic” interaction times  where entanglement distributed to the ends of the chain do not degrade with the length of the chain. 



 

























§ ENTANGLEMENT DISTRIBUTION PROTOCOL


We now describe concretely the entanglement distribution protocol using qubit ensembles. The basic overview of the scheme is shown in Fig. <ref>(b).  In order to keep our presentation independent of a particular physical implementation, we consider generically a system consisting of N qubits located at each node of the quantum network.  We assume that the qubits within the ensemble cannot be controlled individually, but the ensembles may be controlled by total spin operations as we describe below. 

Each of the ensembles is initially prepared in a spin coherent state, which we denote 


    |α,β⟩⟩≡∏_l=1^N (α |0⟩_l +β |1 ⟩_l).


where |0⟩_l and |1⟩_l denotes the computational states for the lth qubit in the ensemble.  Here, α and β are arbitrary complex numbers satisfying |α|^2+|β|^2=1. 

Each ensemble is controlled using total spin operators


    S^γ = ∑_l=1^N σ_l^γ


where γ∈{ x, y, z } and σ_l^γ is the Pauli spin operator for the lth qubit within the ensemble.  The spin operations conserve the particle number, and we assume that N is the same for each ensemble for simplicity. 

Since the spin operators and the initial state are completely symmetric under particle interchange, one can write an equivalent representation using the Schwinger boson formalism <cit.>.  The spin coherent state in the bosonic formulation is 


    |α,β⟩⟩≡1/√(N!)(α a^†+β b^†)^N|0⟩,


where the bosonic creation and annihilation operators a, b obey commutation relations [a,a^†]=[b,b^†]=1. The spin operators are written as


    S^x     = a^† b + b^† a 
    
    S^y     = -i a^† b + i b^† a 
    
    S^z     = n^a - n^b 
    N̂    = n^a + n^b ,


where the number operators for the two qubit states are defined as n^a = a^† a, n^b = b^† b.  The eigenstates of S^z operator, which we call the number states, are defined as


    | k ⟩ =  (a^†)^k (b^†)^N-k/√(k! (N-k)!) | ∅⟩ ,


where | ∅⟩ is the vacuum state.  The eigenvalues are S^z |k ⟩ = (2k - N) | k ⟩. A mapping of the number states in the qubit formulation is given in Ref. <cit.>.  






We consider a chain of M ensembles prepared in the initial state







    |ϕ_0 ⟩=|1/√(2),1/√(2)⟩⟩_1⊗…⊗|1/√(2),1/√(2)⟩⟩_M .


We assume a scenario where the nearest neighbors of the chain can be entangled using a S^z S^z  interaction. More distant pairs cannot be directly entangled.  The Hamiltonian that we will consider in the chain is


    H=  ∑_j=1^M-1 (-1)^j n_j^a n_j+1^a ,


where j labels the  M ensembles. The signs in the Hamiltonian are chosen in this way so that the simplest analytical formulas can be obtained, as shown later.  
 This is the same as a S^z S^z interaction up to local rotations because


    n^a_j n_j+1^a = 1/4 (S^z_j + N̂_j) (S^z_j+1 + N̂_j+1 ).


Numerous proposals for producing a S^z S^z interaction between qubits have been proposed, particular in the context of atomic ensembles and BECs.  In Ref. <cit.> a scheme mediated by photons in an optical fiber was proposed.  This was later adapted to a geometric phase gate <cit.>. Another approach is to use spin-dependent forces on neighboring BECs <cit.>.  
Such an interaction is also produced when one-axis squeezed states are split <cit.>.  This type of entanglement was  experimentally realized in BECs <cit.>.  In this paper, we assume that such an interaction can be induced using suitable methods, and the interaction time t is a controllable parameter. 

Evolving the Hamiltonian for a time t between the ensembles gives the unitary evolution


    U=∏^M-1_j=1 e^- i (-1)^j  n^a_j n^a_j+1 t .


The state after applying the gates is


    |ψ_M ⟩ = U | ϕ_0 ⟩ .




To understand the type of state that is produced by this interaction, consider first the M = 2 case which gives


    |ψ_2 ⟩   =e^i n^a_1 n^a_2 t|1/√(2),1/√(2)⟩⟩_1|1/√(2),1/√(2)⟩⟩_2
       =1/√(2^N)∑_k_1√(C_N^k_1)|k_1⟩ |e^i k_1 t/√(2),1/√(2)⟩⟩_2 .


As discussed in Ref. <cit.>, this produces a type of entangled state where number states in the first ensemble are correlated with spin coherent states around the equator of the Bloch sphere of the second.  The unitary operation U can be viewed as a conditional rotation of ensemble 2 with an angle dependent on the number state on ensemble 1.   

































For a chain of M ensembles, the wavefunction after the entangling gate is 


    |ψ_M ⟩=1/√(2^MN/2)∑_k_1,k_3,…,k_M-1=0^N( ∏_i ∈odd√( C^k_i_N)) 
    × | k_1 ⟩_1  |e^i(k_1-k_3)t/√(2),1/√(2)⟩⟩_2 | k_3 ⟩_3
    |e^i(k_3-k_5)t/√(2),1/√(2)⟩⟩_4  
    …⊗  | k_M-1⟩_M-1 |e^ik_M-1t/√(2),1/√(2)⟩⟩_M










for even M. We have 


    | ψ_M ⟩=1/√(2^(M+1)N/2)∑_k_1,k_3,…,k_M=0^N( ∏_i ∈odd√( C^k_i_N)) 
    × | k_1 ⟩_1  |e^i(k_1 - k_3)t/√(2),1/√(2)⟩⟩_2 | k_3 ⟩_3
    |e^i(k_3 - k_5)t/√(2),1/√(2)⟩⟩_4  
    …⊗ |e^i(k_M-2 - k_M)t/√(2),1/√(2)⟩⟩_M-1 | k_M⟩_M










for M is odd.  In the expressions (<ref>) and (<ref>), the odd numbered ensembles are expanded to number states and the remaining ensembles on even sites are conditionally rotated. We note that the wavefunction can be always written in a way such that on adjacent sites, there is a conditional rotation of the ensemble written as a spin coherent state, dependent on the number states on alternating nodes.  Hence the parity of the number of ensembles M plays important role.  Depending on the parity of M, the last ensemble is written in terms of a spin coherent state or a number state.  Henceforth, we will show expressions for even M and defer the expressions for odd M to Appendix <ref>. 








Next we perform a measurement of the intermediate ensembles in the x-basis.  The measurement operator for this is written 


    M_q^(j) = | q ⟩^(x)_j ⟨ q |^(x)_j ,


where the number state in the x-basis is | k  ⟩^(x) = e^-iS^y π/ 4 | k  ⟩ and q ∈ [0,N] labels the measurement outcome. 
We note that in conventional entanglement swapping, two qubits from different Bell pairs are measured in the Bell basis to create long-distance entanglement. Such entangled basis measurements are implementable in a photonic context but are less convenient in the context of atomic ensembles.  Here, we perform only local measurements on intermediate nodes such as to transfer the entanglement between the first and last ensembles in the chain.  

Later, we show that it is beneficial to introduce a slight angular offset from the x-axis of the measurement.  For this purpose we perform a equatorial rotation before performing the measurement, given by 


    V^(j)(ϕ)  = e^i n^a_j ϕ .


Hence the final state is the projected state


    | Ψ_q⃗⟩ = ( ⊗_j=2^M-1 M^(j)_q_j V^(j)(ϕ) ) | ψ_M ⟩  ,


which is an unnormalized state.  Here we wrote the full set of measurement outcomes of the intermediate ensembles as


    q⃗ = (q_2, q_3, …, q_M-1 ) .





The final unnormalized state after all the measurements is 


    | Ψ_q⃗⟩ =1/√(2^MN/2)∑_ k_1, k_3, …, k_M-1  = 0^N√( C^k_1_N)( ∏_j=2^M-1Ω_q_j^(j))
      
    ×  |k_1⟩|e^i k_M-1t/√(2),1/√(2)⟩⟩_M





































where we defined


    Ω_q^(j) = {[ ⟨ q |^(x) | e^i(k_j-1 - k_j+1)t + i ϕ/√(2),1/√(2)⟩⟩                                             j ∈even;                e^i k_j ϕ√( C^k_j_N)⟨ q|^(x) | k_j ⟩                                              j ∈odd ].   .


The explicit expressions for the matrix elements are 


    ⟨ q |^(x) | e^iα/√(2),1/√(2)⟩⟩ = i^N-q e^i N  α /2√(C_N^q)cos^qα/2sin^N-qα/2


and


    ⟨ q|^(x) | k⟩=1/√(q!(N-q)!2^N)∑^q_l=0∑^N-q_m=0C^l_q C^m_N-q
    
    (-1)^N-q-m√((l+m)!(N-l-m)!)δ_k,l+m


where δ_i,j is Kronecker delta <cit.>. The probability of obtaining this outcome labeled by q⃗ is


    p_q⃗ = ⟨Ψ_q⃗ | Ψ_q⃗⟩ .






























































































§ ENTANGLEMENT DISTRIBUTION



The entanglement for the states can be calculated using the von Neumann entropy


    E=-∑^N_l=0λ_l log_2 λ_l ,


where λ_l are the eigenvalues of the density matrix


    ρ_M = Tr_1  |Ψ_q⃗⟩⟨Ψ_q⃗ |/p_q⃗ =
    1/p_q⃗∑^N_k_1=0⟨ k_1|Ψ_q⃗⟩⟨Ψ_q⃗ |k_1⟩ .




The dynamics of entanglement versus time and the number of nodes in chain is shown in Fig. <ref>. Here we show the case when all measurement results are chosen equal to q_j = N and the offset angle ϕ = 0. We later generalize our analysis to other measurement outcomes, but this case shall be an interesting particular case to examine first.  We firstly note that the form of the graph is similar to the characteristic “devil's crevasse” shape giving sharp dips at fractional multiples of π <cit.>.  The fact that this type of dependence is seen can be understood from the fact that the underlying entanglement between nearest neighbor nodes is generated using  S^z S^z interactions.  It naturally follows that the more entanglement that is present between the neighboring nodes, the larger the amount of entanglement that is distributed to the ends of the chain after the projection is applied.  

An important question is the performance with scaling the protocol with chain length M, such that long distance entanglement distribution can be achieved.  Typically, we observe from Fig. <ref> that the entanglement degrades with increasing chain length (see for example short interaction times t < π/4).  What is interesting is that for some particular times the entanglement does not degrade, and remains constant regardless of the chain length. We call these special points in the graph “magic times”, as these regions are of particular interest in the context of entanglement distribution.  It is also interesting to see that as the chain length increases, in the vicinity of the magic times the entanglement curve flattens out and remains constant for a range of times beyond the magic times.  This is important as it suggests that there is some tolerance allowable for the interaction times, which is beneficial if some errors exist in controlling t.   




To understand the nature of the state that is generated at the magic times,  we evaluate the fidelity with respect to the M = 2 state wavefunction given by (<ref>). The fidelity is given by 


    F = | ⟨ψ_2 | Ψ_q⃗⟩ |^2/p_q⃗  = 1/2^N(M+2)/2 p_q⃗
    ×| ∑_k_1, k_3, …, k_M-1C_N^k_1( ∏_j=2^M-1Ω_q_j^(j)) (e^i(k_M-1 - k_1)t +1/2)^N |^2 .


For an odd number of nodes in Fig. <ref>, we evaluate the fidelity with respect to the M = 3 state wavefunction


    | Ψ_N^(M=3)⟩=∑_k_1,k_3=0^N√( C^k_1_N  C^k_3_N)Ω_N^(2)  |k_1⟩|k_3⟩/√(∑_k_1,k_3=0^N  C^k_1_N  C^k_3_N | Ω_N^(2) |^2 ) .


Explicit expressions for the fidelity are given in Appendix <ref>.  The fidelity versus the interaction time is shown in Fig. <ref> for various chain lengths. We see that at the magic times the fidelity of the state is equal to 1, for all chain lengths.  This shows that the two types of states corresponding to the M = 2 state Eq. (<ref>) the M = 3 state Eq. (<ref>) can be prepared at these magic times without degradation with the length of the chain. 

These results suggest that it is possible to distribute the same type of macroscopic entanglement as that generated using the S^z S^z interaction, as long as the interaction times are in the vicinity of the magic times.  However, for this to be considered a practical protocol, we must also analyze outcomes beyond q_j = N, as this is only a single measurement outcome from many. 
For long chain lengths, there is an exponentially small probability of obtaining q_j = N, and  would be extremely inefficient if only this was the only successful outcome.  If one simply considers other measurement outcomes we do not have the stationary entanglement property as seen in Fig. <ref>, and the particular state that is generated is dependent on the particular measurement outcome.  We now show that by modifying the measurement slightly it is possible to produce the same entanglement between the end nodes of the chain for any measurement outcome.  
































§ SPIN-CAT STATES


To understand what is special about the entanglement at the magic times we perform an analysis of the wavefunction at these times.  For the purposes of our analysis we define the magic times to be at evolution times 


    t_L = 2π/L


where L takes an integer value. We note that there are additional times that have the invariance property with M, e.g. t = 4π/5 (see Fig. <ref>). For the simplicity and generality of the analysis we shall only consider the times (<ref>).  For all times  L ∈ [1, √(N) ] the entanglement exhibits the stationary behavior.  

The reason why times with fractional multiplies of 2 π are special is discussed in Ref.  <cit.> for the M = 2 case.  At these times we see that the the spin coherent states factorize as


    |ψ_2 (t= 2π/L )  ⟩=1/√(2^N)∑_k=0^N √(C_N^k)|k⟩_1  |e^2 π i k/L /√(2),1/√(2)⟩⟩_2 ,  
        = 1/√(2^N)∑_m=0^L-1(  ∑^N_{ k (mod L ) =m }k=0√(C_N^k)|k⟩)  |e^2 π i m/L /√(2),1/√(2)⟩⟩_2 .


Here we made the replacement


    ∑_k=0^N →∑_m=0^L-1∑^N_{ k (mod L ) =m }k=0


where the second summation only runs over those k satisfying k (mod L) = m. 

It is then natural to define the set of states


    | C_m ⟩ = 1/√(Z_m)∑^N_{ k (mod L ) =m }k=0√(C_N^k)|k⟩


where the normalization Z_m ≈ 2^N/L is a good approximation for N ≫ 1.  These states form an orthonormal set ⟨ C_m |  C_n ⟩ = δ_mn. Taking a Fourier transform of these states gives a spin coherent state


    1/√(L)∑_m=0^L-1 e^-2 π i n m/L  | C_m ⟩  =| e^-2 π i n /L /√(2), 1/√(2)⟩⟩ .


The inverse Fourier transform is


    | C_m ⟩ = 1/√(L)∑_n=0^L-1 
    e^2 π i n m/L  | e^-2 π i n /L /√(2), 1/√(2)⟩⟩ .


We see that | C_m ⟩ can be written as a sum of spin coherent states.  This is a spin-cat state which has been considered in various contexts before to encode quantum information <cit.>.  

The M = 2 state at the magic times can be written


    |ψ_2 (t= π/2L )  ⟩ = 1/√(L)∑_m=0^L-1 | C_m ⟩_1 |e^2 π i m/L /√(2),1/√(2)⟩⟩_2 .


For well-separated spin coherent states


    |⟨⟨e^2 π i n/L /√(2),1/√(2) |  e^2 π i m/L /√(2),1/√(2)⟩⟩ |     = cos^N (n-m)π/L 
       ≈ e^-N (n-m)^2 π^2/2 L^2
       ≈δ_nm(if L< √(N)) .


Hence we can consider the state (<ref>) to be an entangled state of L terms where on the first ensemble there is a spin-cat state and on the second ensemble there is a spin coherent state at various positions on the equator of the Bloch sphere.  This fact was already noted in Ref. <cit.> for the case L = 2. 








§ ENTANGLEMENT DISTRIBUTION AT MAGIC TIMES


We now return to the general case of M ensembles and evaluate the quantum state at the magic times.  Using the same methods as in the previous section we obtain the wavefunction before measurement for even M as


    |ψ_M ⟩=1/√(L^M/2)∑_m_1,m_3,…,m_M-1=0^L-1
    × |  C_m_1⟩_1  |e^2 i(m_1-m_3)π/L /√(2),1/√(2)⟩⟩_2 |  C_m_3⟩_3
    |e^2 i(m_3-m_5)π/L /√(2),1/√(2)⟩⟩_4  
    …⊗  |  C_m_M-1⟩_M-1 |e^2 i m_M-1π/L /√(2),1/√(2)⟩⟩_M .


Written in this form we may deduce why there is the entanglement invariance property that was observed in Fig. <ref>.  Let us make the measurement of the intermediate ensembles in two steps, first projecting the even numbered nodes within j ∈ [2,M-1], then the odd nodes. Splitting the measurements into two steps does not affect the final result as all the measurements commute.  Again set the rotation parameter ϕ = 0 for now.  Considering first for 
simplicity obtaining the outcome 


    |q= N ⟩^(x) = | 1/√(2), 1/√(2)⟩⟩


and assume that L < √(N) such that the spin coherent states are approximately orthogonal as in (<ref>).  Then this enforces m_1 = m_3 = … = m_M-1 for even M.  The resulting state for even M is approximately


    ⊗_j=2,4,… M-2 |q= N ⟩_j^(x)⟨ q = N |_j^(x)  |ψ_M ⟩∝
    ∑_m=0^L-1 |  C_m⟩_1 |  C_m⟩_3 … |  C_m⟩_M-1 |e^2 i m π/L /√(2),1/√(2)⟩⟩_M ,


which is valid for N ≫ 1.  
This correlates all the remaining ensembles, in particular the first and last ensemble such that they have the same label m. 

Now looking at the remaining intermediate ensembles, the spin-cat states can be written as a superposition of spin coherent states as shown in (<ref>).  Measurement of the cat states will collapse the superposition as


    ⟨ q = N |^(x)  |  C_m⟩≈1/√(L)


where only the n = 0 term in (<ref>) contributes for well-separated spin coherent states.  The correlations between the first and last ensembles are hence preserved and we obtain


    ⊗_j=2^M-1    |q= N ⟩_j^(x)⟨ q = N |_j^(x)  | ψ_M ⟩∝
       ∑_m=0^L-1 |  C_m⟩_1 |e^2 i m π/L /√(2),1/√(2)⟩⟩_M  ,


which is valid for N ≫ 1.  
This is precisely the same quantum state as (<ref>).  This is obtained for arbitrary chain length M.   This explains the invariance property of the entanglement and fidelity observed in Figs. <ref> and <ref> for the even case. 

For odd M, we can use the same manipulations to obtain


    ⊗_j=2^M-1    |q= N ⟩_j^(x)⟨ q = N |_j^(x)  | ψ_M ⟩∝∑_m=0^L-1 |  C_m⟩_1 |  C_m⟩_M .


This is the same state as (<ref>) which can be written as


    | Ψ_N^(M=3)⟩   ≈1/√(L)∑_m=0^L-1 |  C_m⟩_1 |  C_m⟩_3 , 
        =  1/√(L)∑_n=0^L-1 |e^2 i n π/L /√(2),1/√(2)⟩⟩_1 |e^-2 i n π/L /√(2),1/√(2)⟩⟩_3


which is valid for N ≫ 1.  










§ MEASUREMENTS OF SPIN COHERENT STATES


In the previous section we showed that for the measurement outcome q_j = N at the magic times, an entangled state consisting of spin-cat states is generated.  
We now wish to generalize this result for arbitrary measurement outcomes. One can see from the form of (<ref>) that if one were able to measure in the spin coherent state basis, a similar result would be obtained for all measurement outcomes.  This is however experimentally challenging, since unlike the optical case, homodyne measurements are not easily realizable with spin ensemble systems.  Therefore, an alternative is required to implement the measurements of the last section.  In this section we show that standard spin measurements are sufficient in achieving the same collapse as described in the previous section.  

To illustrate the measurement scheme, first consider performing a measurement of the spin-cat state (<ref>) in the x basis.  Plotting the positions of the spin coherent states involved in this superposition immediately identifies a problem: spin coherent states labeled with n and L-n have the same x position (Fig. <ref>(a)).  Hence projection in the x basis will collapse the superposition, but will catch two spin coherent states together, unless they are located at the extremal positions ⟨ S^x ⟩ = ± N.  To avoid this issue, it is beneficial to slightly offset the spin coherent states such that one measures instead the state


    e^i ϕ n^a  | C_m ⟩ = 1/√(L)∑_n=0^L-1 
    e^2 π i n m/L  | e^-i (2 π n /L - ϕ) /√(2), 1/√(2)⟩⟩ ,


where a suitable angle choice is


    ϕ = π/2L .


This slight rotation offsets the spin coherent states slightly such that a projection on the x-axis results in a collapse of the state to uniquely one of the spin coherent states (Fig. <ref>(b)). 

The effect of the measurement is also evident by examining the probability of the projection


    p_q =| ⟨ q |^(x)  e^i ϕ n^a  | C_m ⟩|^2 .


Figure <ref>(c)(d) shows the probability of the rotated spin-cat state for the same states as shown in Figs. <ref>(a)(b) respectively. 
We see peaks in the probability distribution at the expectation value of S^x each of the spin coherent states involved in the spin-cat state. When the projection involves two spin coherent states, there is a rapid oscillation of the probability distribution (Fig. <ref>(c)).  By applying the rotation ϕ this ambiguity is removed and smooth Gaussian peaks are seen for each of the spin coherent states (Fig. <ref>(d)). 

For a given measurement outcome q, one can deduce which spin coherent state has been projected using the peak position of the Gaussians:


    q_peak = N cos^2 ( π ( n̅-1/4)/L )  ,


where the spin coherent state is labeled by n̅∈ [0, L-1]. For a particular outcome q, by finding the closest q_peak one can deduce which spin coherent state labeled by n̅ the collapse occurred at.  




























§ APPROXIMATE WAVEFUNCTION FOR ARBITRARY MEASUREMENT OUTCOMES



Using the technique in the previous section, we may now examine the remaining projection outcomes in addition to the q_j = N case.  Examining  (<ref>), first consider performing a measurement on the even numbered ensembles.  Assuming that the collapse occurs such as to pick out one and only one of the spin coherent states we obtain the relation



    m_j-1 - m_j+1 = m̅_j (mod L) .


The state is then


    ⊗_j=2,4,… M-2 |q_j ⟩_j^(x)⟨ q_j |_j^(x)  |ψ_M ⟩∝
       ∑_m=0^L-1 |  C_m⟩_1 |  C_m - m̅_2⟩_3 |  C_m - m̅_2 - m̅_4⟩_5 … |  C_m - m̅_even⟩_M-1
       ⊗ |e^2 i (m+ m̅_even)  π/L /√(2),1/√(2)⟩⟩_M  .


where we defined


    m̅_even =  
    ∑_j=2,4, …, M-2m̅_j .


This correlates the first and last ensemble up to an offset in m. In (<ref>), the proportionality factor is


    1/√(L^M/2)∏_j=2,4,…, M-2⟨ q_j|^(x) | e^2i (m_j-1 - m_j+1 ) π/L + i ϕ/√(2), 1/√(2)⟩⟩
        = e^iN m̅_evenπ /L + iN ϕ/2/√(L^M/2)
       ×∏_j=2^M-2√(C_N^q_j)cos^q_j ( m̅_j π/L + ϕ/2 )sin^N-q_j ( m̅_j π/L + ϕ/2 )


which has no dependence on m hence does not affect the wavefunction.  

Measurement of the odd numbered intermediate ensembles will give factors


    ⟨ q|^(x) |  C_m ⟩    = 1/√(L)∑_n=0^L-1 e^2 π i nm/L ⟨ q|^(x)  |  e^-2 π i n/L + i ϕ/√(2),  1/√(2)⟩⟩
       ≈1/√(L) e^i πn̅ (2m - N)/L + i ϕ N/2 √(C_N^q)
       ×cos^q ( n̅π/L - ϕ/2 )sin^N-q_j ( n̅π/L - ϕ/2 )


where in the second line we used the result of the previous section that the measurement will collapse the spin-cat state to one of the spin coherent states with label n̅.  Here the important factor that affects the wavefunction is the phase dependence on m.  The total m dependence is given by projecting the odd sites is


    ⟨ q_3|^(x) |  C_m -m̅_2⟩⟨ q_5|^(x) |  C_m -m̅_2 - m̅_4 ⟩…⟨ q_M-1 |^(x) |  C_m - m̅_even⟩
    ∝ e^2 π i m n̅_odd/L


where 


    n̅_odd = ∑_j=3,5,…, M-1n̅_j  .


We thus finally obtain the final state after projection 


    | Ψ_q⃗^approx⟩ =  ⊗_j=2^M-1  |q_j ⟩_j^(x)⟨ q_j |_j^(x)  |ψ_M ⟩∝
       ∑_m=0^L-1 e^2 π i m n̅_odd/L  |  C_m⟩_1 |e^2 i (m - m̅_even)  π/L /√(2),1/√(2)⟩⟩_M .



Figure <ref> show a comparison of the approximate wavefunction (<ref>) with the exact wavefunction (<ref>).  In Fig. <ref>(a) the amplitude of the coefficients in the number basis is shown.  Excellent agreement is seen, with the coefficents having nearly perfect agreement.  The fidelity of the approximate wavefunction improves with N, as seen in Fig. <ref>(b).  The reason that the fidelities improve with N is due to the measurement scheme that we implement as described in the previous section.  In order to collapse the state to a single spin coherent state, the Gaussians for each one should be well-separated, as discussed in Fig. <ref>.  This requires L < √(N) typically. With increasing N, the Gaussian become better separated and approaches the ideal collapse to a single spin coherent state.  

We note that in Fig. <ref> perfect fidelity F = 1 was obtained, whereas in Fig. <ref> only near-unit fidelity is obtained.  The reason is that in Fig. <ref> the offset angle ϕ = 0 and only the outcome q_j = N was considered.  For this case, the projection on the spin coherent state is perfect, as can be seen in Fig. <ref>(a) and <ref>(c).  However, there is no guarantee that the outcome q_j = N is obtained, in fact for long chains it has an exponentially small probability.  Using the offset measurement of Fig. <ref>(b)(d) is advantageous in this sense, as all measurement outcomes lead to the generation of a nearly ideal state (<ref>).  The small price to pay for this is a non-unit fidelity, although as can be seen in Fig. <ref>(b) the fidelity quickly approaches 1.  










Therefore, using the offset measurement include ϕ = π/2L, one may obtain consistently obtain (<ref>) for all measurement outcomes.  The final step to turn this into a deterministic scheme for preparing entanglement is to peform conditional rotations to eliminate the n̅_odd and  m̅_even terms.  Specifically, knowledge of n̅_odd and  m̅_even  allows for a correction of the state to the state (<ref>) using the rotations


    e^i 2 π n^a /L  |  C_m ⟩    = e^i 2 π m /L  |  C_m ⟩
    
    e^i ϕ n^a  |e^ i ϕ' /√(2),1/√(2)⟩⟩    = |e^ i (ϕ+ϕ')  /√(2),1/√(2)⟩⟩ .


Then by applying the corrections


    e^-2 π i n̅_odd  n^a_1/L   e^2 π i m̅_even  n^a_M/L   | Ψ_q⃗⟩≈ | ψ_2 (t= π/2L ) ⟩


we obtain a deterministic preparation of the state (<ref>) which is the aim of this study.  



















§ RESOURCE SCALING AND EXPERIMENTAL IMPLEMENTATION


We now comment on the resource costs required to execute the protocol.  This question is intimately connected to the experimental implementation, since the resources required depend on the way that the operations are implemented. 
The key aspect that makes this scheme efficient is that the protocol has no dependence on N, the number of qubits in the ensembles, in any of the operations that need to be performed.  Despite this, the amount of entanglement that is distributed is of order log_2 N, as can be seen in Fig. <ref>.  The reason for this is that only collective spin operations are used to control the ensembles in all the steps of the protocol. 

To make this point clear, let us review the steps of the protocol and examine them each:



  * Prepare the initial state | ϕ_0 ⟩ = |1/√(2), 1/√(2)⟩⟩^⊗ M. 

  * Apply the Hamiltonian (<ref>) for a time t. 

  * Apply the rotation (<ref>) to the intermediate qubits. 

  * Measure the intermediate qubits in the x-basis (<ref>). 

  * Based on the measurement results, apply the correction (<ref>).  


Specifically, the initial state preparation in Step 1 can be routinely performed in atomic ensembles using optical pumping to populate one of the atomic levels, followed by a π/2 pulse implemented with radio frequency (RF) pulses. Both of these operations are independent of N since the optical and RF pulses applied to the ensembles are common to all atoms in the ensemble.  The unitary operations in Steps 2, 3, 5 are all in terms of total spin operators S^γ, hence are at the ensemble level. Again these can be implemented using RF pulses on the hyperfine ground states of the atoms.  The S^z S^z interaction of Step 2 can be implemented using techniques discussed in Ref. <cit.>, where the interaction is mediated using an optical channel.  Finally, the measurement in Step 4 is performed using atomic counting of the ensemble, which is routinely performed in works such as Refs. <cit.>.  All these operations only require operations at the ensemble level, not at the microscopic level of the individual qubits.  For this reason there is no N dependence to the scheme.  The only scaling is with respect to the chain length M, where the M - 1 links of the chain must be entangled and M - 2 intermediate nodes must be projected out.  


















§ CONCLUSIONS


We have proposed a protocol to distribute macroscopic entanglement on ensembles of qubits, by first generating entanglement in a chain configuration then projecting out the intermediate ensembles to leave the ends of the chain entangled.  Such a protocol is non-trivial in the ensemble case since the entanglement is of a complex many-body nature, which causes the entanglement tends to diminish with longer chain lengths.  We have found “magic” interaction times where the entanglement can be transferred without degradation.  The nature of the states at these magic times was found due to an encoding of the entangled states in terms of generalized spin-cat states.  This allowed us to derive the explicit form of the states for any ensemble size. The final wavefunction (<ref>), which has the approximate form (<ref>), takes a similar form to the M = 2 ensemble case, up to simple spin rotations.  By correcting for these known rotations, it allows for the deterministic preparation of the equivalent state generated by an S^z S^z interaction.  

The amount of entanglement that is generated using this scheme is of order log_2 N, and hence is macroscopic by our definition.   The resources required to distribute this do not involve N, and only depend upon M, the chain length.  The reason is that only total spin operations are used in the protocol from start to finish. In this sense this has a considerable scaling advantage over preparing the entanglement microscopically.  The parallelization occurs since the resources to control an ensemble are the same as that of a qubit. The use of total spin operators means that the control pulses illuminate the entire ensemble simultaneously, which is the origin of the parallelization.  The type of entanglement that is distributed is suitable for various protocols that are based on the S^z S^z interaction, since this is ultimately the form of entanglement that is distributed.  This includes schemes such as quantum teleportation <cit.>, Deutsch-Jozsa algorithm <cit.>. Such tasks could be implemented on a distributed quantum computing platform.  The distribution of entanglement at macroscopic quantities is a task that we anticipate will become more important as the quantum internet gains more prominence.  Our protocol offers the possibility of distribute entanglement at scale, which can serve as a fundamental primitive in quantum network infrastructure.  





















§ ACKNOWLEDGMENTS


A. N. P and I. D. L.  are supported by the Russian Science Foundation (grant no 23-21-00507). T.B. is supported by the National Natural Science Foundation of China (62071301); NYU-ECNU Institute of Physics at NYU Shanghai; Shanghai Frontiers Science Center of Artificial Intelligence and Deep Learning; the Joint Physics Research Institute Challenge Grant; the Science and Technology Commission of Shanghai Municipality (19XD1423000,22ZR1444600); the NYU Shanghai Boost Fund; the China Foreign Experts Program (G2021013002L); the NYU Shanghai Major-Grants Seed Fund; Tamkeen under the NYU Abu Dhabi Research Institute grant CG008; and the SMEC Scientific Research Innovation Project (2023ZKZD55).






§ EXPRESSIONS FOR ODD M



Here we show the corresponding expressions for odd M for the even M expressions given in the main text. 

The odd M counterpart of (<ref>) is 


    | Ψ_q⃗⟩=1/√(2^(M+1)N/2)∑_k_1,k_3,…,k_M=0^N√( C^k_1_N  C^k_M_N)
    ×( ∏_j=2^M-1Ω_q_j^(j)) |k_1⟩|k_M⟩ .


In order to calculate the fidelity for odd case, we take the overlap with the projected M = 3 state (<ref>). The fidelity in this case equals 


    F = | ⟨ψ_3 | Ψ_q⃗⟩ |^2/p_q⃗   =1/2^N(M+5)/2 p_q⃗
    ×|∑_k_1, k_3, …, k_M  
     C_N^k_1C_N^k_MΩ_N^(2)^* ( ∏_j=2^M-1Ω_q_j^(j))   |^2 /∑_k_1,k_3=0^N  C^k_1_N  C^k_3_N | Ω_N^(2) |^2


when we measure N on all  intermediate ensembles. 



























































































The odd M counterpart of (<ref>) is


    | ψ_M ⟩=1/√(L^(M+1)/2)∑_m_1,m_3,…,m_M=0^L-1
    × |  C_m_1⟩_1  |e^2 i(m_1-m_3)π/L /√(2),1/√(2)⟩⟩_2 |  C_m_3⟩_3
    |e^2 i(m_3-m_5)π/L /√(2),1/√(2)⟩⟩_4  
    …⊗ |e^2 i(m_M-2-m_M)π/L /√(2),1/√(2)⟩⟩_M-1 |  C_m_M⟩_M .


Following similar steps to that in Sec. <ref> we get the odd M counterpart of (<ref>)


    | Ψ_q⃗^approx⟩ =  ⊗_j=2^M-1  |q_j ⟩_j^(x)⟨ q_j |_j^(x)  |ψ_M ⟩∝
       ∑_m=0^L-1 e^2 π i m n̅_odd/L  |  C_m⟩_1 |  C_m - m̅_even⟩_M  .


where for (<ref>) we use the modified definitions


    m̅_even    =  
    ∑_j=2,4, …, M-1m̅_j  
    n̅_odd    = ∑_j=3,5,…, M-2n̅_j  .































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 Paper Title: Quantum Thermodynamics of Open Quantum Systems: Nature of Thermal Fluctuations.







neha@ctp-jamia.res.in

devvrat.1@iitj.ac.in


subhashish@iitj.ac.in

Department of Physics, Indian Institute of Technology, Jodhpur, India



§ ABSTRACT 

We investigate the thermodynamic behavior of open quantum systems through the Hamiltonian of Mean Force, focusing on two models: a two-qubit system interacting with a thermal bath and a Jaynes-Cummings Model without the rotating wave approximation. By analyzing both weak and strong coupling regimes, we uncover the impact of environmental interactions on quantum thermodynamic quantities, including specific heat capacity, internal energy, and entropy. Further, the ergotropy and entropy production are computed. We also explore the thermodynamic uncertainty relation, which sets an upper bound on the signal-to-noise ratio. 


Quantum Thermodynamics of Open Quantum Systems: Nature of Thermal Fluctuations
    Subhashish Banerjee
    
1Fusion@ÖAW, Institute for Theoretical Physics - Computational Physics, Graz University of Technology, 8010 Graz, Austria

2 Max-Planck-Institut für Plasmaphysik, 17489 Greifswald, Germany

3Institute of Plasma Physics, National Science Center “Kharkov Institute of Physics and Technology”, 61108 Kharkov, Ukraine

===============================================================================================================================================================================================================================================================================================================================



§ INTRODUCTION

The study of thermodynamics within the realm of open quantum systems is a burgeoning area that seeks to understand how classical thermodynamic principles apply when quantum effects are significant  <cit.>. This field is particularly relevant for systems interacting with their surroundings, leading to complex behaviors and phenomena not observed in isolated systems. 
A fundamental development in the theory of open quantum systems is the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) master equation <cit.>, which offers a framework for describing the time evolution of a system S under the influence of weak interactions with a large external environment. Advances in theoretical and experimental domains implore one to go beyond the scope of GKLS dynamics, specifically where strong-coupling non-Markovian effects are prominent <cit.>.

Open quantum systems that are non-Markovian demonstrate the intricate connection between a quantum system and its surroundings. Unlike their Markovian counterparts, non-Markovian systems retain a memory of their past, resulting in complex dynamical behavior  <cit.>. A prominent feature of non-Markovian evolution is the resurgence of quantum properties  <cit.>, the examination of which is crucial for comprehending the system's dynamics under varying levels of coupling with the environment. Studying the dynamics of thermodynamic quantities when the system has strong coupling with the environment in the non-Markovian regime presents a formidable challenge  <cit.>. Numerous techniques have been developed recently to address the dynamics of such systems. These encompass the Hamiltonian of mean force (HMF)  <cit.>, the method of hierarchical equations of motion, a numerical method utilizing the influence-functional formalism <cit.>, the reaction coordinate method, which explores strong coupling effects through mapping the Hamiltonian with a reaction coordinate  <cit.> and pseudomodes technique  <cit.>. 

Here, we make use of the Hamiltonian of Mean Force to investigate the strong-coupling non-Markovian effects. This concept of HMF has been previously studied in the Caldeira-Leggett Model <cit.>, Spin-Boson Model <cit.>, Quantum Brownian Motion <cit.> to ascertain the system's effective dynamics and thermodynamics and also in molecular systems <cit.> to study solvation dynamics and chemical reaction rates in complex environments. It has been observed that HMF provides consistent generalizations of the laws of thermodynamics and fluctuation relations <cit.>.

Recent advances in the study of open quantum systems have yielded significant insights into the behavior of quantum thermodynamic quantities <cit.>, and with technological advancements, this is being leveraged towards quantum thermal devices <cit.>. The specific heat capacity of the systems coupled to thermal reservoirs demonstrates non-trivial temperature dependencies influenced by quantum effects <cit.>. Along with ubiquitous entropy, another quantity prominent in quantum thermodynamics is entropy production, which, along with system dynamics, also depends on global state evolution <cit.>. The Fluctuation Dissipation Relation (FDR), which connects response functions to correlation functions, is fundamental in understanding dissipation and fluctuations  <cit.>. It turns out that in the strong coupling regime of thermodynamics, FDR has modifications coming from the specific heat capacity and another term that is an essential ingredient in computing the system's internal energy <cit.>. Additionally, the concept of ergotropy, representing the maximum extractable work from a quantum system via unitary transformations, was introduced  <cit.> and is a crucial ingredient in characterizing the thermodynamic behavior of a system. These studies collectively advance our understanding of the complex behavior and properties inherent to open quantum systems.

Another development in this field is a bound on signal-to-noise ratio for estimation of temperature  <cit.>. This involves tools of quantum metrology  <cit.> in the form of the quantum Cramer-Rao inequality for an unbiased estimation of the temperature of the system. As is well known, the quantum Cramer-Rao inequality is bounded by the Fisher information  <cit.>. The other ingredient in this bound is the modified fluctuation-dissipation relation involving the specific heat capacity of the system as well as the Wigner-Yanase-Dyson skew information and another term that is pertinent to computing the system's internal energy  <cit.>.

Dynamics of two-qubit systems interacting with a squeezed thermal bath has revealed significant insights into entanglement dynamics, coherence, and quantum correlations <cit.>. The Jaynes-Cummings model  <cit.>—which describes a single two-level system interacting with a single-mode resonator—has seen significant progress in improving quantum measurement techniques specifically in quantum sensing and metrology and also in understanding the interplay between coherence and quantum thermal states <cit.>.
In this article, we discuss the quantum thermodynamics of the following open quantum systems models: (i) a two-qubit model interacting with a thermal bath and (ii) the Jaynes-Cummings Model without the rotating wave approximation, using the Hamiltonian of Mean force across all coupling regimes. 


The structure of the paper is as follows. The preliminary section (Sec. II) includes an introduction to the Hamiltonian of Mean Force, certain thermodynamical potentials, thermodynamic quantifiers like ergotropy and entropy production, and a brief explanation of quantum information theoretic quantities such as WYD skew-information and Quantum Fisher information. In Sec. III, we have briefly introduced the models under study and discussed their quantum thermodynamic properties using the Hamiltonian of Mean Force. This is followed by the conclusion in Sec. IV.


§ PRELIMINARIES

This section is divided into two categories. The first category focuses on Quantum Information-theoretic quantities, where we define quantum uncertainty, skew information, and Fisher information. The second category pertains to quantum statistical mechanics, introducing the Hamiltonian of Mean Force and thermodynamic potentials such as entropy and specific heat capacity. Additionally, we provide a brief introduction to entropy production and ergotropy.



 §.§ Quantum Information-Theoretic Quantities



  §.§.§ Quantum Uncertainty and Wigner-Yanase-Dyson Skew Information

Skew information <cit.>, quantifies the quantum uncertainty in a state relative to an observable defined by I(ρ, Y) = -1/2 Tr ([√(ρ), Y]^2). It measures the non-commutativity between the state and the observable and has been extensively studied and generalized. The concept of skew information was extended through the definition I_η (ρ, Y)= -1/2([ρ^η,Y][ρ^1-η,Y]), commonly referred to as WYD entropy <cit.>, where 0<η<1. When η = 1/2, it simplifies to the well-known skew information. This generalized form can be expressed as:

    I_η (ρ, Y)= (ρ Y^2)- (ρ^η Y ρ^1-η Y).

In comparison, the usual variance of an observable Y for the quantum state ρ is given by <cit.>,

    Var(ρ, Y) = (ρ Y^2) - ((ρ Y))^2= Q[ρ̂, Ŷ]+ K[ρ̂, Ŷ].

The variance of the observable Y is influenced by both quantum and classical uncertainties inherent in the system. The measures of quantum and classical uncertainties, Q[ρ̂, Ŷ] and K[ρ̂, Ŷ], respectively, are defined as follows <cit.>

    Q[ρ̂, Ŷ] = ∫_0^1 Q_a[ρ̂, Ŷ]   da,

where Q_a[ρ̂, Ŷ] = - 1/2( [Ŷ, ρ̂^a] [Ŷ, ρ̂^1-a] ) with a ∈ (0,1). This measure Q[ρ̂, Ŷ] quantifies the quantum uncertainty associated with the observable Ŷ in the state ρ̂. The negative sign reflects the non-commutativity of ρ̂ and Ŷ, capturing the essence of quantum fluctuations.
Similarly,

    K[ρ̂, Ŷ] := ∫_0^1 K_a[ρ̂, Ŷ]   da,

where K_a[ρ̂, Ŷ] := ( ρ̂^aδŶρ̂^1-aδŶ) with a ∈ (0,1). Here, δŶ = Ŷ - ⟨Ŷ⟩ represents the deviation of Ŷ from its expectation value. The measure K[ρ̂, Ŷ] captures the classical uncertainty, reflecting the statistical spread of outcomes due to classical probabilities. The parameter a indicates that there is no unique way of decomposing the variance into quantum and classical components. These quantum and classical uncertainty parameters play a very important role in the thermodynamics of quantum systems which is explored in the later sections.



  §.§.§ Quantum Fisher Information

Quantum Fisher Information (QFI) is a crucial concept in quantum metrology and information theory, representing the sensitivity of a quantum state to changes in a parameter. It is a fundamental tool for determining the ultimate precision limits in parameter estimation tasks, such as those encountered in quantum thermometry. Mathematically, QFI is defined through the symmetric logarithmic derivative (SLD), Λ_θ

    F(θ) = Tr[ρ(θ) Λ_θ^2],
 where ρ(θ) is the density matrix of the quantum state dependent on the parameter θ.
This quantity is pivotal as it establishes the quantum Cramér-Rao bound (QCRB), expressed as

    Δθ≥1/√(N F(θ)),

where N is the number of measurements, dictating that the variance of any unbiased estimator θ is inversely proportional to the QFI. It has been recently proved  <cit.> that for an exponential state ρ̂_θ= exp(-Φ̂_θ)/𝒵_θ the QFI concerning the parameter θ is bounded by,

    F(θ) ≤ K[ρ̂_θ, Ŷ_θ],
 where, Ŷ_θ is the thermodynamic observable defined as Ŷ_θ=∂_θΦ̂_θ ,and K[ρ̂_θ, Ŷ_θ] is the classical uncertainty defined in Eq. (<ref>).
The spectral decomposition of the state ρ̂_θ= exp(-Φ̂_θ)/𝒵_θ is given by ρ̂_θ= ∑_n p_n|ψ_n⟩⟨ψ_n| where the eigenstates satisfy 
Φ̂_θ| ψ_n⟩  = λ_n| ψ_n⟩. From this spectral decomposition, the QFI can be expressed as  <cit.>,

    F(θ) = 2 ∑_n,m| ⟨ψ_n| ∂_θρ̂_θ| ψ_m⟩|^2/p_n + p_m.
 

In this work, we have considered this unbiased estimator parameter θ as Temperature. For a quantum thermal state, the Fisher Information for the temperature is proportional to the specific heat capacity of the system as F(T)= C_S/T^2  <cit.>.



 §.§ Quantum Statistical Mechanics Related Quantities



  §.§.§ Hamiltonian of Mean Force

A quantum system 𝒮 interacting with a thermal bath ℬ is described by the Hamiltonian,

    Ĥ_𝒯 = Ĥ_S ⊗𝕀̂_B  + 𝕀̂_S ⊗Ĥ_B + 𝒱̂_SB,

where Ĥ_S and Ĥ_B are the hamiltonian of system and bath respectively, while 𝒱̂_ SB is an interaction term of arbitrary strength. Here, we will consider situations where the environment is huge compared to the system, i.e. the operator norms fulfill ||Ĥ_B|| ≫ || Ĥ_S||, ||𝒱̂_ SB||. The Hamiltonian of mean force takes into account the average effects of the environment on the subsystem.
The global equilibrium state of the system and the reservoir at temperature T is denoted by ζ̂_SB, which is of Gibbs form,

    ζ̂_SB := exp(-βĤ_𝒯)/𝒵_SB,

where, 𝒵_SB = _SB[exp(-βĤ_𝒯)]
is the partition function for SB. Also, β = (k_bT)^-1, the Boltzmann constant k_b is set to unity throughout. The interaction term causes the reduced state of the system S, ζ̂_S(T) =  Tr_B[ζ̂_SB(T)] to deviate from the thermal state unless the coupling is very weak (the magnitude of interaction term 𝒱̂_SB is much lesser than the magnitude of the bare system hamiltonian H_S). Thus, for the purpose of calculating internal energy in the strong coupling regime, the system's bare Hamiltonian would not be useful. 

This issue can be resolved by rewriting the state of the system S as an effective Gibbs state ζ̂_S(T) = exp(-βĤ_𝒮^* (𝒯))/𝒵_S^*, where the partition function for the system S can be expressed as the ratio 

    𝒵_S^* = 𝒵_SB/𝒵_B,
 where 𝒵_SB= _SB[e^-βH_SB] and 𝒵_B= _B[e^-βĤ_B] and,

    Ĥ_𝒮^* (𝒯) := - 1/βln(  Tr_B[exp(-βĤ_̂𝒯̂) ]/ Tr_B[exp(-βĤ_̂ℬ̂)]),

is the Hamiltonian of Mean Force  <cit.>. 
This operator can be interpreted as an effective Hamiltonian describing S, and unlike the bare Hamiltonian Ĥ_S, it implicitly depends upon both the temperature T and the interaction 𝒱̂_SB. In the weak coupling regime, this reduces to the bare system Hamiltonian. HMF is essential for understanding thermodynamic properties such as free energy, entropy, and heat capacities in systems where direct interactions with an environment cannot be ignored. It allows for the formulation of generalized thermodynamic potentials and facilitates the study of equilibrium and non-equilibrium processes, providing insights into phenomena such as quantum dissipation and decoherence.



  §.§.§ Thermodynamic Potentials

The partition function in statistical mechanics contains information regarding the distribution of occupation probabilities among the various microstates of the system S. It enables the calculation of important thermodynamic potentials like free energy, internal energy, and entropy.
The internal energy of the system S can  be written as <cit.>,

    U_S(T) = U_SB(T) - U_B^'(T).

This U_S(T) is simply the difference between the total energy U_SB(T)= -∂_βln𝒵_SB and the energy of the reservoir, U_B^'(T)= -∂_βln𝒵_B.
U_S(T) can be expressed as an expectation value <cit.>, 

    U_S(T)    = ⟨Ê_S^*(T) ⟩ =  Tr[ Ê_S^*(T). ζ̂_S ] 
       =  Tr[∂_β ( βĤ_𝒮^* (𝒯) ). ζ̂_S ] ,

of the observable,

    Ê_S^*(T) = ∂_β [ βĤ_𝒮^* (𝒯) ] 
        = Ĥ_S+  ∂_β[β (Ĥ_𝒮^* (𝒯)- Ĥ_S)].

This Ê_S^*(T) can be interpreted as the effective energy operator describing the system whose eigenstates are referred to as the system's energy eigenstates.  
With the inclusion of this operator, the fluctuations in the internal energy are considered as:

    U_S= √( Var[ζ̂_S, Ê_S^*]).

It is significant to remember that Ê_S^*(T) depends on both the temperature T and the system-environment coupling 𝒱̂_ SB. This Ê_S^*(T) differs from both the bare system hamiltonian Ĥ_S and hamiltonian of mean force Ĥ_S^*(T) <cit.>. From Eq. (<ref>) we can observe that only in those cases where Ĥ_S^*(T) becomes temperature independent will Ĥ_S^*(T) be almost equivalent to Ê_S^*(T). One example is the bilinear coupling model for Quantum Brownian motion  <cit.>. 

The entropy of the system is calculated as

    𝒮_S = -  Tr[ (lnζ̂_S) . ζ̂_S] + β^2  Tr[ (∂_βĤ_S^*).ζ̂_S  ].

Here, we recognize the first contribution as the von Neumann entropy for the system, while the second term stems from the temperature dependence of the HMF. This implies that the thermodynamic entropy is generally not equivalent to the information in the equilibrium state when interactions are present.

In the weak-coupling limit, it is well established that the specific heat capacity C_S is proportional to the variance in internal energy, expressed as  <cit.>

    C_S = β^2 Δ U_S^2.

This is a standard result of the fluctuation-dissipation relation, which connects the rate of change of the system's energy with temperature to the energy fluctuations at equilibrium. 
Recent research has indicated that for open quantum systems of the form given in Eq. (<ref>), the heat capacity can become negative at low temperatures, suggesting that it is not generally proportional to a positive variance. Further, in the presence of strong coupling, the fluctuation-dissipation relation (FDR) includes two additional contributions <cit.>,

    C_S(β) = β^2 Δ U_S^2 - β^2 Q[ζ̂_S, Ê_S^*] - 1/β^2⟨∂_βÊ_S^* ⟩,

where all the terms are as defined above.
This equation implies that C_S(β) can be less than β^2 Δ U_S^2 and even negative.
The first correction term arises from quantum fluctuations in energy, represented by the average WYD information related to the observable Ê_S^*. The second correction term ⟨∂_βÊ_S^* ⟩ accounts for the temperature dependence of Ê_S^* and remains in the classical limit. In the weak coupling limit, where Ê_S^* = Ĥ_S^*, these correction terms disappear, making the heat capacity proportional to the energy variance.
When considering temperature as the unbiased parameter θ, the Cramer-Rao bound inequality in Eq. (<ref>) can be written as Δβ≥1/√(F(β)) for a single measurement N=1. Thus, Eq. (<ref>) can be reformulated as F(β) ≤ K[ζ̂_S, Ê_S^*]. From this, a modified thermodynamic uncertainty relation was obtained as <cit.>,

    Δβ≥1/√(Δ U_S^2 - Q[ζ̂_S, Ê_S^*]) .

Comparing Eqs. (<ref>) and (<ref>), the uncertainty in the temperature in the strong coupling regime can be expressed as

    Δ T ≥T/√(C_S(T) - ⟨∂_TÊ_S^* ⟩) .

This expression sets a lower bound on the thermal fluctuations Δ T. In the scenario of strong coupling, the best possible signal-to-noise ratio T/Δ T for determining the temperature of the system S is constrained by both the specific heat capacity and an additional dissipation term ⟨∂_T Ê^*_S ⟩, which can be negative or positive. In the case of weak coupling, this additional dissipation term is zero. Therefore, in that case, the optimal signal-to-noise ratio T/Δ T depends solely on the specific heat capacity, indicating that accurate temperature measurement requires a large heat capacity.
 


 §.§ Quantum thermodynamic quantifiers

Here, we briefly discuss quantum thermodynamic quantifiers, particularly ergotropy and entropy production. 


  §.§.§ Ergotropy

Ergotropy is the maximum work that can be extracted from a quantum system <cit.>. Consider a quantum state ρ, whose spectral decomposition is given by 

    ρ = ∑_i r_i|r_i⟩⟨r_i|,

where r_i's (in the order r_1>r_2>r_3>...) are the eigenvalues of ρ, and the spectral decomposition of the system's Hamiltonian H_S is given by

    H_S = ∑_j e_j|e_j⟩⟨e_j|,

where e_j (in the order e_1<e_2<e_3...) are eigenvalues of the system's Hamiltonian. The ergotropy for this system is given by 

    𝒲[ρ(t)] = ∑_j, i r_ie_j(|⟨r_i||e_j⟩|^2 - δ_ij).




  §.§.§ Entropy production

The transfers undergone by the system during its evolution are alluded to as entropy production. It accounts for the irreversibility that arises due to discarding the bath degrees of freedom. 
Consider a system bath (S-B) evolution of an arbitrary initial state ρ_SB(0) dictated by the unitary operator U, such that ρ_SB(t) = Uρ_SB(t)U^†. Entropy production <cit.> is given by 

    Σ = ℐ(S:B) + S[ρ_B(t)||ρ_B(0)],

where ℐ(S:B) is the mutual information given by the von Neumann entropies (S(ρ) = - Tr(ρlnρ)) of the subsystems S and B and the joint system SB, that is, 

    I(S:B) = S(ρ_S) + S(ρ_B) - S(ρ_SB).

S(ρ||σ) =  Tr(ρlnρ - ρlnσ) is the quantum relative entropy. The above relation can further be simplified as

    Σ = S[ρ_SB(t)||ρ_S(t)⊗ρ_B(0)],

which is the Kullback-Leibler divergence between the joint system-bath evolved state and the state ρ_S(t)⊗ρ_B(0), where ρ_S(t) =  Tr_B(Uρ_SB(t)U^†). 



§ ILLUSTRATIVE OPEN QUANTUM SYSTEM MODELS

Here, we briefly discuss the models under consideration in this work. This is followed by a study of their corresponding quantum thermodynamic properties.




 §.§ Two-qubit system interacting with a thermal bath

We start with a two-qubit system interacting with a thermal bath via a dissipative interaction. 
The dissipative interaction between 2-qubits (a two-level atomic system) and the bath (represented as a three-dimensional electromagnetic field (EMF)) via the dipole interaction  <cit.> is described by the Hamiltonian,

    H    = H_S + H_B + H_SB
       = ∑_j=1^2 ħω_j S_j^z 
                     + ∑_k⃗sħω_k ( α_k⃗s^†α_k⃗s + 1/2) 
          - iħ∑_k⃗s⃗∑_j=1^2 [ μ⃗_j ·g⃗_k⃗s⃗( r⃗_j )(S_j^+ + S_j^-) α_k⃗s - h.c. ]

The two-qubits are modeled as two-level systems with excited state |e_j⟩, ground state |g_j⟩, transition frequency ω_j, and transition dipole moments μ_j. We assume that the qubits are located at different atomic positions r_j. The transition dipole moments are dependent on r_j.
S_j^+ = |e_j⟩⟨ g_j| and S_j^- = |g_j⟩⟨ e_j| are the dipole raising and lowering operators, respectively, satisfying the commutation relation and 
S_j^z = 12 (|e_j⟩⟨ e_j| - |g_j⟩⟨ g_j|)
is the energy operator of the jth-atom. 
α_k⃗s^†, α_k⃗s are the creation and annihilation operators of the field mode k⃗s with wave vector k⃗, frequency ω_k and the polarization index s. The system-reservoir coupling constant is

    g_k⃗s (r⃗_j) = (ω_j/2ϵħ V)^1/2e_k⃗s e^i k⃗.r⃗_j.

V is the normalization volume, and e_k⃗s is the unit polarization of the field. This equation implies that the system-reservoir coupling constant depends on the atomic position r_j. This leads to two regimes: one where the inter-qubit spacing is less than the length scale set by the bath, that is, the collective regime, and the other where the inter-qubit spacing is comparable with the length scale set by the bath, the independent regime <cit.>. In this work, the collective regime is considered for further analysis. 

For simplification, we assume that both the qubits have the same transition frequency and the first qubit is at the origin while the distance between them is ξ. Further, k⃗ is taken to be along r⃗_12. The dipole moments μ⃗_j's are taken to be equal for both qubits ( μ_1 = μ_2 = μ) and along the direction of a polarization vector, such that μ⃗_j.e⃗_k⃗s = μ. Incorporating these quantities, the Hamiltonian in Eq. (<ref>) (for ħ = c = 1) can be rewritten as

    H    = ω_0(S_1^z + S_2^z) + ∑_k⃗ω_k ( α_k^†α_k + 1/2) 
       -i ∑_kλ√(ω_k)[(S_1^+ + S_1^-)(α_k - α_k^†).
       . +(S_2^+ + S_2^-)(α_ke^iω_kξ - α_k^† e^-iω_kξ) ],

where λ  = (μ/√(2ϵ_0V)). 
The reduced density matrix of the two-qubit system in the interaction picture and the standard Born-Markov rotating wave approximation (RWA) was obtained in <cit.> by assuming separable initial system-bath correlation and taking a trace over the bath. 
When λ√(ω_k)∼ω_0, the interaction strength between the system and the bath is comparable to the system's natural frequency ω_0. This characterizes a strong coupling regime, wherein the system and the bath can efficiently exchange energy, resulting in significant mutual influence on each other's dynamics. 
Conversely, when λ√(ω_k)≪ω_0, the interaction strength is considerably smaller than the system's natural frequency. This corresponds to a weak coupling regime, where the interaction between the system and the bath is minimal. Consequently, the system's dynamics are less perturbed by the bath, and energy exchange between the two is limited.

HMF for this model is calculated using Eqs. (<ref>) and (<ref>), where ℋ_𝒯 is the total Hamiltonian of the system and bath H. The numerical form of the HMF for this model is given by 

    H^*_S = [ a 0 0 e; 0 b f 0; 0 g c 0; h 0 0 d ].

It is observed that in the weak coupling limit, the off-diagonal elements e, f, g, h tend to become zero, and the above Hamiltonian becomes equal to the bare Hamiltonian of the system H_S.   
The specific heat capacity and the corresponding variables in its expression are plotted in Fig. <ref>.

As can be observed, the quantum uncertainty and the quantity pertaining to E^*_S are zero in the weak coupling regime but non-zero in the strong coupling regime. This is consistent with the generalized fluctuation-dissipation relation, Eq. (<ref>).

An upper bound on the signal-to-noise ratio, obtained from the modified thermodynamic uncertainty relation, Eq. (<ref>), is depicted in Fig. <ref>. 

The bounds are well satisfied for both the weak and strong coupling regimes, with the weak coupling regime depicting the tighter bound, in fact, saturating the bound.

The entropy of the reduced dynamics of the system is plotted in Fig. <ref>. 

The weak coupling behavior of the entropy is along the expected line, whereas for strong coupling, the entropy is negative at lower temperatures. However, it approaches the entropy in the weak-coupling regime as the temperature is raised. 

The ergotropy is calculated using the state ζ_S(T) = e^-β H^*_S/Z^*, and is plotted in Fig. <ref>.

From this figure, one can infer that in the weak coupling regime, the HMF coincides with the bare Hamiltonian of the system, and the ergotropy becomes zero, as the state ζ_S(T) becomes the Gibbs state for the bare Hamiltonian, which is not the case in the moderate and strong coupling regimes. Further, ergotropy is composed of coherent and incoherent parts. It is observed that for the state ζ_S(T) of the two-qubit model, the incoherent contribution is zero. The non-zero ergotropy comes from the coherent part. This is due to the fact that the state ζ_S(T) has coherence terms present in it at stronger system-bath couplings, which are absent in the weak coupling regime.

Considering the single-mode resonator approximation of the above mode (for single k in Eq. (<ref>)), we calculate the entropy production, Eq. (<ref>), using the Bell state 1/√(2)(|00⟩ + |11⟩) as the initial state of the two-qubit system, and ρ_B(0) = e^-β H_B/ Tr(e^-β H_B) as the initial state of the bath. The dynamics of the total and the reduced systems are obtained using U {ρ_S(0) ⊗ρ_B(0)} U^† and Tr_B [U {ρ_S(0) ⊗ρ_B(0)} U^†], respectively, where U = e^-iHt. 

The variation of entropy production Σ as a function of temperature, at time t = 1, is plotted in Fig. <ref>. As can be observed from Fig. <ref>, in this case, the hierarchy of the entropy production with respect to the coupling strength is similar to that of the ergotropy. 



 §.§ A single two-level system interacting with a single-mode resonator

Here, we take an example of a two-level system interacting with a single-mode field, that is, the Jaynes-Cummings (JC) model without rotating wave approximation <cit.>. The Hamiltonian (for ħ=1) of the total system is given by 

    H    = H_S + H_B + H_SB
       = ω_0/2σ^z + ω_c â^†â + λσ^x(â + â^†).

We find out the HMF numerically for this model using Eq. (<ref>). It is found to be diagonal in form, though not equal to H_S in the strong coupling regime. It is, however, the same as the bare Hamiltonian in the weak coupling regime. The specific heat capacity C_S(T) and related quantities as a function of temperature are plotted in Fig. <ref>. 

It is observed that the quantum uncertainty is zero in both weak and strong coupling limits. The term ⟨∂_T Ê^*_S⟩ is non-zero for strong coupling but vanishes in the weak coupling limit. The generalized fluctuation-dissipation relation, Eq. (<ref>), is satisfied.
Further, C_S(T) is negative in the strong-coupling regime, which was also observed in <cit.>.


The upper bound on the signal-to-noise ratio is depicted in Fig. <ref> and is observed to be tight both for the weak and strong coupling regimes. From this, one could infer the local temperature attained by the system S. The entropy of the system using HMF is plotted in Fig. <ref> and is along expected lines. 


The ergotropy is calculated using the state ζ̂_S(T) = e^-β H^*_S/Z^*_S, which is zero due to the absence of coherence, and the diagonal terms having Tr[σ^zζ̂_S(T)]<0, that is, the population of the excited state is always lower than the ground state of the system in the state ζ_S(T) <cit.>. 

The entropy production for the JC model is calculated using the initial states |ψ⟩_S = 1/√(2)(|0⟩ + |1⟩) and ρ_B(0) = e^-β H_B/ Tr[e^-β H_B] of the system and the bath, using Eq. (<ref>), where the unitary evolution is given by U = e^-iHt. The variation of entropy production as a function of temperature is shown in Fig. <ref> and exhibits the same pattern as in the earlier model.



§ CONCLUSIONS

In this work, we studied the quantum thermodynamic behavior of open quantum system models. To understand the effect of weak and strong coupling, use was made of the Hamiltonian of mean force, which could be interpreted as an effective Hamiltonian of the system instead of the system's bare Hamiltonian in the strong coupling regime. The thermodynamic potentials, particularly specific heat capacity, internal energy, and entropy, were studied using this Hamiltonian of mean force.  Further, we also discussed the thermodynamic uncertainty relation that provides an upper-bound to signal-to-noise ratio. This upper bound was computed using the specific heat capacity of the system of interest and a quantity used for computing the internal energy of the system. The behavior of the above thermodynamic quantities was illustrated in two models. First, we considered a two-qubit model interacting with an electromagnetic field, and next the Jaynes-Cumming model without the rotating wave approximation. The generalized fluctuation-dissipation relation was satisfied for both models for all coupling limits, ranging from weak to strong coupling. The upper bound on the signal-to-noise ratio was tight for the two-qubit model in the strong coupling regime. In the weak coupling regime of the two-qubit model and for the single-qubit model for all coupling strengths, the inequality was saturated. Further, in the case of the two-qubit model, the Gibbs state found using the Hamiltonian of mean force had a non-zero capacity for doing work, as its coherent ergotropy was non-zero in strong and moderate coupling limits due to the presence of coherence in this state. The entropy production for both systems was greater in the strong coupling regime, reducing with decrease in the coupling strength. Our results provide insights into the complex behavior of open quantum systems in the context of quantum thermodynamics.




apsrev



 Paper Title: Preparation of high fidelity entangled cat states with composite pulses.





Fujian Key Laboratory of Quantum Information and Quantum Optics, Fuzhou University, Fuzhou 350108, China
Department of Physics, Fuzhou University, Fuzhou 350108, China


Fujian Key Laboratory of Quantum Information and Quantum Optics, Fuzhou University, Fuzhou 350108, China
Department of Physics, Fuzhou University, Fuzhou 350108, China

yehong.chen@fzu.edu.cn
Fujian Key Laboratory of Quantum Information and Quantum Optics, Fuzhou University, Fuzhou 350108, China
Department of Physics, Fuzhou University, Fuzhou 350108, China
Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research, RIKEN, Wako-shi, Saitama 351-0198, Japan

hbh@fzu.edu.cn
Fujian Key Laboratory of Quantum Information and Quantum Optics, Fuzhou University, Fuzhou 350108, China
Department of Physics, Fuzhou University, Fuzhou 350108, China



xia-208@163.com
Fujian Key Laboratory of Quantum Information and Quantum Optics, Fuzhou University, Fuzhou 350108, China
Department of Physics, Fuzhou University, Fuzhou 350108, China

§ ABSTRACT 
We propose a protocol for the preparation of high-fidelity entangled cat states with composite pulses. The physical model contains two Kerr-nonlinear resonators and a cavity. By properly designing the parameters, each Kerr-nonlinear resonator is confined in the cat-state subspace and the entangled cat states can be generated efficiently. We introduce composite two-photon drives with multiple amplitudes and frequencies to improve the fidelity of the entangled cat states in the presence of parameter errors. The performance of the protocol is estimated by taking into account the parametric errors and decoherence. Numerical simulation results show that, the protocol is insensitive to timing error and detuning error, and has strong robustness to decoherence. We hope the protocol may provide a method for preparing stable entangled cat states.


Preparation of high fidelity entangled cat states with composite pulses
    Yan Xia
    August 5, 2024
=======================================================================





§ INTRODUCTION

Quantum computers, which utilize entanglement superposition properties for information prosessing, possess the potential to outperform classical computers on some certain problems <cit.>, such as searching unsorted databases <cit.>.
However, quantum computers also face many challenges. For example, the experimental operating and environment noise may cause errors, especially on large-scale quantum computing <cit.>. Thus, quantum computing and its scaling will be limited.
To reduce error rate, protocols for quantum error correction have been developed in the past decades <cit.>. Encoding quantum information onto bosonic systems is beneficial to quantum error correction in some aspects <cit.>.
For example, bosonic systems can provide infinitely large Hilbert space, which could be used to effectively protect and process quantum information <cit.>.

Cat-state qubits are one of the promising bonsic quantum qubits in quantum information processing <cit.>, whose two logical qubits are usually represented by two orthogonal cat states (i.e., superposition of coherent states) <cit.>. Cat-state qubits have many unique advantages. For instance, they are noise biased <cit.>. The phase-flip error can be effectively suppressed, only the bit-flip error needs to be concerned in error correction. Thus, the number of building blocks for error correction can be significantly reduced <cit.>. In addition, cat qubits have an enhanced lifetime with error corrections <cit.>. Therefore, cat-state qubits have received much attention and many protocols in view of cat-state qubits have been proposed, such as nonadiabatic geometric quantum computation with cat-state qubits <cit.> and one-step parity measurement of N cat-state qubits <cit.>. Entangled cat states are also important for the demonstration of the fundamentals of quantum physics, and have wide applications in modern quantum technologies <cit.>. Protocols have been presented for producing entangled cat states, such as the generation of giant entangled cat states <cit.> and the preparation of multidimensional entangled cat states in cavity quantum electrodynamics (QED) <cit.>.

In this paper, we consider to use the Mølmer-Sørensen (MS) entangling gate for the preparation of entangled cat states <cit.>. The MS entangling gate has some specific characteristics. Firstly, it makes qubits independent of the decoherence of the motion mode  <cit.>. Secondly, it possesses a built-in noise-resilience feature against certain types of local noise <cit.>.
Therefore, the MS entangling gate has become an important resource  and inspired a wide range of research interests <cit.>.
For example, in 2021, Chen et al. proposed that entangled cat states can be prepared by using cat-code MS gate <cit.>. However, the protocol <cit.> could be sensitive to timing error. We note that some kind of composite pulses can play a role in robust control of quantum system and reduce errors <cit.>. For example, Haddadfarshi et al. pointed out that properly designed polychromatic control pulses can greatly reduce errors <cit.> and then Webb et al. demonstrated the protocol experimentally <cit.>. Shapira et al. generalized the MS entangling gate by using additional frequency components in the laser drive <cit.>. Wang et al. used suitably designed laser pulse with modulated amplitude and phase to improve the fidelity of the MS entangling gate <cit.>.

Inspired by the literatures above in this paper, we propose to use cat-code MS gate to prepare high-fidelity entangled cat states. We introduce a time-independent drive which is constructed by composite two-photon drives with different amplitudes and frequencies. Under such a drive, we can obtain more stable entangled cat states with appropriate parameters. Meanwhile, the sensitivities of the entangled cat states against errors are significantly reduced, and the robustness to decoherence are obviously improved.

The article is organized as follows. In Sec. <ref>, we describe the physical model, and give the concrete form of the Hamiltonian. The effective Hamiltonian is obtained by setting appropriate parameters, and the maximum entangled cat states are prepared with the MS entangling gate. In Sec. <ref>, we introduce the composite drives. Then we prepare the entangled cat states under the effective Hamiltonian, and numerically simulate the feasibility of the protocol. In Sec. <ref>, we evaluate the performance of the protocol against errors, and discuss the fidelity of the entangled cat states in the presence of the dephasing and single-photon loss. Finally, conclusions are given in Sec. <ref>.



§ CAT-CODE MS ENTANGLING GATE
 
The physical model consists of two Kerr-nonlinear cavities (𝒜_1,𝒜_2) with the same frequency ω, and another cavity (𝒜_0) with frequency ω_0. The Hamiltonian of this system in the interaction picture is given by (assume ħ=1)

    H=∑_k=1,2H_k^Kerr+H_I.

The interaction Hamiltonian H_I is <cit.>

    H_I=∑_k=1,2Ja_ka^†_0e^iΔ t+H.c.,

where J is the coupling strength, a_k(k=1,2) is the annihilation operator of the Kerr-nonlinear resonator, a_0 is the annihilation operator of the cavity, and Δ=ω_0-ω is the detuning. Each Kerr-nonlinear resonator is resonantly driven by a single-mode two-photon squeezing drive with frequency ω_p=2ω and amplitude Ω_p. The Hamiltonian of the Kerr parametric oscillators (KPOs) H_k^Kerr in Eq. (<ref>) is <cit.>

    H_k^Kerr=-Ka^†2_k a^2_k+(Ω_p a^2_k+Ω^*_p a^†2_k),

where K is Kerr nonlinearity coefficient <cit.>. To construct cat-state qubits, the Hamiltonian H_k^Kerr in Eq. (<ref>) can be rewritten as:

    H_k^Kerr=-K(a^† 2_k-Ω^*_p/K)(a_k^2-Ω_p/K)+|Ω_p|^2/K.

Equation (<ref>) clearly shows that when α=√(Ω_p/K), the two coherent states |±α⟩ are degenerate eigenstates of the annihilation operator a_k. Therefore, the superpositions of the coherent states <cit.>

    |C_±⟩_k=N_±(|α⟩_k± |-α⟩_k),

are also the degenerate eigenstates of H_k^Kerr. Here N_± are normalization coefficients. For simplicity, we set the parameters {K,Ω_p,J,Δ}>0 and α=α^*>0.


The subspace composed by |C_±⟩_k is called the cat-state subspace, which is separated from the other subspaces with an energy gap E_gap≃4Kα^2 <cit.>.
For large α, the annihilation operator a_k only causes the flip between the two cat states, i.e.,

    a_k|C_±⟩_k≃α|C_∓⟩_k.

When the condition E_gap≫ J is satisfied, the transition probability from the ground states |C_±⟩_k to the excited states is extremely small <cit.>.
Therefore, the dynamics of the system can be restricted in the cat-state subspace with the effective Hamiltonian

    H_eff   ≈   ∑_k=1,2Ω^2_p/K(|C_-⟩_k⟨C_-|+|C_+⟩_k⟨C_+|)            +Jα[|C_+⟩_k⟨C_-|(a_0e^-iΔt+a^†_0e^iΔt)+H.c.].

The first term in the right-hand side of Eq. (<ref>) can be dropped because it is the identity matrix (|C_-⟩_k⟨C_-|+|C_+⟩_k⟨C_+|) of the subspace. We define Pauli matrices σ^+_k=|C_-⟩_k⟨ C_+| and σ^-_k=|C_+⟩_k⟨ C_-|. Then the Hamiltonian H_eff in Eq. (<ref>) reduces to

    H_eff=2Jα S_x(a_0e^-iΔ t+a^†_0 e^iΔ t),

where S_x=1/2∑_k=1,2(σ^+_k+σ^-_k).

To well understand the evolution of the system, the exact solution of the evolution operator based on the effective Hamiltonian H_eff can be calculated with the Magnus expansion <cit.>:

    U(t)=exp{-i[(χ(t)a^†_0+H.c.)S_x+β(t)S^2_x]},


    χ(t)   =   2iJα/Δ[1-exp(iΔt)],      β(t)   =   (2Jα/Δ)^2(sinΔt-Δt).

Note that a large detuning Δ is necessary for the Magnus expansion to proceed. Setting t=T=2π/Δ, there has χ(T)=0 and β(T)=(2Jα/Δ)^2·(-2π). When choosing the parameter Δ=4Jα, then β(T)=-π/2. Thus the evolution operator at time T reads

    U(T)=exp(iπ/2S_x^2).

The U(T)  in Eq. (<ref>) is the well-known MS entangling gate <cit.>. The MS entangling gate is an important physical resource in quantum computing, for example, it can be applied to the steps of marking and state amplification in Grover's quantum search algorithm <cit.>. Under the action of U(T), we can realize the preparation of the entangled cat states. As shown in Eq. (<ref>), the left side of Eq. (<ref>) are the four different initial states (the input states), the right side are the corresponding maximum entangled cat states (the output state) obtained via U(T) <cit.>.

    |C_+⟩|C_+⟩|0⟩      →1/√(2)(|C_+⟩|C_+⟩|0⟩+i|C_-⟩|C_-⟩|0⟩),
    |C_-⟩|C_-⟩|0⟩      →1/√(2)(|C_-⟩|C_-⟩|0⟩+i|C_+⟩|C_+⟩|0⟩),
    |C_+⟩|C_-⟩|0⟩      →1/√(2)(|C_+⟩|C_-⟩|0⟩-i|C_-⟩|C_+⟩|0⟩),
    |C_-⟩|C_+⟩|0⟩      →1/√(2)(|C_-⟩|C_+⟩|0⟩-i|C_+⟩|C_-⟩|0⟩).

For convenience, the initial state of the system is set as |Ψ_in⟩=|C_+⟩|C_+⟩|0⟩, and the corresponding entangled cat state is |Ψ_out⟩=1/√(2)(|C_+⟩|C_+⟩|0⟩+i|C_-⟩|C_-⟩|0⟩) in this manuscript. All numerical simulation results in this section are obtained by using the full Hamiltonian given by Eq. (<ref>). Here, the fidelity of entangled state is given by F=⟨Ψ|ρ|Ψ⟩, where |Ψ⟩ is the target entangled cat state i.e., |Ψ_out⟩, and ρ is the density operator. The experimental parameters are chosen as α=2, the Kerr nolinearity K/2π=20MHz <cit.>, Ω_p=Kα^2, J=2πMHz, and Δ=4Jα.


In Fig. <ref>, we plot the time evolutions of the states |C_+⟩|C_+⟩|0⟩ and |C_-⟩|C_-⟩|0⟩. As is shown, the populations of both states are close to 50% around the gate time T. This means that the protocol is feasible, and the maximum entangled cat state can be successfully prepared when t=T. However, the high-fidelity entangled cat state can only exist when the time is extremely close to T. When the timing error δ_t deviates from 0, the fidelity of the entangled cat state |Ψ_out⟩ drops significantly, which can be seen in Fig. <ref>. Here, the relative error of the parameter x is defined via δ_x=(x'-x)/x, in which x' denotes the actual value and x is the ideal value.



§ PREPARATION OF HIGH-FIDELITY ENTANGLED CAT STATE
 

As shown in the second section, the fidelity of the entangled cat state is relatively sensitive to time error. In order to obtain stable and high-fidelity entangled cat state, we introduce composite two-photon squeezing drives with frequency ω_n and amplitude Ω_n. Then the total Hamiltonian H in Eq. (<ref>) changes to

    H'=∑_k=1,2H_k^Kerr'+H_I,


    H_k^Kerr'=   -   Ka^†2_k a^2_k+(Ω_p a^2_k+Ω^*_pa^†2_k)         +   ∑_n=1^N(Ω_n a^†2_k e^iδ_n t+Ω_n^*a^2_k e^-iδ_n t).

The last term in Eq. (<ref>) is the additional composite drives, and δ_n=ω_n-ω is the detuning of the nth two-photon drive. According to effective Hamiltonian theory <cit.>, the effective Hamiltonian in the limit of large detunings can be written as

    H_eff,N(t)=∑_n=1^N4JΩ_n/ε_n
    [a_k^†a_0^†e^i(Δ-δ_n)t+H.c.],

where 1/ε_n=(1/Δ+1/δ_n).
Similar to the derivation from Eq. (<ref>) to Eq. (<ref>), we can get a new effective Hamiltonian by restricting the Hamiltonian H_eff,N(t) to the cat-state subspace

    H_eff,N'(t)   =   2Jα S_x∑_n=1^N4Ω_n/ε_n
    [a_0^†e^i(Δ-δ_n)t+H.c.].

For simplicity, we define 4Ω_n/ε_n=r_n and ζ=(Δ-δ_n)/n with n=1,2,3,...,N. Then the effective Hamiltonian H_eff,N'(t) can be expressed as

    H_MS,N(t)=2Jα S_x∑_n=1^N r_n(a_0e^inζ t+a_0^† e^-inζ t).



To understand the construction of composite pulses, we introduce the dimensionless position operator x=1/√(2)(a_0+a_0^†)  and the momentum operator p=i/√(2)(a_0^†-a_0). The effective Hamiltonian in Eq. (<ref>) can be reformulated as

    ℋ_MS,N(t)=f(t)S_x x+g(t)S_x p.

where f(t)=2√(2)Jα∑_n=1^Ncos(nζ t) and g(t)=-2√(2)Jα∑_n=1^Nsin(nζ t). The propagator for the Hamiltonian in Eq. (<ref>) can be expressed as <cit.>

    𝒰(t)=e^-iℱ(t)S_x xe^-i𝒢(t)S_x pe^-i𝒜(t)S_x^2,

where

    ℱ(t)=      ∫_0^tf(t')dt'=2√(2)Jα/ζ∑_n=1^Nr_n/nsin(nζ t),𝒢(t)=      ∫_0^tg(t')dt'=2√(2)Jα/ζ∑_n=1^Nr_n/n[cos(nζ t)-1],𝒜(t)=      -∫_0^tℱ(t')g(t')dt',

and {r_n}^N_n=1 is a numerical set of r_n when N takes different values <cit.>

    r_n=(-1)^N-nN!/2^N√(2√(π)/(N-1)Γ(N+1/2))(N-1)!/(N-n)!(n-1)!.

Here Γ(a)=∫_0^∞ e^-xx^a-1dx.

As seen in Eq. (<ref>), the evolution of the operator 𝒰(t) follows the trajectory with dimensionless coordinates ℱ(t) and 𝒢(t) in the x-p phase-space. Equation (<ref>) shows that ℱ(t) and 𝒢(t) are respectively the integrations of f(t) and g(t) <cit.>. Ideally, when the time t=τ=2π/ζ, ℱ(τ)=0, 𝒢(τ)=0, and the phase-space trajectory of (ℱ(t), 𝒢(t)) is closed as seen in Fig. <ref>(a). Meanwhile, the operator  𝒰(τ)=e^-i𝒜(τ)S_x^2, that is, only the internal spin evolution is retained. However, errors will lead to the phase-space trajectory not closed completely, which in turn can causes infidelity of entangled cat states <cit.>. We can see from Fig. <ref>(b), the circular trajectory becomes more complete with the increase of N. In other words, we can reduce errors of the entangled cat states by increasing N.


The full Hamiltonian of the optimized protocol is written as

    H_N(t)=      ∑_k=1,2[-Ka^†2_k a^2_k+
    (Ω_p a^2_k+Ω^*_p a^†2_k)]            +∑_k=1,2[J_N(t)a_ka^†_0exp(iΔ t)+H.c.],

where J_N(t)=J e^-iΔ t∑_n=1^Nr_n e^inζ t is the composite drive. If the condition Δ=ζ is satisfied, J_1(t)=J, the case N=1 corresponds to the unoptimized protocol, which is disscussed in the second section.

In Fig. <ref>, we plot the population evolutions of the cat states with different N (N=1, 2, 4, 8). We can see from Fig. <ref> that the curves become more flat near time T with the increase of N. This means that more stable entangled cat state can be obtained by simply increasing N. Here, the parameters {α, K, Ω_p, Δ} are same as Fig. <ref>, and ζ=Δ.



§ ERROR AND DECOHERENCE ANALYSIS

In this section, we will investigate the performance of the optimized protocol. Specifically, we discuss the effects of parameters errors on the fidelity of entangled cat state in the Sec. <ref>. Then we analyze the robustness of the entangled cat state to dephasing and single-photon losses in Sec. <ref>. Here, we choose the values N∈[1,4,8].


 §.§ Robustness against parametric errors

It is inevitable that some parametric errors will cause infidelity in the entangled cat states because of the imperfect experimental operations. In this part, with the help of the numerical simulation, we will show the sensitivity of the protocol against parameter errors i.e., timing error δ_t, detuning error δ_Δ, and coupling error δ_J.

In Fig. <ref>, we numerically study the robustness of the  protocol for timing error δ_t. The error range is δ_t∈[-0.1,0.1] and the curve for N=1 corresponds to the unoptimized one. As expected, when N takes different value, the fidelity of the entangled cat state changes. Comparing the curves with N=1 and N=4, we can see the fidelity has been greatly improved. When N is set to a larger value i.e., N=8, the fidelity of entangled cat state is almost unaffected by timing error. That is, the protocol can enhance the robustness against timing error.

In actual experiments, imperfect experimental operations may cause inevitable detuning errors. We consider the range of detuning error δ_Δ as [-0.1, 0.1], and plot the fidelity versus detuning error δ_Δ in the Fig. <ref>. It can be seen from the figure that the fidelity of the entangled cat state for N>1 is much higher than that with N=1. The inset in Fig. <ref> shows that when the error range is [-0.05, 0.05], the fidelities are large than 0.99 when N>1 . This indicates that the increase of N can reduce the sensitivity to detuning errors.

Figure <ref> shows the fidelity against coupling error δ_J. As shown in Fig. <ref>, the fidelity is unaffected by the values of N, and the value of fidelity always greater than 0.97. That shows that the protocol cannot improve the robustness against errors in the coupling.




 §.§ Robustness against decoherence



In the process of gate operation, decoherence can cause infidelity. For the system discussed, we consider two types of noise: single-photon losses and pure dephasing. The master equation of the system can be written as:

    ρ̇   =   -i[H,ρ]+κ_0D[a_0]ρ+γ_0D[a^†_0a_0]ρ            +∑_k=1,2κ_k D[a_k]ρ+γ_kD[a^†_k a_k]ρ,

where ρ is the density operator of the system, D[o]ρ=oρ o^†-1/2(o^†oρ+ρ o^†o). For simplicity, we assume that κ_k=κ (k=1,2) is the single-photon losses rate, and γ_k=γ is the pure dephasing rate. Note that the influence of decoherence in the cavity mode 𝒜_0 is different from those in the KPOs. The decoherence of the cavity mode 𝒜_0 can be adiabatically eliminated for large Δ  <cit.>, so the system is insensitive to the decoherence of 𝒜_0. In the numerical simulation, we set κ_0=κ and γ_0=γ for convenience. To well understand the effect of decoherence, we project the system onto the eigenstates of H_k^Kerr. Here, the projection operator of the KPOs is

    P=∑_k=1,2[|C_±⟩_k⟨C_±|+∑_v=1^∞|ψ_±^e,v⟩_k⟨ψ^e,v_±|],

where |ψ^e,v_±⟩_k=N^e_±[D_k(α)∓| D_k(-α)]|v⟩_k, v represents the level of the excited states.

Then the master equation becomes

    ρ̇   ≃   -i[PHP,ρ]+κ_0D[a_0]ρ+γ_0D[a^†_0a_0]ρ            +∑_k=1,2κ_k D[Pa_kP]ρ+∑_k=1,2γ_kD[Pa^†_k a_kP]ρ.

When γ_k, κ_k are much smaller than the energy gap E_gap, the dynamics of the system is still well confined to the cat-subspace <cit.>.
For large α, the influence of the single-photon losses in the KPOs is described by the penultimate term in Eq. (<ref>)

    D'[a_k]=      D[Pa_kP]≃      α^2D[√(tanhα^2)|C_+⟩_k⟨ C_-|+√(α^2)|C_-⟩_k⟨C_+|]      +D[√(N_+/N_+^e)|C_+⟩_k⟨ψ^e,1_+|
    +√(N_-/N_-^e)|C_-⟩_k⟨ψ^e,1_-|]      +α^2D[√(N^e_-/N_+^e)|ψ^e,1_-⟩_k⟨ψ^e,1_+|
    +√(N_+^e/N_-^e)|ψ^e,1_+⟩_k⟨ψ^e,1_-|],

where |ψ^e,1_±⟩_k are the first-excited eigenstates of the Hamiltonian H_k^Kerr.
Note that the highly excited eigenstates of KPOs are not excited in the presence of single-photon losses. Therefore, we only consider the first-excited eigenstates in Eq. (<ref>). The second term in Eq. (<ref>) means that single-photon losses can only cause the transition from the excited eigenstate |ψ^e,1_±⟩_k to the ground state |C_±⟩_k. If the KPOs are initially in the cat-state subspace, they always remain in the cat-state subspace. Therefore, we can ignore the last two terms of the Eq. (<ref>) and obtain

    D'[a_k]ρ≃α^2/√(1-e^-4α)
    D[σ^x_k+ie^-2α^2σ_k^y]ρ,

where σ^x_k=|C_+⟩_k⟨ C_-|+|C_-⟩_k⟨ C_+|, σ^y_k=i(|C_-⟩_k⟨ C_+|+|C_+⟩_k⟨ C_-|). Bringing the Eq. (<ref>) into the Eq. (<ref>), we find that in the computational subspace, single-photon losses mainly lead to a bit-flip error σ^x_k, accompanied by an exponentially small phase flip error σ^y_k.

In Fig. <ref>, we plot the fidelity of the single-photon losses for different values of N. The single-photon losses rate is assumed as κ∈[0,0.1] MHZ, and the dephasing rate is assumed as γ=0. As can be seen from Fig. <ref>, the increase of N does not affect the fidelity too much. The fidelities can all reach above 0.9 when κ=0.1MHZ, which means that the optimized protocol still maintains a high fidelity in the presence of the single-photon losses. We also numerically simulate the dependence of the fidelity on the pure dephasing rate in Fig. <ref> and a similar result can be obtained.

Since single-photon losses and dephasing coexist in the system, so we numerically investigate the effect on the fidelity when single-photon losses and dephasing are present together. In Fig. <ref>, we plot the fidelity of the optimized protocol with N=8 versus κ=γ∈[0,0.05] MHz. The result shows that the optimized protocol maintains a high fidelity more than 0.9 even when photon losses and dephasing exist simultaneously.



§ CONCLUSION


In conclusion, we have investigated a protocol to use photonic cat-state qubits for preparing high-fidelity entangled cat states. The entangled cat states can be successfully obtained under the action of the cat-code MS gate. We further adopt composite pulses to improve the fidelity of the entangled cat states. The modulated time-independent composite drives can be realized by introducing composite two-photon squeezing drives. Numerical simulation results show that the protocol can provide strong robustness against timing error. The protocol also exhibits high insensitivity to other parametric errors with proper parameters. Furthermore, under the influence of decoherence, the protocol still maintains a high fidelity. That is, the higher fidelity of the entangled cat states are possible with the larger N of the parameter. On the other hand, employing composite pulses to prepare entangled states is meaningful to investigate. We hope that the proposal could offer a simple and feasible method for preparing stable and high-fidelity entangled cat states in quantum computation.


Y. Xia was supported by the National Natural Science Foundation of China under Grants Nos. 11575045, 11874114, the Natural Science Funds for Distinguished Young Scholar of Fujian Province under Grant 2020J06011 and Project from Fuzhou University under Grant JG202001-2. Y.-H. Chen was supported by the National Natural Science Foundation of China under Grants No. 12304390.





