Rapid thermalization of dissipative many-body dynamics of commuting Hamiltonians
Jan Kochanowski Alvaro M. Alhambra Angela Capel Cambyse Rouzé



1,2]Jan Kochanowskijan.kochanowski@inria.fr
3]Álvaro M. Alhambraalvaro.alhambra@csic.es
4,5]Ángela Capelac2722@cam.ac.uk
2,6]Cambyse Rouzérouzecambyse@gmail.com
[1]Max-Planck-Institute of Quantum Optics, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany
[2]Inria, Télécom Paris - LTCI, Institut Polytechnique de Paris, 91120 Palaiseau, France
[3]Instituto de Física Teórica UAM/CSIC, C/ Nicolás Cabrera 13-15, Cantoblanco, 28049 Madrid, Spain
[4]Fachbereich Mathematik, Eberhard Karls Universität Tübingen, Germany
[5]Department of Applied Mathematics and Theoretical Physics, University of Cambridge, United Kingdom
[6]Department of Mathematics, Technische Universität München, Germany


Rapid thermalization of dissipative many-body dynamics of commuting Hamiltonians
    [
    April 29, 2024
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Quantum systems typically reach thermal equilibrium rather quickly when coupled to a thermal environment. The usual way of bounding the speed of this process is by estimating the spectral gap of the dissipative generator. However the gap, by itself, does not always yield a reasonable estimate for the thermalization time in many-body systems: without further structure, a uniform lower bound on it only constrains the thermalization time to grow polynomially with system size.

Here, instead, we show that for a large class of geometrically-2-local models of Davies generators with commuting Hamiltonians, the thermalization time is much shorter than one would naïvely estimate from the gap: at most logarithmic in the system size. This yields the so-called rapid mixing of dissipative dynamics. The result is particularly relevant for 1D systems, for which we prove rapid thermalization with a system size independent decay rate only from a positive gap in the generator. We also prove that systems in hypercubic lattices of any dimension, and exponential graphs, such as trees, have rapid mixing at high enough temperatures. We do this by introducing a novel notion of clustering which we call “strong local indistinguishability” based on a max-relative entropy, and then proving that it implies a lower bound on the modified logarithmic Sobolev inequality (MLSI) for nearest neighbour commuting models.  

This has consequences for the rate of thermalization towards Gibbs states, and also for their relevant Wasserstein distances and transportation cost inequalities.
Along the way, we show that several measures of decay of correlations on Gibbs states of commuting Hamiltonians are equivalent, a result of independent interest. At the technical level, we also show a direct relation between properties of Davies and Schmidt dynamics, that allows to transfer results of thermalization between both. 









        








§ INTRODUCTION
 

Physical systems in nature are most often coupled to an external environment, with which they eventually equilibrate. For quantum ones, that coupling implies that their dynamics are described by Quantum Markov Semigroups (QMS) of the form {e^tℒ}_t≥ 0, which are generated by a Lindbladian super-operator ℒ(·).

This so-called dissipative evolution monotonically converges to a unique fixed point under a weak set of conditions <cit.>, which, roughly speaking, are satisfied as long as the evolution induced by the external coupling is sufficiently ergodic. Additionally, when that external coupling is very weak, and to an environment with a fixed temperature β, that unique fixed point is the Gibbs state

    σ^Λ=e^-β H_Λ/[e^-β H_Λ],

where H_Λ is the Hamiltonian of the system. The QMS describing those thermalization processes are then referred to as Davies maps <cit.>.

The Davies evolution is a Markovian approximation of the reduced state dynamics of a many-body spin system weakly-coupled to an infinite-dimensional environment in thermal equilibrium. This type of open system dynamics described by a master equations, which always has a QMS as a solution, is of high interest in the fields of quantum optics, condensed matter, chemical physics, statistical physics, quantum information, and mathematical physics. The interest in Markovian descriptions of open system dynamics has been further motivated by developments in quantum information theory and the study of decoherence. Davies evolutions, originally studied in <cit.>, frequently feature in the literature concerning thermalization of quantum systems, both from the physical and computational perspectives <cit.>. 

One of the more important aspects to understand about these processes is: if the Gibbs state is always reached, independently of the initial conditions, how quickly does that happen? The speed of convergence to equilibrium or thermalization can be expressed through the notion of the mixing time.  Write ρ_t:=e^tℒ_*(ρ), and let 𝒟(ℋ) be the set of normalised density operators. Then for ϵ>0, it is defined as

    t_mix(ϵ) := inf{t≥ 0   |   ∀ρ∈𝒟(ℋ)  ρ_t-σ_1≤ϵ}.


The most frequent way of estimating this mixing time, both in quantum and classical scenarios, is through the spectral gap of the generator λ(ℒ). 
This can be expressed variationally through the Poincaré inequality as 

    λ^_σ(X_t) ≤ -d/dt|_t=0^_σ(X) = -⟨ X,ℒ(X)⟩^_σ,
 where ⟨ X,Y⟩_σ^ := [√(σ)X^*√(σ)Y] and  ^_σ(X):=  ⟨ X-[σ X],X-[σ X]⟩_σ^ (see <Ref>).  The spectral gap is the largest constant λ which satisfies this inequality for all X∈ℬ(ℋ) <cit.>, i.e.

    λ(ℒ) := inf_X∈ℬ(ℋ)-⟨ X,ℒ(X)⟩^_σ/^_σ(X).

This directly implies exponential decay of the variance, i.e.
^_σ(X_t)≤ e^-λ(ℒ)t^_σ(X),
from which it follows that ρ_t-σ_1≤σ^-1^-1/2e^-λ() t, so that

    t_mix(ϵ)≤1/λ(ℒ)log(ϵ^-1σ^-1/2) .
 
While this inequality is often a good approximation in small systems, it can be an enormous overestimation of the mixing time in many-particle settings. In that case, σ^-1^-1=e^𝒪(|Λ|) with |Λ| the system size and the upper bound of Eq. (<ref>) scales as poly(|Λ|). However, when the interactions among the particles have an underlying local structure, we expect that very often the mixing time in the worst case will be of the form t_mix(ϵ) ≤1/γ(ℒ)(log(ϵ^-1)+log |Λ|), for γ(ℒ) possibly another constant depending on the Lindbladian. 

Heuristically, the reason for this is that the local structure of the interactions, both among the many particles and with the environment, may cause the effective dissipation to be local. In that case, the Lindbladian can be written as a sum of local jump operators, such that we can think of the thermalization of the whole system as a sum of roughly independent processes localized among regions of 𝒪(1) many particles. Since there are polynomially many such regions, the total convergence error ϵ should not be more than the sum of that of the individual regions. This “divide and conquer" line of thought then suggests a convergence error ϵ∼ |Λ| e^-Ω (t)  [In fact, this is the scaling that one can trivially find when there are no interactions between all the particles.]. When this scaling holds, the mixing time grows at most as log |Λ| and we say the system displays rapid mixing.

Rapid mixing is a defining feature of dissipative many-body dynamics, and comes along with a number of important consequences. The fact that an evolution has rapid mixing can be associated to properties of the correlations of the fixed point: for systems to reach a steady state quickly, it must be the case that the fixed points do not have features akin to long-range order. As such, the study of rapid mixing, both in the classical and quantum case, is very closely linked to the study of the correlation properties of their (thermal) fixed points. 

Along these lines, we know that dissipative evolutions with the rapid mixing property are stable under perturbations <cit.>, and their fixed points have decay of correlations <cit.>, display concentration properties <cit.>, and equivalence of ensembles, among various other features associated with standard statistical ensembles. Additionally, rapid mixing signals the absence of dissipative phase transitions <cit.> and rules out the usefulness of models as self-correcting quantum memories <cit.>. It is thus of great interest to understand when such property holds. 

While rapid mixing may be an intuitive feature of thermalizing dynamics, proving it is in general highly non-trivial. Nevertheless, progress has been made in recent years <cit.>, mostly in the context of commuting interactions, through the concept of the MLSI (modified logarithmic Sobolev inequality, see Section <ref>) constant, α≡α(ℒ)>0. This quantity directly yields the estimate

    t_mix(ϵ) ≤1/α𝒪(log1/ϵ+log|Λ|),

so that rapid mixing can be proven via lower bounds on α.



For Davies evolutions of 1-dimensional systems with uniform geometrically-local, commuting, and translation invariant Hamiltonians, it was shown in <cit.> that there exists a strictly positive MLSI constant α(ℒ)=Ω((log|Λ|)^-1) at any temperature. While this guarantees rapid mixing with a polylogarithmic scaling, it does not yet reach the optimal 𝒪(1) constant rate of exponential decay with time that is expected on physical grounds. So far, this optimal scaling was only known for on-site depolarizing noise <cit.> and, in a more general context, for the Schmidt generators (which are a less physically motivated thermalization process, see  <Ref>) of a system in hypercubic latices in dimensions D≥ 2 above a threshold temperature, and uniform nearest neighbour commuting Hamiltonian <cit.>. 



 §.§ Summary of results


In this paper, we prove rapid mixing for the Davies dynamics of a large class of lattice models of commuting Hamiltonians. We show that in 1D the property of rapid mixing follows directly from the presence of a gap in the Davies generator (which can be proven from first principles), and that it can also be proven at high temperatures with a great degree of generality in higher degree lattices. To do this, we derive relations between different measures of correlations at the fixed point, including a novel notion that we term “strong local indistinguishability", which is based on the max-relative entropy between the marginals of the fixed point and we find to be directly linked to the strategies for proving rapid mixing.    

Our first main result is for 1D systems, where we achieve the optimal Ω(1) scaling for the MLSI constant.

In 1D, for Davies generators of commuting, local Hamiltonians, having a positive gap is equivalent to the existence of a system-size independent positive MLSI constant α(ℒ^D_Γ)=Ω(1)_|Γ|→∞. This yields optimal rapid mixing at all positive temperatures β^-1 for these models. 

This is a strict strengthening of the previous 1D results <cit.>  with an additional extension to the non translation-invariant setting due to <cit.>. The formal version of this result can be found in Theorem <ref> and Corollary <ref>. Under the same assumption on the gap, we also get an, over the simple gap assumption, square-root improved mixing time for 2D lattices assuming the decay of correlations is strong enough.

In 2D, for Davies generators of commuting, nearest-neighbour Hamiltonians, having a positive gap and a sufficiently small correlation length is equivalent to the existence of a strictly positive square root decreasing MLSI constant α(ℒ^D_Γ)=Ω(|Γ|^-1/2)_|Γ|→∞. This implies a mixing time that scales at worst with the square root of the system, up to a logarithmic correction. 


The formal version of this result can also be found in Theorem <ref>.
For higher dimensional lattices in the high temperature regime we also give strict improvement in the following.
 Nearest neighbour, commuting potentials at sufficiently high temperature satisfy a MLSI with 

    
  1) system-size independent constant α(ℒ^D_Γ)=Ω(1)_|Γ|_→∞ when on a sub-exponential graph, e.g. ℤ^D for any D∈ℕ, or
    
  2) log-decreasing constant α(ℒ^D_Γ)=Ω((log|Γ|)^-1)_|Γ|_→∞ when on an exponential graph, e.g. a b-ary tree 𝕋_b for b>1.
 
In both of these cases the Davies dynamics displays rapid mixing.

For trees the bound on mixing times is novel within the quantum setting, while for hypercubic lattices it generalises the result of <cit.> from the less physically motivated Schmidt dynamics to the Davies generators. The formal version of this result can be found in Theorem <ref>. An overview of these mixing time results can also be found in <ref>.



A key ingredient in obtaining these results is by showing equivalence between several different notions of clustering, meaning that for fixed-size regions, exponential decay of one measure is equivalent to exponential decay of others. More concretely, we have the following implications.

For regions of fixed finite size, an exponential decay in the following notions of clustering of Gibbs states of geometrically-local, commuting Hamiltonians is equivalent, and implied by uniform gap of the Davies generator:

    
  1. Uniform 𝕃_2-clustering (<Ref>)
    
  2. Uniform decay of covariance (𝕃_∞-clustering) (<Ref>)
    
  3. Uniform local indistinguishability (<Ref>)
    
  4. Uniform decay of mutual information (<Ref>)
    
  5. Uniform strong local indistinguishability (<Ref>)
    
  6. Uniform mixing condition (<Ref>)


These last two notions together (5.,6.) also imply uniform q𝕃_1→𝕃_∞ clustering (<Ref>), which will be instrumental in the proof of the main result. For the formal definition of these notions see the respective definitions in <Ref> and for a more detailed picture of the implications see <Ref>. 


In the 1D setting, <Ref> also holds for arbitrary regions, allowing us to go directly from gap to system-size invariant MLSI. However, for higher dimensional lattices the decay functions of 4.,5.,6., assuming 1.,2., or 3. have prefactors depending exponentially on the boundaries of the regions, which is why for rapid mixing of commuting systems on higher dimensional lattices we require the stronger assumption of high temperature.



§ PRELIMINARIES
 
 



 §.§ Spin systems on graphs

We now describe the graphs that underlie the interactions among the particles. A graph is a tuple Λ = (V,E_V) of vertex set V and edge set E_V⊂ V× V. A complete subgraph Γ⊂Λ is a tuple (G,E_G), where G⊂ V and E_G contains all edges in E_V which contain the vertices in G. Abusing notation slightly, we call complete subgraphs subsets, writing Γ⊂Λ. For simplicity of notation we associate the graph with its vertex set. Hence we may write x∈Λ, A⊂Λ, or x∈Γ for an x∈ V, when the edge set E_V of Λ=(V,E_V) is clear from context. 

We define the size of a graph Λ, or of a subset Γ⊂Λ, denoted as |Λ|, |Γ|, respectively, as the number of vertices it contains. When emphasizing that Γ is a finite subset of Λ, i.e. |Γ|<∞, we write Γ⊂⊂Λ.
We write CD⊂Λ for the complete subgraph containing all of the vertices of C and D, so in this sense CD = C∪ D. Note that this does not require C,D to be disjoint.
We call a subset of vertices Γ⊂Λ connected, if for any two vertices x,y∈Γ there exists a sequence of pairwise overlapping edges in E_G, such that the first overlaps with x and the last with y. 

The graph distance d (on Λ) between two vertices x≠ y∈Λ is defined as the minimal length of a connected subset of edges which overlap both with x and y. We also set d(x,x)=0   ,  ∀ x∈Λ. The length of a subset of edges is given by the number of edges it contains. The distance between two subsets A,B⊂Λ is defined as the minimal graph distance between pairs of points in A and B, respectively. It is denoted, with slight abuse of notation, with the same symbol d. 
We define the diameter of a set A⊂Λ as diam(A):=sup_x,y∈ Ad(x,y). 

The graph has growth constant ν>0 defined as the smallest positive number such that, for any m∈ℕ, the number of connected subsets of size m containing some edge, for any edge, is bounded by ν^m:

    n_m:= sup_e∈ E_V|{F⊂ E_V  connected  |  F∋ e, |F|=m }| ≤ν^m.

Note that any regular graph, i.e. one where every vertex has the same number of neighbours as every other, has finite growth constant. For example, the growth constant of the D-dimensional hypercubic lattice (ℤ^D) is bounded by 2D e, where e is Euler's number
<cit.>. We say a graph is 2-colorable if there exists a labeling of the graph with labels 0 and 1, i.e. a map which assigns each vertex one label, such that adjacent vertices, i.e. ones which are connected by some edge, have different labels.

 For an infinite graph Λ we define N(l):=sup_x∈Λ|B_l(x)|, where B_l(x):={v∈Λ  |  d(x,v)≤ l } is the ball of radius l around vertex x.
We call a graph sub-exponential if there exists a δ∈(0,1) s.t. N(l)≤exp(l^δ) holds eventually, i.e. if log N(l)=𝒪(l^δ)_l→∞. Analogously, we call it exponential if no such δ exists, i.e. if log N(l)=Ω(l)_l→∞.

First note that all graphs with finite growth constant are in either of these two classes, since we can crudely bound |B_l(x)|≤ν^l and hence N(l)≤ν^l. Hypercubic lattices are sub-exponential under this definition, whereas b-ary trees are exponential. 

We denote the complement of some set A⊂Λ as A^c:=Λ∖ A.
We will often also consider geometrically-r-local interactions, with r>1 some integer, on such graphs. For some fixed r we define the boundary of a subset A⊂Λ, denoted with ∂ A, to be all vertices in Λ∖ A that are within graph distance r-1 from vertices in A 
    ∂ A    := {x∈Λ∖ A   |  d(x,A) < r}, 
    
        A∂   :=A∪∂ A.

It will be clear from context what r and hence the set-boundary ∂ is.
Hence, for nearest neighbour interactions (r=2), ∂ A coincides with the usual set-boundary. Whenever we consider Λ =ABC with B shielding A from C, we denote by ∂_A B the boundary of B in A.
An important class of graphs considered here are hypercubic lattices of dimension D∈ℕ, Λ=ℤ^D, with the graph distance equal to the Hamming distance.  Another example is the complete infinite b-ary tree 𝕋_b, for some integer b≥ 1. These are loop-free, exponential, and two-colorable graphs. where each vertex has exactly b neighbours. Each tree has one vertex, called the root, from which the tree extends, and whose b neighbours are called its children or leaves. Every other vertex has exactly b-1 children or leaves. 



 §.§ General Notation


A quantum spin system on a finite graph Γ = (V,E_V) is described by the Hilbert space 
    ℋ_Γ := ⊗_x∈ Vℋ_x,
 where each local Hilbert space ℋ_x has dimension d<∞, i.e. describes a qudit system. Hence the global dimension of the system is (ℋ_Γ)=d^|Γ|. We will only be considering finite-dimensional Hilbert spaces in this work.
We denote the algebra of bounded linear operators over ℋ_Γ by ℬ(ℋ_Γ) and the set of density operators with 𝒟(ℋ_Γ):={ρ∈ℬ(ℋ_Γ)   |  ρ≥ 0, [ρ]=1 }. Note that this algebra is *-homeomorphic to 𝒜_Γ*, the pre-dual of 𝒜_Γ=ℬ(ℋ_Γ) w.r.t. to the Hilbert-Schmidt inner product induced by the canonical trace on the finite-dimensional Hilbert space ℋ_Γ, i.e. the map ⟨ X,Y⟩=[X^*Y]. 
We recall that we can associate to each normalized state (a positive, linear functional) its density operator representation, i.e. for ω∈𝒜_Γ* there exists a ρ∈𝒟(ℋ_Γ), s.t. ω(X) =[ρ X], and the other way around. We denote the trace-class operators on a Hilbert space ℋ with ℬ_1(ℋ). The norm on ℬ(ℋ) is the usual operator norm, denoted by A≡A_∞ for A∈ℬ(ℋ). The norm on 𝒟(ℋ) is the usual trace-norm, denoted by ρ_1:=[|ρ|] for ρ∈𝒟(ℋ), where the trace on the full Hilbert space ℋ_Γ is denoted as [·]. Moreover, the partial trace on the Hilbert space corresponding to a region A⊂Γ is denoted as _A[·]: ℬ_1(ℋ_Γ)→ℬ_1(ℋ_Γ∖ A).

We denote the identity operator on ℋ as ≡_ℋ∈ℬ(ℋ) and the identity map ℬ(ℋ)→ℬ(ℋ) as 𝕀≡𝕀_.
Given a linear map Φ :ℬ(ℋ)→ℬ(ℋ) we denote its pre-dual with respect to the Hilbert-Schmidt inner product as Φ_*. We call such a map Φ a unital CP map if it is completely positive and identity-preserving Φ()=.   Their pre-duals Φ_*:ℬ_1(ℋ)→ℬ_1(ℋ) are completely positive, trace-preserving maps (CPTP or quantum channels). 

We denote the spectrum of an operator A∈𝒜 with spec(A). 
We define the subregion of the graph on which the operator A is non-trivially supported, denoted as (A)⊂Γ as the smallest subregion X⊂Γ s.t. A=A^'_X⊗_Γ∖ X, for a suitable A^'_X∈ℬ(ℋ_X).

We will employ the following “big-O”-notation 𝒪(g(x))_x→∞ when meaning that f(x)=𝒪(g(x)) for x→∞, i.e. to indicate in which limit the scaling 𝒪(g(x)) holds for a function f(x), and we use the “big-Ω" notation similarly.



 §.§ Weighted non-commutative 𝕃_p,σ-spaces and inner products
 
We will make use of so-called (weighted) non-commutative 𝕃_p spaces in this work.
For a general overview and construction of such spaces on von Neumann algebras, see e.g. <cit.>. 
In our case of finite-dimensional Hilbert spaces, the non-weighted ones are just the p-Schatten spaces on ℋ.

The non-commutative p-Schatten norm of X∈ℬ(ℋ) is defined as

    X_p:=[|X|^p]^1/p 1≤ p < ∞, 
       X_∞ = X.

Given a full-rank state σ∈𝒟(ℋ), we define the weighted non-commutative 𝕃_p,σ norm of X as

    X_p,σ   :=[|σ^1/2pXσ^1/2p|^p]^1/p  1≤ p<∞, 
    X_∞,σ   := X_∞≡X,
 respectively. These norms turn the 𝕃_p,σ spaces into Banach spaces for p∈[1,∞] and satisfy the usual Hölder-type inequality, Hölder duality, and monotonicity in p for fixed σ, see e.g. <cit.>.
Similarly, 𝕃_2,σ is a Hilbert space with respect to the KMS-inner product

    ⟨ X,Y⟩_σ^ := [√(σ)X^*√(σ)Y].

There exists a natural embedding Γ_σ:𝕃_1,σ→𝕃_1 via
Γ_σ(X):= √(σ)X√(σ).
Hence the weighted p,σ-norm can also be expressed as X_p,σ=Γ_σ^1/p(X)_p. 
For completeness we define the modular operator of σ here as

    Δ_σ(X):=σ X σ^-1,
 and the modular group of σ as {Δ_σ^is}_s∈ℝ. 
Additionally, the GNS-inner product on ℬ(ℋ), for a finite dimensional Hilbert space ℋ is

    ⟨ X,Y⟩_σ^ := [σ X^*Y].
 





 §.§ Uniform families of Hamiltonians
 
In this work we consider families of many-body Hamiltonians {H_Γ}_Γ⊂⊂Λ, such that

    H_Γ=∑_X⊂ΓΦ_X
 for a given fixed potential Φ:{X⊂Λ}→𝒜_Λ:X↦Φ_X.[Here 𝒜_Λ represents the closure of the algebra created by local operators on Λ, sometimes also called the algebra of quasi-local observables, when |Λ|=∞ and B(ℋ_Λ) when it is finite |Λ|<∞.] Note that for each X⊂Λ, Φ_X is a self-adjoint operator acting only non-trivially on the sub-region X. 
The potential is called commuting (on Λ) if for each X,Y⊂Λ, Φ_X and Φ_Y commute. 
It is said to have bounded interaction strength J:=max_X⊂Λ{Φ_X} and interaction range r:=max{diam(X)   |   X⊂Λ, Φ_X≠ 0}, where diam(X) stands for the diameter of region X with respect to the graph distance. We will call potentials with interaction range r geometrically-r-local. When r=2, we will interchangeably use the terms geometrically-2-local and nearest-neighbour.
We call this family {H_Γ}_Γ⊂⊂Λ the  associated family of Hamiltonians to (Λ,Φ). It is called a uniform J-bounded, geometrically-r-local, commuting family if the potential is J-bounded, geometrically-r-local and commuting.[Hence the constants J,r do not depend on the regions Γ, and explicitly on |Γ|, on which the local Hamiltonians are defined.] In this work, we will only consider such uniform families, unless explicitly stated otherwise.

The associated Gibbs state of the local Hamiltonian on A⊂Γ at inverse temperature β is denoted by

    σ^A:=e^-β H_A/[e^-β H_A]∈𝒟(ℋ_Γ),
 while the reduced state onto some subregion A⊂Γ is denoted by

    σ_A:=_Γ∖ Aσ^Γ∈𝒟(ℋ_A),
 where σ≡σ^Γ=σ_Γ.
Given a potential Φ on Λ and some inverse temperature β>0 we call the family of Gibbs states {σ^Γ}_Γ⊂⊂Λ, where the Hamiltonian H_Γ is given as in (<ref>) w.r.t this Φ, the family of Gibbs states associated to (Λ,Φ,β).
We will employ the convenient notation E_X,Y:=e^-H_XYe^H_X+H_Y for Araki's expansionals for two disjoint subsets X,Y⊂Λ from <cit.>. 



 §.§ Quantum Markov semigroups and Lindbladians
 
A quantum Markov semigroup (QMS) is a strongly continuous one-parameter semigroup of unital CP maps {Φ_t}_t≥ 0:ℬ(ℋ)→ℬ(ℋ). 
This is a family such that Φ_0=𝕀_, Φ_s+t=Φ_s∘Φ_t   ,  ∀ s,t≥ 0, and lim_t↓0(Φ_t-𝕀)(X)=0  ,  ∀ X∈ℬ(ℋ). 
By the Hille-Yosida theorem there exists a densely defined generator, called the Lindbladian

    ℒ(X):=lim_t↓01/t(Φ_t-𝕀)(X),
 such that the semigroup is given as Φ_t= e^tℒ  ,  ∀ t≥ 0.
In our case of a finite-dimensional Hilbert space, the Lindbladian is defined on all of ℬ(ℋ) and its pre-dual on all of ℬ_1(ℋ). 

A QMS with generator  gives the unique solution to the master equation d/dtρ(t)=ℒ(ρ(t)). A state σ is invariant (or stationary) if Φ_t*(σ)=σ for all t≥0, which is equivalent to _*(σ)=0. 
We call a QMS and its generator faithful if the QMS admits a full-rank invariant state σ∈𝒟(ℋ) and primitive if this state is unique.

We call a QMS and its generator reversible or KMS-symmetric w.r.t. a state σ if the QMS is symmetric w.r.t the KMS-inner product and similarly for
GNS-symmetric.
In the latter case, we also say that the QMS satisfies the detailed balance condition, which is equivalent to

    [σ X^*(Y)] = [σ(X)^*Y] ∀ X,Y∈.

If a QMS is GNS-symmetric w.r.t a state σ, then this state is necessarily a stationary one.

Given a graph Λ and a finite subset Γ⊂⊂Λ, we consider a family of Lindbladians _Λ={_Γ}_Γ⊂⊂Λ, such that

    _Γ = ∑_X⊂ΓL_X,
 where {L_X}_X⊂Λ is a fixed family of local Lindbladians, such that J̃:=sup_X⊂ΛL_X*_1→1,cb<∞ and L_X*=0 whenever diam(X)>r̃.[Here Φ_*_1→1,cb is the completely bounded 1→1 norm, i.e. 
    Φ_*_1→1,cb := sup_n∈ℕsup_ρ∈𝒟(ℂ^n⊗ℋ)(𝕀_n⊗Φ_*)(ρ)_1 .
 ] Hence we call these a uniform  geometrically-r̃-local, J̃-bounded family of (bulk) Lindbladians, in analogy with the Hamiltonian case. 
The family is called locally reversible, if each element ℒ_Γ is reversible (KMS-symmetric) w.r.t. the Gibbs state of any larger region σ^Π for Γ⊂Π. 
Note that this implies that it also is frustration-free, that is for any two finite subsets Γ⊂Π⊂⊂Λ, the stationary states of _Π are also stationary under _Γ, i.e. ker(_Π)⊂ker(_Γ).





Note that if a QMS is GNS-symmetric, i.e. satisfies detailed balance, then it is also KMS-symmetric, i.e. reversible.
For a region Γ⊂⊂Λ, we can write the projection onto the fixed point subalgebra of _Γ as 

    E_Γ(·):=lim_t→∞e^tℒ_Γ(·).

It turns out that for a primitive, frustration-free uniform family, these projections are conditional expectations w.r.t the family of stationary states. See <Ref> for more details.

Here we will be working with the Davies generators ^D_Λ={^D_Γ}_Γ⊂⊂Λ, which is a physically motivated uniform family of Lindbladians associated to a uniform family of Hamiltonians. These are introduced in <Ref>.
In the setting we are considering, they are a uniform geometrically-bounded, locally GNS-symmetric 
family of Lindbladians which describe thermalization of a spin system.

We call a uniform family of Lindbladians which are locally GNS-symmetric 
and frustration-free w.r.t to a set of Gibbs states {σ^Γ}_Γ⊂⊂Λ a quantum Gibbs sampler of the system (Λ,Φ,β) when {σ^Γ}_Γ⊂⊂Λ is the uniform family of Gibbs states associated to (Λ,Φ,β). The Davies generators (see <Ref>), the Heat-bath generators <cit.> and the Schmidt generators <cit.> are examples of quantum Gibbs samplers. 



 §.§ Conditional Expectations
 

Given a von Neumann subalgebra 𝒩⊂ℬ(ℋ), a conditional expectation onto 𝒩 is a completely positive, unital map E_𝒩:ℬ(ℋ)→𝒩, such that

    E_𝒩(X)=X   ,  ∀  X∈𝒩
       E_𝒩(aXb)= aE_𝒩(X)b   ,  ∀  a,b∈𝒩, X∈ℬ(ℋ).

By complete positivity and unitality, it follows that the preadjoint of any conditional expectation with respect to the Hilbert-Schmidt inner product E_𝒩*:𝒩_*→ℬ(ℋ)_* is a completely positive, trace-preserving map, i.e. a quantum channel.
Any conditional expectation onto 𝒩 for which there exists a full-rank state σ∈𝒟(ℋ) which satisfies

    E_𝒩*(σ)=σ⟺[σ E_𝒩(X)]=[σ X]   ,  ∀  X∈ℬ(ℋ),
 is said to be with respect to the state σ 
<cit.>.
Let E be a conditional expectation with respect to a full-rank state σ onto 𝒩, then from the definition it follows that it is self-adjoint with respect to the σ-KMS inner product, i.e.

    σ^1/2E(X)σ^1/2 = E_*(σ^1/2Xσ^1/2)
 holds for any X∈ℬ(ℋ). 

Furthermore, it can be shown that E commutes with the modular automorphism group of σ, i.e.

    Δ_σ^is∘ E = E ∘Δ_σ^is  ,  ∀ s∈ℝ.
  
Moreover, given a *-subalgebra 𝒩⊂ℬ(ℋ) and a full-rank state σ∈𝒟(ℋ), the existence of a conditional expectation w.r.t. σ onto 𝒩 is equivalent to the invariance of 𝒩 under the modular automorphism group {Δ_σ^is}_s∈ℝ. Furthermore, in the case that the  *-subalgebra 𝒩 is invariant under the modular automorphism group of said full-rank state σ, this conditional expectation is uniquely determined by σ <cit.>.

Conditional expectations with respect to some full-rank state σ between finite-dimensional matrix algebras, as all the ones in this work, can be given in an explicit form, see e.g. <cit.>. Any finite dimensional subalgebra 𝒩⊂ℬ(ℋ) is unitarily isomorphic to the algebra

    𝒩=⊕_i=1^n ℬ(ℋ_i)⊗ℂ_𝒦_i,  where ℋ = ⊕_i=1^nℋ_i⊗𝒦_i.

Now there exist density operators {τ_i∈𝒟(𝒦_i)}_i=1^n and projections {P_i∈ℬ(ℋ_i)}_i=1^n, respectively, onto {ℋ_i⊗𝒦_i} such that

    E_𝒩(X) = ⊕_i=1^n _𝒦_i[P_iXP_i(_𝒦_i⊗τ_i)]⊗_𝒦_i ⟺ E_𝒩*(ρ) = ⊕_i=1^n_𝒦_i[P_iρ P_i]⊗τ_i,
 for X∈ℬ(ℋ) and ρ∈𝒟(ℋ) <cit.>. 
An important example of conditional expectations are the following.

  
Let Γ be some finite graph. 
Let ℒ_Γ ={_A}_A⊂Γ 
be a uniform locally reversible family of Lindbladians with respective stationary states {σ^A}_A⊂Γ. The local Lindbladian projector associated with the family ℒ_Γ on A⊂Γ is given by

    E_A(X):=lim_t→∞e^tℒ_A(X)

for X∈ℬ(ℋ_Γ). Notice that each E_A acts only non-trivially on A^c and since _Γ is frustration-free, E_A is a conditional expectation with respect to the stationary state σ^Γ onto the subalgebra _A∂⊗ℬ(ℋ_(A∂)^c).

For a proof of these claims see e.g. <cit.>. In this case the expectation value of any observable w.r.t. the invariant state on the full system is

    [σ X]= [σ E_Γ(X)] = [σ E_A(X)]   ∀ A⊂Γ.

On the other hand, given a family of local conditional expectation E_A: ℬ(ℋ)→_A∂⊗ℬ(ℋ)_(A∂)^c w.r.t. the same state σ∈𝒟(ℋ), then

    ℒ_A:= (E_A-𝕀_A),
 is a family of locally reversible, frustration-free Lindbladians with invariant state σ. 




 §.§ The relative entropy and strong data processing
 

The Umegaki relative entropy <cit.> between two finite-dimensional quantum states given by their density operators ρ,σ∈𝒟(ℋ) is defined as

    D(ρσ) := [ρ(logρ-logσ)]
           if supp(ρ)⊂supp(σ) 
    ∞   else,
 where the logarithm here is the natural logarithm to base e. 
Pinsker's inequality (<ref>) gives an upper bound on the trace-distance in terms of the relative entropy:

    ρ-σ_1^2 ≤ 2D(ρσ).


The relative entropy satisfies the data processing inequality, so that no quantum channel, i.e. CPTP map Φ_*, can increase it between any two states,

    D(Φ_*(ρ)Φ_*(σ))≤ D(ρσ).

The core part of the entropic-inequalities approach to thermalization relies upon a strengthening of this inequality. We say a quantum channel Φ_* satisfies a non-trivial strong data processing (sDPI)  with contraction coefficient η≡η(Φ_*)<1 if for any pair (ρ,σ) of states, with ρ≠σ, it holds that

    D(Φ_*(ρ)Φ_*(σ))≤η(Φ_*)D(ρσ).

More formally, we define the contraction coefficient for a GNS symmetric QMS Φ_*≡Φ_t_0* as

    η(Φ_*):=sup_ρ∈𝒟(ℋ)inf_σ∈ΣD(Φ_*(ρ)Φ_*(σ))/inf_σ∈ΣD(ρσ),
 where Σ is the set of stationary states of Φ_*. These are all the density operators which are left invariant under the action of the channel.[In the case of a general quantum channel (instead of QMS) one would need to replace Σ by the decoherence-free subalgebra. If Φ_*(X)=∑_kA_kXA_k^* is the Kraus representation of Φ_*, then the decoherence-free subalgebra is Σ :=⋂_k∈ℕ𝒩(Φ_*^k), where 𝒩(Φ_*):=Alg{X∈ℬ(ℋ)  |   [X,A_i^*A_j]=0  ∀ i,j}] Assume we have some channel Φ_* which has a contraction coefficient η(Φ_*)<1 and a unique invariant state σ, i.e. Φ_*(σ)=σ. Then sDPI immediately induces an exponential decay of the relative entropy in the number of times the channel is applied.

    D(Φ_*^n(ρ)σ)=  D(Φ_*^n(ρ)Φ_*^n(σ)) ≤η^n D(ρσ).

Since conditional expectations are, by definition, projections on closed-*-subalgebras (which are convex), the following chain rule holds for states ρ,σ∈𝒟(ℋ), whenever E_𝒩*(σ)=σ <cit.>

    D(ρσ)= D(ρE_𝒩*(ρ))+D(E_𝒩*(ρ)σ).


For completeness we define the max-relative entropy <cit.> between two finite dimensional quantum states given by their density operators ρ,σ∈𝒟(ℋ) as

    D_max(ρσ):= loginf{λ∈| ρ≤λσ} = logσ^-1/2ρσ^-1/2   if supp(ρ)⊂supp(σ) 
    ∞   else,

where the inverses are taken as generalised inverses. It also satisfies the DPI and a triangle inequality and it holds that

    D(ρσ) ≤ D_max(ρσ).


The relative entropy also gives rise to the quantum mutual information I, which measures correlations between two regions. Given a finite graph Γ=ABC, the mutual information of a state ρ∈𝒟(ℋ_Γ) between the reduced state on the region A and the one on the region C  is defined as

    I_ρ(A:C):=D(ρ_ACρ_A⊗ρ_C).
 
Analogously, the max-mutual information I_max of the state
ρ∈𝒟(ℋ_Γ) on Γ=ABC between the reduced state on the region A and the one on the region C  is defined as

    I_max,ρ(A:C):=D_max(ρ_ACρ_A⊗ρ_C).
 



 §.§ Relative entropy decay via the complete modified logarithmic Sobolev Inequality

Assuming that our QMS has at least one full-rank invariant state σ with respect to which it is GNS-symmetric (as is the case for all Davies maps), we can establish the sDPI (<ref>)  for time-continuous QMS {e^tℒ}_t≥0. 

The way to do this is via a differential version of the strong data processing inequality (<ref>) for the channel Φ_t*:=e^tℒ_* in which we set η(e^tℒ_*)=e^-tα, yielding

    -d/dt D(e^tℒ_*(ρ)E_*(ρ)) |_t=0 =: _ℒ(ρ)  ≥α D(ρE_*(ρ)).

Here E_*:=lim_t→∞e^tℒ_* is the projection onto the stationary states (<ref>), see also Example <ref>.
We call this inequality (<ref>) the modified logarithmic Sobolev inequality (MLSI) and _ℒ(ρ) the entropy production of the QMS {e^tℒ}_t≥ 0.
The optimal constant α satisfying the MLSI is called the modified logarithmic Sobolev constant (MLSI constant) α(ℒ). It is hence given by

    α(ℒ):= inf_ρ∈𝒟(ℋ)_ (ρ)/D(ρE_*(ρ)).

By integration and use of Gronwall's inequality it follows that any QMS {e^t}_t≥0 which satisfies the MLSI with strictly positive MLSI constant α≡α(ℒ)>0 induces exponential convergence in relative entropy to its stationary states, i.e.

    D(e^t_*(ρ)E_*(ρ))≤ e^-α tD(ρE_*(ρ)).

One important way to establish the existence of such constants in the classical setting is to exploit its stability under tensorization. This allows us to describe the dynamics of large composite systems via their dynamics on small subregions. This is, however, often not straightforward in the quantum setting, i.e. if we have two QMS {e^tℒ}_t≥0, {e^t𝒦}_t≥ 0, then the joint evolution, given by {e^t⊗ e^t𝒦}_t≥ 0 is not necessarily as quickly mixing as the slower individual one, i.e. α(+𝒦)≱min{α(),α(𝒦)}. <cit.>.
In order to recover the stability under tensorization we introduce the so-called complete MLSI (cMLSI) and the cMLSI constant 

    α_c(ℒ):=inf_n∈ℕα(ℒ⊗𝕀_n),
 where 𝕀_n:ℬ(ℂ^n)→ℬ(ℂ^n) is the identity channel <cit.>.
Hence, we say that the QMS {e^t}_t≥ 0 satisfies the cMLSI if the QMS {e^t⊗𝕀_n}_t≥ 0 satisfies the MLSI for all ancillary systems of arbitrary dimension with the same constant.
In <cit.> it was shown that for two QMS with commuting generators ℒ,𝒦, respectively, it holds that

    α_c(+𝒦)≥min{α_c(),α_c(𝒦)}.





Next, the following important result from <cit.> guarantees the existence of positive cMLSI constants for a sufficiently large class of QMS.

For any GNS-symmetric QMS {e^tℒ}_t≥0 on the algebra ℬ(ℋ) of bounded linear operators over some finite dimensional Hilbert space ℋ, we have α_c(ℒ)>0, with α_c(ℒ)= Ω(λ(ℒ)/logℋ). 
In particular, for many-body quantum systems, this bound on the cMLSI constant gives a `trivial' lower bound that is decreasing at best as Ω(|Γ|^-1) as long as the gap λ(ℒ_Γ) is constant.
 

Local existence of a strictly positive cMLSI constant is a promising starting point. However, on its own it does not give good bounds for systems in the thermodynamic limit. The strategy used in this work, when showing existence of a cMLSI constant with a better scaling than Ω(|Λ|^-1), is to use approximate tensorization results to geometrically break down the lattice into finite-size parts. Then, we apply Theorem <ref> to each of these small regions and put them back together into the whole lattice. Such an approach is sometimes called a “divide-and-conquer strategy", or a global-to-local reduction.





§ DYNAMICAL PROPERTIES: LOCAL GENERATORS AND CONDITIONAL EXPECTATIONS




In this work we consider two classes of dissipation dynamics. The first one, commonly known as Davies dynamics, is a physically motivated Quantum Markov Semigroup. It is typically used to model thermalization of finite-dimensional quantum systems weakly coupled to their environment. The main result of this paper is formulated with respect to it, so
we devote Section <ref> to the introduction of the Davies evolution and to some technical results concerning it. 


The second class of semigroups we explore are the so called Schmidt generators, see Section <ref> for its definition and the notation used. It serves as a mathematically more tractable model that will be useful in the proofs, even if it lacks a clear physical interpretation. We will use these dynamics as a proxy to derive rigorous bounds on the mixing time of the former through establishment of the modified logarithmic Sobolev inequality. 




 §.§ Davies dynamics
 
The main dynamics considered here are the Davies dynamics introduced in <cit.>, see also <cit.>, <cit.> for a more modern derivation, and <cit.>, whose notation we will be mostly using, in the context of Gibbs sampling.
On a finite subsystem Γ⊂⊂Λ, its generators, the Davies Lindbladians or generators (in the Heisenberg picture) are given by

    _Λ^D(X) = i[H_Λ,X] + ∑_x∈Λℒ^D_x(X)   ,
 where H_Λ is the lattice Hamiltonian and dissipative terms 

    _x^D(X) =∑_ω,α(x)χ_α(x)(ω)(S^*_α(x)(ω)XS_α(x)(ω)-1/2(S^*_α(x)(ω)S_α(x)(ω)X+XS^*_α(x)(ω)S_α(x)(ω)))   .
 Here ω∈spec(H_Λ)-spec(H_Λ) are the Bohr frequencies, χ_α(x)(ω) a function satisfying the KMS-condition χ_α(x)(-ω)=e^-βωχ_α(x)(ω) and S_α(x)(ω) operators related to the interaction between our lattice and the thermal environment <cit.>. 
If we assume that our system is described by a uniformly bounded, geometrically-r-local, commuting family of Hamiltonians {H_Γ}_Γ⊂⊂Λ, as in (<ref>), then the above generator reduces to a local Davies generator

    ^D_Γ(X) = i[H_Γ,X]+∑_x∈Γℒ^D_x(X)   ,
 for any Γ⊂⊂Λ. 
We call these a (family of) Davies generators associated to  (Λ,Φ,β), whenever the family of Hamiltonians that describe our system is the one associated to (Λ,Φ) and the inverse temperature of the environment is β>0. Note that these are not unique, and depend on the details of the environment through χ_α(x)(ω). However, they always correspond to a uniformly bounded, geometrically-local family of Lindbladians satisfying the following properties:

(<cit.>)
For some inverse temperature β, some finite graph Λ, a subset Γ⊂Λ and Φ a uniformly bounded, geometrically-r-local, commuting potential on Λ, the associated local Davies generators defined in (<ref>-<ref>) satisfy

    
  * For any subset Γ⊂Λ, {e^t^D_Γ}_t≥ 0 is a CP unital semigroup with generator ^D_Γ.
    
  * The family ^D={^D_Γ}_Γ is geometrically-local, in the sense that each individual term ^D_x acts only nontrivially on the region B_r(x) for some fixed radius r≤r̃≤ 2r.
    
  * The family ^D={^D_Γ}_Γ is locally reversible and satisfies detailed balance w.r.t the global Gibbs state σ^Λ.
    
  * The family ^D={^D_Γ}_Γ is frustration-free.


Recalling <Ref>, we see that E^D_Γ(X):=lim_t→∞e^t^D_Γ(X) is a conditional expectation, called the Davies conditional expectation.



 §.§ Dynamics generated by Schmidt conditional expectations


Although the Davies dynamics are our target due to their physicality, their corresponding conditional expectations are not always mathematically tractable. For this reason it will be very useful to consider the so-called Schmidt conditional expectation, as introduced in <cit.> inpired by <cit.>. As we show below, their conditional expectation are much more tractable, yet the dynamics they induce are closely related to the Davies. The construction and results in this section work for any 2-colorable graph of finite growth constant. 


Let Λ = (V,E_V) be a quantum spin system with bounded, nearest-neighbour, commuting potential Φ:(i,j)↦ h_i,j, such that the Hamiltonians on finite sub-lattices Γ=(G,E_G)⊂⊂Λ can be written as

    H_Γ=∑_(i,j)∈ E_G h_i,j,
 where each term h_i,j acts only non-trivially on vertices i and j.
The Gibbs state of H≡ H_Γ with inverse temperature β is

    σ = e^-β H/[e^-β H] = ∏_{i,j}∈ E_Ve^-β h_i,j/[∏_{i,j}∈ E_Ve^-β h_i,j].

Given some A⊂⊂Λ we will define a suitable *-algebra 𝒩_A and conditional expectation E_A^S onto it, whose pre-adjoint has the Gibbs state as an invariant state. For simplicity of notations, we do this for a singleton A={a}. However, this construction works similarly for all A⊂⊂Λ.
Given some a∈Λ, we enumerate the sets

    ∂{a}   :={x∈Λ|(x,a)=1} = {b_i}_i∈ I_a, 
    ∂{b_i}   :={x∈Λ|(x,b_i)=1} = {c_i,j}_j∈ J^(i), s.t. a=c_i,0 ∀ i.
 
Hence ∂(∂{a})∖{a}={y∈Λ| (a,y)=2}={c_i,j}_i∈ I_a,j∈ J^(i)∖{0}. See Figure <ref> for a graphical example of these definitions. 




We will drop the index a of I≡ I_a, the labeling of all the neighbours of a in the following. 
Now we Schmidt-decompose

    e^-β h_b_ic_ij=∑_sX^j,s_b_i⊗ X^s_c_ij
 for i∈ I, where the operators {X^j,s_b_i}_j,s⊂ℬ(ℋ_b_i) and for j∈ J^(i), {X^s_c_ij}_s⊂ℬ(ℋ_c_ij).
Let us define the *-algebra 𝒜^j_b_i to be generated by all {X^j,s_b_i}_s, for all i∈ I. <cit.>.

Any two non-identical of these algebras {𝒜^j_b_i}_i∈ I,j∈ J^(i) commute.


Consider 𝒜^j_b_i and 𝒜^n_b_m. If i≠ m, then the statement is obviously true, since their generators act on different Hilbert spaces ℋ_b_i, ℋ_b_m, respectively. If i=m, see that

    0= [e^-β h_b_ic_ij,e^-β h_b_ic_in] = [∑_s X_b_i^j,s⊗ X_c_ij^s,∑_r X_b_i^n,r⊗ X^r_c_in] j≠ n=∑_s,rX^s_c_ij⊗ X^r_c_in⊗[X_b_i^j,s ,X_b_i^n,r] 
       [X_b_i^j,s ,X_b_i^n,r]=0  ∀ s,r ,
 where the last implication follows since {X^s_c_ij}_s,{X^r_c_in}_r form a set of linear independent operators by Schmidt decomposition.

Therefore these algebras and the underlying Hilbert spaces admit the following joint decomposition

    ℋ_b_i:=⊕_α_i⊗_j∈ J^(i)ℋ_j^α_i⊗ℋ_c^α_i=:⊕_α_iP^α_iℋ_b_i,
 where P^α_i are orthogonal projectors such that

    P^α_iℋ_b_i=⊗_j∈ J^(i)ℋ_j^α_i⊗ℋ_c^α_i,
 and {ℋ_j^(α_i),ℋ_c^(α_i)}_α_i are such that

    𝒜^j_b_i=⊕_α_iℬ(ℋ_j^α_i)⊗_⊗_k∈ J^(i)∖{j}ℋ_k^α_i⊗ℋ_c^α_i∀ j∈ J^(i).


For any {a}∈Λ, define the *-subalgebra 𝒩_a:=_{a}⊗⊗_i∈ I⊗_j∈ J^(i)∖{0}𝒜^j_b_i⊗ℬ(ℋ_({a}∂))^c ⊂ℬ(ℋ_Λ).

 
The modular group of the Gibbs state σ leaves this algebra invariant, for any a∈Λ, i.e.

    Δ^it_σ(𝒩_a)⊂𝒩_a ∀ t∈ℝ,
 where Δ_σ^it(X):=σ^itXσ^-it,  ∀ X∈ℬ(ℋ_Λ).


We can set β=1 without loss of generality. Let us fix a∈Λ. It is enough to show that e^ith_kl𝒩_ae^-ith_kl⊂𝒩_a holds for any pair (k,l)∈ E_V (for examples of the following cases see also <Ref>):




Case i) For (k,l)⊂ ({a}∂)^c, then this is obvious, since 𝒩_a|_ℋ_({a}∂)^c=ℬ(ℋ_({a}∂)^c).




Case ii) For
(k,l)⊂{a}∂, assume w.l.o.g. k=a, hence l=b_i for some i∈ I. Let Y∈𝒩_a then via the spectral theorem [e^-ith_ab_i,Y]=0 ⇔ [e^-h_ab_i,Y]=0. It is enough to show this for the generators Y=_{a}⊗ X^j,s_b_i for all s and j∈ J^(0)∖{0}, by closedness of the algebra.

    [e^-h_ab_i,_{a}⊗ X^j,s_b_i] = [∑_r X_a^r⊗ X^0,r_b_i,_{a}⊗ X^j,s_b_i] = ∑_r X_a^r ⊗[X_b_i^0,r,X_b_i^j,s] = 0,
 ∀ s,j∈ J^(i)∖{0} since the algebras 𝒜^0_b_i and 𝒜^j_b_i commute for j∈ J^(i)∖{0}. 



Case iii) For (k,l)⊂∂{a}∪({a}∂)^c. W.l.o.g. k=b_i for some i∈ I, hence l=c_ij for some j∈ J^(i)∖{0}. Then e^-β h_kl=e^-β h_b_ic_ij⊗_ℋ_{k,l}^c=∑_s X^j,s_b_i⊗ X_c_ij^s⊗_ℋ_{k,l}^c∈𝒩_a. Thus by the spectral theorem e^± ith_kl⊗∈𝒩_a and hence by closedness of the algebra e^ith_kl𝒩_ae^-ith_kl⊂𝒩_a.


Thus by Takesaki's theorem <cit.>, see also <cit.> due to Proposition <ref>, there exists a conditional expectation

    E^S_a : ℬ(ℋ_Λ) →𝒩_a,
 such that its pre-adjont has the Gibbs state σ, which is full-rank, as an invariant state, i.e. E^S_a*(σ)=σ.
This is called the Schmidt conditional expectation.

The above construction works exactly the same for any subregion A⊂Λ in place of {a}⊂Λ, yielding a *-subalgebra 𝒩_A. Hence, we equally define the family of conditional expectations {E_A^S}_A⊂Λ on ℬ(ℋ_Λ). Similarly, for these it holds that E^S_A*(σ)=σ.
We can think of these conditional expectations as replacing any given observable on the local subset with the identity, in such a way that is consistent with the invariance of the Gibbs state under its pre-adjoint.
Hence, the family of Schmidt conditional expectations still has the desirable properties of the Davies expectations, i.e. that the Gibbs state is invariant, but their structure is easier to analyze since we can give an explicit expression for the conditional expectations which does not depend on system-environment couplings <cit.>. 

Before giving their explicit form, we highlight another important property of the Schmidt conditional expectations.

For any two subsets A_1,A_2⊂Λ, such that (A_1,A_2)≥ 2, the Schmidt conditional expectations E^S_A_1 and E^S_A_2 satisfy

    E^S_A_1∘ E^S_A_2   =  E^S_A_2∘ E^S_A_1 = E^S_A_1∪ A_2, 
    
        E^S_A_1∘ E^S_A_1 ∩ A_2   = E^S_A_1 ∩ A_2∘ E^S_A_1 = E^S_A_1.


This follows from the fact that the conditional expectation E^S_A is a local map, acting only non-trivially on A∂, and the following Lemma. 
 
For any two subsets A_1,A_2⊂Λ, s.t. (A_1,A_2) ≥ 2, or such that one is a subset of the other it holds that 

    
  1) 𝒩_A_1∩𝒩_A_1 = 𝒩_A_1∩ A_2 .
    
  2) 𝒩_A_1∪𝒩_A_2 = 𝒩_A_1∪ A_2.

Here, 𝒩_A_1∪𝒩_A_2 denotes the *-algebra generated by 𝒩_A_1 and 𝒩_A_2. 𝒩_A_1∩𝒩_A_2 denotes the *-algebra generated by all elements in both 𝒩_A_1 and 𝒩_A_2.


The proof is elementary from the definition of the algebras 𝒩 and may be found in Appendix <ref>.


For a subset A⊂Λ, we call a set (α):={α_i}_i∈ I_A a boundary condition.


For A⊂Λ, let (α):={α_i}_i∈ I_A be a fixed boundary condition for the subset A and denote P^(α):=⊗_i∈ I_AP^α_i. We set

    ℋ^(α)_A_   :=ℋ_A⊗⊗_i∈ I_Aℋ_0^α_i⊗ℋ_c^α_i≡ℋ_A⊗ℋ^(α)_∂_ A, 
    ℋ^(α)_A_   := ℋ_(A∂)^c⊗⊗_i∈ I_A⊗_j∈ J^(i)∖{0}ℋ_j^α_i≡ℋ_(A∂)^c⊗ℋ^(α)_∂_ A,





i.e. ℋ_Λ=⊕_(α)ℋ^(α)_A_⊗ℋ^(α)_A_, and write _ℋ_A^(α)_≡_A_,(α), respectively, _ℋ_A_^(α)=_A_,(α) for simplicity. 

Since every element of the algebra 𝒩_A is block diagonal w.r.t the sets (α), we can decompose the Schmidt conditional expectation E^S_A and its pre-adjoint E^S_A* along those blocks as well, yielding

    E_A^S(X)   :=⊕_(α) E^S,(α)_A(X), 
    
        E_A*^S(ρ)   :=⊕_(α) E^S,(α)_A*(ρ),

for any X∈ℬ(ℋ_Λ) and ρ∈𝒟(ℋ_Λ).  E^S,(α)_A and E^S,(α)_A have the following expressions <cit.>, defined on the block Hilbert space P^(α)ℋ_Λ

    E^S,(α)_A(X)    = P^(α)(_A_,(α)[τ_A_^(α)X]⊗_A_,(α))P^(α), 
    
        E^S,(α)_A*(ρ)    = _A_,(α) [P^(α)ρ P^(α)]⊗τ^(α)_A_,
 where the state τ_A_^(α) is given by

    τ^(α)_A_ := 1/[...]_A_,(α)[P^(α)σ^(A∂) P^(α)] = 1/[...]_∂_ A,(α)[P^(α)σ^(A∂) P^(α)],
 where the prefactors 1/[...] contain the trace on ℬ(ℋ^(α)_A_) and ensure proper trace normalization [τ_A_^(α)]=1, and the partial trace in the last expression traces out the Hilbert space ℋ^(α)_∂_ A=⊗_i∈ I_A⊗_j∈ J^(i)∖{0}ℋ_j^α_i = ℋ^(α)_A_∖(A∂)^c,.

It is easy to check that the expressions (<ref>) and (<ref>) are dual to each other w.r.t. the Hilbert-Schmidt inner product on ℬ(ℋ_Λ). 
The expression (<ref>) follows from the invariance of the local Gibbs states σ^(A∂) via P^(α)σ^(A∂)P^(α)=E^S(α)_A*(P^(α)σ^(A∂)P^(α))=_A_,(α)[P^(α)σ^(A∂)P^(α)]⊗τ_A_^(α). Taking the partial trace of this expression on ℋ_A_^(α) gives the above expression.

















One can think of the α_i as labeling the boundary conditions at site i∈ I_A, and all (α):={α_i}_i∈ I_A give a complete labeling of all boundary conditions of some subset A.
Hence, one can think of the effect of the Schmidt-conditional expectation on states as effectively replacing the state ρ locally on A with the Gibbs state σ, where the boundary conditions are set to (α)={α_i}_i∈ I_A.

As in the Davies evolution, there exists a uniform family of Lindbladians for which the Schmidt conditional expectations are given by the local Lindbladian projectors.
For the Schmidt conditional expectation, the corresponding family of Lindbladians is

    _A^S(X):=∑_x∈ AE^S_x(X)-X.
  We call them the (family of) Schmidt generators associated to (Λ,Φ,β)[Explicitly, when {σ^Γ}_Γ⊂⊂Λ is the family of Gibbs states associated to (Λ,Φ,β) is the one used to construct these Schmidt conditional expectations. 
] 
<cit.>. It is straightforward to see that the projection onto their kernel is given by the Schmidt conditional expectation. From the properties established above, they are uniform families of locally primitive, locally GNS-symmetric, frustration-free Lindbladians. Through this definition we immediately obtain additivity in the region. That is, ℒ^S_A+ℒ^S_B=ℒ^S_AB for non-overlapping regions A,B. 



 §.§ Relating Davies and Schmidt dynamics

An important observation is that, since the Schmidt and Davies families of conditional expectations almost have the same fixed-point algebras, we can relate the relative entropy distance of any given state to the fixed point subalgebra of one to the other: 

Let X⊂Λ, ρ∈𝒟(ℋ_Λ). For E^S_X*(ρ) the Schmidt conditional expectation of ρ and E^D_X*(ρ) its Davies conditional expectation, it holds that

    D(ρE^D_X*(ρ)) ≤ D(ρE^S_X*(ρ)) ≤ D(ρE^D_X∂*(ρ)).




First recall that for some region X⊂Λ, E_X^D is the projection onto the kernel of ℒ^D_X, and call it ℱ^D_X:=Fix(E^D_X). It is also the projection onto the largest *-subalgebra of ℬ(ℋ_X^c)⊗_X⊂ℬ(ℋ_Λ) which is invariant under the modular group of the Gibbs state σ, {Δ^it_σ}_t∈ℝ <cit.>.
Now, E^S_X is a projection onto, say ℱ^S_X:=Fix(E^S_X). This is by construction a *-subalgebra of ℬ(ℋ_X^c)⊗_X⊂ℬ(ℋ_Λ) invariant under the modular group {Δ^it_σ}_t∈ℝ, see <Ref>. Thus ℱ_X^S⊂ℱ_X^D. This implies that
D(ρE^D_X*(ρ)) ≤ D(ρE^S_X*(ρ)) for any state ρ∈𝒟(ℋ_Λ), since

    D(ρE^S_X*(ρ))-D(ρE^D_X*(ρ))    = [ρ(log E^D_X*(ρ)-log E^S_X*(ρ))_∈ℱ^D_X] 
       = [E^D_X*(ρ)(log E^D_X*(ρ)-log E^S_X*(ρ))] 
       = D(E^D_X*(ρ)E^S_X*(ρ)) ≥ 0.

Here we used that if ω_1,ω_2 are fixed points of some conditional expectation, then so is logω_1-logω_2.
Since ℱ_X∂^D⊂_X∂⊗ B(ℋ_(X∂)^c)⊂ℱ_X^S holds by construction of the Schmidt conditional expectation, it equally follows from the calculation above that D(ρE^S_X*(ρ)) ≤ D(ρE^D_X∂*(ρ)). 



This is a crucial lemma that allows us to analyse the MLSI for Davies generators in terms of entropic inequalities associated to Schmidt generators, which are easier to analyse. The Schmidt generators hence serves as a proxy QMS to the Davies. Such a comparison is a well known technique for classical Markov chains <cit.>. 








§ STATIC PROPERTIES: CLUSTERING OF CORRELATIONS ON GIBBS STATES




In this section we study the static properties of quantum spin systems. These are used to measure how correlations decay between spatially separated regions on a Gibbs state.

We review various types of spatial clustering and spatial mixing properties ranging from the more commonly considered notions, such as exponential decay of covariance, to stronger ones, such as exponential decay of mutual information. Clustering properties and their refinement “mixing conditions" are central to the geometric divide-and-conquer arguments that establish rapid mixing for spin systems, both in the quantum <cit.> and the classical setting  <cit.>. While the various notions are useful at different steps along the proofs, we will also show equivalence relations between them in the context of Gibbs states of commuting Hamiltonians. For our current purposes these will, however, only be interchangeable on low dimensional lattices, e.g. 1D and 2D hypercubic ones.
In the rest of this section we will assume that Λ is some fixed graph with finite growth constant.



 §.§ Decay of correlations
 

The most commonly used quantifier of correlations on Gibbs states is the covariance. Given two operators X,Y ∈ and a full rank state σ∈𝒟(ℋ), it is defined as

    _σ^,(X,Y) :=⟨ X-[σ X],Y-[σ Y]⟩_σ^,  ,

where the inner product considered is either the KMS (see (<ref>)) or the GNS (see (<ref>)) one. When X=Y, the covariance reduces to the variance

    ^,_σ(X) := ^,_σ(X,X)   .

This allows us to introduce the first notion of clustering, the exponential decay of covariance, or simply exponential decay of correlations (termed 𝕃_∞-clustering in <cit.>). Since all conditions of clustering of correlations considered in this paper exhibit exponential decay, we will omit the reference to this on the names of the properties hereafter. We will define uniform versions of the following notions of clustering. This guarantees that we have clustering independently of the sequence of sub-lattices Γ⊂⊂Λ, since the function l↦ϵ_X,Y(l) and its decay rate do not depend on Γ. It may, however, still depend on the regions of X or Y or their boundaries. Instead, a non-uniform notion of clustering would be one in which we are only guaranteed the existence of a function ϵ_Γ(l) for each σ^Γ, with an explicit dependence on Γ. 

  
Let Φ be a geometrically local potential on Λ and consider an inverse temperature β>0. We say that the family of Gibbs states {σ^Γ}_Γ⊂⊂Λ associated to (Λ,Φ,β) 
satisfies uniform exponential decay of covariance if there exist a function (X,Y,l)↦ϵ_X,Y(l), exponentially decaying in l, with decay rate independent of X,Y such that for any subregion Γ⊂⊂Λ and any A,C⊂Γ with (A,C)=l, it holds that 

    _σ^Γ^(f,g)≤fgϵ_A,C(l),
 for any self-adjoint f,g∈ℬ(ℋ_Γ) with support on A,C, respectively. Uniformity here refers to the fact, that this function does not explicitly depend on Γ. 


This condition is also frequently rewritten in terms of 

    _σ^Γ^(A,C):= 
    f ∈ℬ(ℋ_A), f≤ 1
    
    g ∈ℬ(ℋ_C), g≤ 1sup_σ^Γ^(f,g),
 
so that _σ^Γ^(A,C)≤ϵ_A,C(l). We will use this notation when we want to emphasise the systems between which we are studying correlations. Moreover, since we will mostly focus on the GNS covariance in the rest of the text, we will drop the superscript whenever this is the case. 



    The decay length of the function ϵ_A,C(l) is called the (𝕃_∞)-correlation length ξ, which is the standard decay rate of thermal two-point correlation functions, i.e. -logϵ_A,C(l)=𝒪(l/ξ)_l→∞. It is by the uniformity assumption independent of A,C.



The decay of covariance is a rather weak notion of clustering, hence sometimes referred to as weak clustering. In 1D, it was proven to hold in infinite spin chains, for translation invariant interactions, first in the finite-range regime <cit.> at every inverse temperature β >0, and subsequently at high enough temperature for exponentially-decaying interactions in <cit.> (with a critical temperature tending to zero when approaching the finite-range case). This was extended to the finite chain regime in <cit.> and <cit.> for finite-range and exponentially-decaying interactions, respectively. More recently, <cit.> proved this decay for commuting Hamiltonians without the translation-invariant assumption, as well as slower decays in other regimes. The high dimensional result appears frequently in the literature, with proofs for the finite-range case in e.g. <cit.> and for exponentially-decaying interactions in <cit.>. Additionally, in relation to dynamical properties of QMS, the exponential decay of correlations is known to hold for steady states of rapidly mixing QMS <cit.>.

For completeness, we also introduce the very related property of 𝕃_2-clustering.  

 
Let Φ be a geometrically local potential on Λ and consider an inverse temperature β>0. We say that the family of Gibbs states {σ^Γ}_Γ⊂⊂Λ associated to (Λ,Φ,β) 
satisfies uniform 𝕃_2-clustering if there exists a function (X,Y,l)↦ϵ_X,Y(l), exponentially decaying in l, with decay rate independent of X,Y such that for any subregion Γ⊂⊂Λ and any A,C⊂Γ with (A,C)=l, it holds that

    _σ^Γ^(f,g)≤f_2,σ^Γg_2,σ^Γϵ_A,C(l),
 for any self-adjoint f,g∈ℬ(ℋ_Γ) with support on A,C, respectively. 


This property immediately implies the previous exponential decay of correlations by the monotonicity of 𝕃_p,σ-norms in p. Moreover, the 𝕃_2-clustering is important in gapped primitive QMS.  In <cit.>, it was proven that steady states of gapped primitive QMS satisfy 𝕃_2-clustering, using the detectability lemma <cit.>. 

We now discuss the notion of local indistinguishability <cit.>, which quantifies the influence of the state at spatially separated regions in that of a fixed subregion A.

 
Let Φ be a geometrically local potential on Λ and consider an inverse temperature β>0. We say that the family of Gibbs states {σ^Γ}_Γ⊂⊂Λ associated to (Λ,Φ,β) 
satisfies uniform local indistinguishability <cit.> 
if there exists a function (X,Y,l)↦ϵ_X,Y(l), exponentially decaying in l, with decay rate independent of X,Y such that for any subregion Γ⊂⊂Λ and any A,C⊂Γ with (A,C)=l, it holds that

    _BC[σ^ABC] -_B[σ^AB] _1 ≤ϵ_A,C(l)   .
 



In <cit.> it was shown that Gibbs states of geometrically-local, bounded, possibly non-commuting Hamiltonians, which satisfy uniform decay of covariance, also satisfy uniform local indistinguishability (see also <cit.>), where the exponentially decaying function in the latter l↦ϵ_A,C(l) has an additional factor of |∂ C|, the boundary of the region that is `cut' away, compared to the exponential decay function of the former.
There it was also shown that if the Hamiltonian is commuting then the decay length of the function in the local indistinguishability can be controlled by the thermal correlation length ξ.
For a more recent extension of the latter to exponentially-decaying interactions, and an overview on the relation between these properties and the locality of the Hamiltonian, we refer the reader to <cit.>.

We now also introduce a stronger version of local indistinguishability, which will be crucial for our proofs later on. 

  
Let Φ be a geometrically local potential on Λ and consider an inverse temperature β>0. We say that the family of Gibbs states {σ^Γ}_Γ⊂⊂Λ associated to (Λ,Φ,β) 
satisfies uniform strong local indistinguishability if there exists a function (X,Y,l)↦ϵ_X,Y(l), exponentially decaying in l, with decay rate independent of X,Y such that for any subregion Γ⊂⊂Λ and any A,C⊂Γ with (A,C)=l, it holds that

    (_B[σ^AB])^-1/2_BC[σ^ABC](_B[σ^AB])^-1/2 - _A_∞≤ϵ_A,C(l)   .
  


We will show that the family of Gibbs states associated to (Λ,Φ,β) satisfies uniform strong local indistinguishability with an exponential dependence on the boundaries if it satisfies uniform decay of covariance whenever Λ has finite growth constant, Φ is a bounded, geometrically-local, commuting potential. The requirement for the commuting property may be dropped in 1D systems (see Theorem <ref>). For commuting Hamiltonians, we also give quantitative results in terms of the correlation length ξ and inverse temperature β in <Ref>.

 
Let Φ be a geometrically local potential on Λ and consider an inverse temperature β>0. We say that the family of Gibbs states {σ^Γ}_Γ⊂⊂Λ associated to (Λ,Φ,β) 
satisfies the uniform mixing condition 
if there exists a function (X,Y,l)↦ϵ_X,Y(l), exponentially decaying in l, with decay rate independent of X,Y such that for any subregion Γ⊂⊂Λ and any A,C⊂Γ with (A,C)=l, it holds that

    σ_AC(σ_A ⊗σ_C)^-1 - _AC_∞≤ϵ_A,C(l)   .
 



For Gibbs states of geometrically-local, possibly non-commuting, bounded, and translation invariant Hamiltonians on a 1D lattice this condition was shown to follow qualitatively from uniform exponential decay of covariance in <cit.> at any inverse temperature β >0, and subsequently in <cit.> for arbitrary dimensions at high enough temperature. In <cit.>, this was crucial to establish the existence of a log-decreasing MLSI constant α=Ω(log|Γ|)^-1 for commuting quantum spin chain systems . 

The last quantity to measure correlations explored in this section is the mutual information. We recall that, given a bipartite space _A ⊗_B and ρ_AB a density matrix on it, it is defined as 

    I_ρ(A:B) := [ρ_AB(logρ_AB - logρ_A ⊗ρ_B)] = D(ρ_ABρ_A⊗ρ_B)   .


 
Let Φ be a geometrically local potential on Λ and consider an inverse temperature β>0. We say that the family of Gibbs states {σ^Γ}_Γ⊂⊂Λ associated to (Λ,Φ,β) 
satisfies uniform exponential decay of mutual information 
if there exists a function (X,Y,l)↦ϵ_X,Y(l), exponentially decaying in l, with decay rate independent of X,Y such that for any subregion Γ⊂⊂Λ and any A,C⊂Γ with (A,C)=l, it holds that

    I_σ^Γ(A:C) ≤ϵ_A,C(l),
 holds. 


By a standard use of Pinsker's and Hölder's inequalities, the decay of the mutual information directly implies decay of the covariance in <Ref> with the decay rate halved, since 
    I_σ(A:C)=D(σ_ACσ_A⊗σ_C)≥1/2σ_AC-σ_A⊗σ_C^2_1.
 
Additionally, by <cit.>, the following inequality holds

    I_σ(A:C)≤σ_AC(σ_A⊗σ_C)^-1-_AC_∞  .

so that the mixing condition from <Ref> also implies the usual decay of covariance (𝕃_∞-clustering).  
We show in the next subsections that these implications can be reversed in the case of commuting, finite-range Hamiltonians. This allows us to conclude that to establish the mixing condition, or the mutual information decay for the above considered systems, it is enough to establish the decay of the covariance.






 §.§ A useful relation



To simplify the notation in the rest of this section, we first define the following relation.


Given a finite lattice Γ and two states ω,τ∈𝒟(ℋ_Γ) with the same support (ω)=(τ), the relation ∼ is

    ωϵ∼τ :⟺ω^1/2τ^-1ω^1/2-≤ϵ<1,

where the identity ≡_(ω)=_(τ) is on the support of the states. The inverse represents the generalized inverse, i.e. the inverse on the support times the projection onto it.
 
 
    When restricted to states, it holds that the notion of similarity induced by this relation is equivalent to the one induced by the max-relative divergence. I.e. for ω,τ∈𝒟(ℋ) such that ωϵ∼τ it holds that
    
    D_max(ωτ)≤log(1+ϵ) ≤ϵ = ω^1/2τ^-1ω^1/2-≤ D_max(ωτ)exp(D_max(ωτ)).


A proof of this can be found in Appendix <ref>.

It directly follows that ωϵ∼τ D(ωτ)≤ϵ and by Hölder's inequality it follows that ωϵ∼τω-τ_1≤ϵ, but the converse is in general not true. Hence, this relation quantifies a stronger form of similarity between a pair of states than closeness in relative entropy and 1-norm. 
Despite the equivalence to the max-relative entropy, when restricting to states, it will often be simpler to work directly with the relation instead of the max-relative entropy, and thereby avoid a logarithm. 


We will often have ϵ be some exponentially decaying function depending on the supports of ω,τ, as the clustering functions discussed above. Thus, by a slight abuse of notation, we may write ω∼τ when meaning that there exists some exponentially decaying function ϵ(l) with ωϵ∼τ, where l is a suitable parameter (most often a distance between regions).
The mathematical terminology relation is justified as per the following proposition. 


Given a bipartite system 𝒦=ℋ⊗ℋ^', we consider positive[In fact self-adjoint would be sufficient, but for simplicity we restrict ourselves to positive operators here.] A,Ã,B,C∈ℬ(ℋ), D,E∈ℬ(𝒦), and F,F̃∈ℬ(ℋ^') and P∈ℬ(ℋ^') a projection. 
The above defined relation ϵ∼ satisfies the following properties:

    
  1)  Reflexivity: A0∼ A.  
    
  2) Symmetry:  Aϵ∼ B  Bϵ(1-ϵ)^-1∼ A, B^-1ϵ∼ A^-1. 
    
  3) Transitivity: Aϵ_1∼ B, Bϵ_2∼ C  Aη∼ C, where η = ϵ_1ϵ_2+ϵ_1+ϵ_2.
    
  4) Tensor multiplicativity:  Aϵ_1∼A, Fϵ_2∼F A⊗ F η∼A⊗F, where η =ϵ_1ϵ_2+ϵ_1+ϵ_2.
    
  5) Locality preservation:  Dϵ∼ E _ℋ^'(⊗ P)D(⊗ P) ϵ∼_ℋ^'(⊗ P)E(⊗ P). 
    
  6) Normalization preservation: Dϵ∼ E _ℋ^'(⊗ P)D(⊗ P)/[(⊗ P)D(⊗ P)]ϵ(2+ϵ)∼_ℋ^'(⊗ P)E(⊗ P)/[(⊗ P)E(⊗ P)]. 

    

For notational simplicity, we may write Aϵ_1∼Bϵ_2∼C, implying transitivity, when we mean Aϵ_1∼B, Bϵ_2∼C. The following corollary can be derived as a consequence of the previous proposition.  

 
 If A_i ϵ∼ A_i+1 for i= 0,...,K-1, then A_0η∼A_K  with η = (1+ϵ)^K-1.

The above corollary plays an important role in the estimation of the mixing condition between separate regions composed of several connected components.
The proofs of Proposition <ref> and Corollary <ref> are elementary, but we include them for completeness in <Ref>.


    Using the relation introduced in this subsection, we can rewrite the strong local indistinguishability of a state σ^ABC as 
    
    _BC[σ^ABC] ϵ_A,C(l)∼_B[σ^AB] ⟺ D_(_BC[σ^ABC]_B[σ^AB])≲ϵ_A,C(l)    ,

while the mixing condition between A and C can be expressed as 

    σ_ACϵ_A,C(l)∼σ_A⊗σ_C ⟺ I_(A:C)≲ϵ_A,C(l)   .
 




 §.§ Relation between measures of correlations


We are now in a position to prove the main result of this section: the equivalence between the seemingly weaker notion of decay of correlations provided by the covariance and stronger notions such as strong local indistinguishability and the mixing condition. Note that this result will only hold in full generality for Gibbs states of commuting Hamiltonians, and that the equivalence is up to exponential pre-factors that grow with the boundaries of the relevant regions. As such, the results are more useful for low-dimensional systems, such as in 1D and 2D, where these boundaries are not too large compared to the distance between regions.

 
Let Φ be a bounded, geometrically-r-local, commuting potential and β>0. Then if the family of Gibbs states {σ^Γ}_Γ⊂⊂Λ associated to (Λ,Φ,β) satisfies uniform exponential decay of covariance with decay rate ξ, it satisfies, 

1) Uniform strong local indistinguishability with decay length ξ. That is for each ABC=Γ⊂⊂Λ, where A,B,C are disjoint and l=(A,C)>r, it holds that

    _BC[σ^ABC]ϵ(l)∼_B[σ^AB]  with ϵ_A,C(l)= e^𝒪(βmin{|∂ A|,|∂_A B|})𝒪(|∂ C|)exp(-l-r/ξ).
 Here ∂_AB=∂ B∩ A.



 2) Uniform mixing condition with decay length ξ. That is for each ABC=Γ⊂⊂Λ, where A,B,C are disjoint and l=(A,C)>3r, σ^Γ satisfies

    σ_ACη(l)∼σ_A⊗σ_C  with η_A,C(l)= e^𝒪(β(|∂ A|+|∂ C|))exp(-l-2r/ξ).

In particular, by (<ref>), it also satisfies 
uniform exponential decay of mutual information with

    I_σ^ABC(A:C) ≤η_A,C(l)   .







In the statement of the theorem we have an exponential decay w.r.t (A,C) but we also have a spurious pre-factor which is growing exponentially in the size of the boundary of A for the strong local indistinguishability and A and C for the mixing condition. This is a consequence of the proof technique we are using and we expect it to be not physically tight. 


We will later be using these clustering results on connected and growing sets A,C. For the 1 dimensional spin chains these prefactors are just constant and thus negligible. In the 2-dimensional square lattice this exponentially increasing pre-factor can still be dominated by the decay in the distance if the decay length ξ is short enough. 

Since by <cit.> the existence of a gap in the QMS with fixed point σ implies uniform exponential decay of covariance, we immediately have strong local indistinguishability, mixing condition, and exponential decay of the mutual information from the gap property. This implies that 1-dimensional quantum spin chains satisfy these properties at any temperature for geometrically-local, commuting, bounded Hamiltonians and in D-dimensional regular latices, although with exponential prefactors in the boundaries of local regions. 

With this in mind, implication 1) above is a strict strengthening of the local indistinguishability result in <cit.> for commuting Hamiltonians. Implication 2) can be viewed as an extension of the results in <cit.> to any lattice with finite growth constant under the additional condition of commutativity of the Hamiltonian. 

Before proving Theorem <ref>, we note that for 1-dimensional quantum spin chains we can also establish the above results and implications for non-commuting Hamiltonians. In this case, 2) is the main result of <cit.>, and 1) is as follows. 


Let Φ be a geometrically-r-local and J-bounded potential on  and β>0 such that the to (,Φ,β) associated family of Gibbs states {σ^Γ}_Γ⊂⊂ satisfies uniform exponential decay of covariance.
Then there exist constants K,a>0, such that for each interval I=ABC⊂⊂ℤ, where B shields A away from C and 2l:=|B|= (A,C)>r it holds that

    (_BC[σ^ABC])(_B[σ^AB])^-1-≤ Ke^-al.

Note that K,a>0 depend only on the interaction range r and effective strength β J.



<Ref> will be proven in <Ref>. Its proof is analogous to the one for <Ref>, using some additional technical prerequisites from <cit.>.
For the latter we will first need the following technical Lemma.
Let us use the standard notation

    E_A,B^β:=exp(-β H_AB)exp(β (H_A+H_B))

to denote Araki's expansionals. We omit the superscript with the inverse temperature β >0 when it is unnecessary or clear from the context. 




Let Φ be a geometrically-r-local, J-bounded, commuting potential on a quantum spin system Λ with finite growth constant ν. Let Γ=ABC⊂⊂Λ, with B shielding A from C. Then, if we denote by ∂_A B the boundary of B in A, i.e. ∂_A B := (∂ B) ∩ A, similarly ∂_BA and ∂_A,B:=min{|∂_B A|,|∂_A B|} the following bounds hold with K_A,B:=exp(𝒪(β |∂_A,B|)) independent of l:=(A,C).

    
  1) For every β>0, we have
    
    E_A,B^±1≤ K_A,B  .

    
  2) For any strictly positive Q∈ℬ(ℋ_Λ), we have 
    
    Q^∓1^-1≤_B[σ^BQ]^±1≤Q^±1  .
 
    
  3) The following bounds also hold respectively
    
    {_B[σ^BE_A,B^±1]^±1 ,  _B[σ^BE_B,C^±1]^±1 ,  [σ^ABE_A,B^±1]^±1}≤{K_A,B,K_B,C,K_A,B}  .

    
    _B[σ^BE_A,B^±1E_AB,C^±1]^±1≤ K_A,BK_B,C  .


The same inequalities hold when exchanging the order of σ and the expansionals inside the partial traces. 
Note that the big-𝒪 notation refers to the dependence in β and the boundaries and omits dependence on J,d,r,ν. 



The proof of this result is deferred to <Ref>.


Our proof of Lemma <ref> requires the commutativity of the Hamiltonian, since we require E_A,B≥0, see 2). If positivity could be proven without this assumption, 
or 3) directly some other way, we believe that we could establish <Ref> without the additional assumption of commutativity. 

By the combined use of Lemma <ref>, clever rewritings inspired by the proofs in <cit.>, and repeated application of local indistinguishability, we can prove the main theorem of this section. 


We first note that the following holds: 

    ω^1/2τ^-1ω^1/2-≤ωτ^-1- ,

by e.g. <cit.>.
Set l:=(A,C)>r. Since we are assuming uniform exponential decay of covariance with correlation length ξ, we may write

    Cov_σ(A:C) ≤K̃exp(-l/ξ)

for some constant K̃>0. 



1) To show strong local indistinguishability (<ref>), first note that

    _BC[σ^ABC] =  e^-H_A_BC[σ^BCE_A,BC]  Z_BC/Z_ABC  ,

with Z_χ :=[e^-H_χ] for any χ⊂Γ. Then, we start by rewriting

    (_BC[σ^ABC])(_B[σ^AB])^-1 = _BC[σ^BCE_A,BC]_B[σ^BE_A,B]^-1λ_ABC^-1  ,

where 

    λ_ABC=Z_ABC Z_B/Z_AB Z_BC = [σ^ABE^-1_A,B]/[σ^ABCE_A,BC^-1]  .

Note that whenever we omit the subscript in [X], we are referring to a total trace in the subsystems where X has non-trivial support.
Thus, we need to upper bound 

    _BC[σ^BCE_A,BC]_B[σ^BE_A,B]^-1λ_ABC^-1 -   .

We do this by splitting the proof into several claims, proven independently.




Claim 1: |λ_ABC^∓1-1| is exponentially decaying in l with decay length ξ. 




Proof: Considering first λ_ABC, we have

    |λ_ABC-1|   = 1/[σ^ABCE_A,BC^-1]|[σ^ABE_A,B^-1]-[σ^ABCE_A,BC^-1]| 
       ≤E_A,BC^+1|[σ^ABE_A,B^-1]-[σ^ABCE_A,BC^-1]|,
 
where the last inequality follows from Lemma <ref> 2). Now, set B=B_1B_2 with B_1:=∂ A, B_2=B∖ B_1, s.t. (A,B_2)=r,  (B_1,C)≥ l-r. We also set A=A_0A_1, where A_1=∂_AB and A_0=A∖ A_1 Then E_A,BC=E_A,B=E_A_1,B_1. Therefore, 

    |[σ^ABE_A,B^-1]-[σ^ABCE_A,BC^-1]|    = |_AB_1[_B_2(σ^ABE_A,B^-1)]-_AB_1[_B_2C(σ^ABCE_A,BC^-1)]| 
       =|_AB_1[(_B_2[σ^AB]-_B_2C[σ^ABC])E_A_1,B_1^-1]| 
       ≤_B_2[σ^AB]-_B_2C[σ^ABC]_1E_A_1,B_1^-1 ,

where the last line follows from Hölder's inequality, and where we write the subscript in  to emphasize the systems over which we are tracing out. 
By <cit.>, uniform exponential decay of correlations implies local indistinguishability with the same decay and an additional factor |∂ C|. So we have with Lemma <ref> 1) that

    (_B_2[σ^AB]-_B_2C[σ^ABC])_1E_A_1,B_1^-1   ≤ K_A,B |∂ C|K̃exp(-1/ξ(B_1,C)) 
       =K_A,BK̃ |∂ C|exp(-l-r/ξ).

This allows us to conclude 

    |λ_ABC-1|≤ K_A,B^2 K̃|∂ C|exp(-l-r/ξ)   .

The same follows for |λ^-1_ABC-1| analogously or by application of the geometric series to the above. 
This concludes the proof of Claim 1. 
Now we can rewrite

    _BC[σ^BCE_A,BC]_B[σ^BE_A,B]^-1λ_ABC^-1 -

to

    (_BC[σ^ABC])(_B[σ^AB])^-1-   =_BC[σ^BCE_A,BC]_B[σ^BE_A,B]^-1λ_ABC^-1 - 
    
          ≤_BC[σ^BCE_A,BC](_B[σ^BE_A,B])^-1|λ_ABC^-1-1| 
          +(_B[σ^BE_A,B])^-1_BC[σ^BCE_A,BC]-_B[σ^BE_A,B].



Claim 2: _BC[σ^BCE_A,BC]-_B[σ^BE_A,B] is exponentially decaying in l with decay rate ξ. 



Proof:
Again set B=B_1B_2, B_1:=∂ A, B_2:=B∖ B_1, and split A into A_1:= ∂_A B and A_0 := A ∖ A_1. Thus (B_1,C)≥ l-r and E_A,BC=E_A,B=E_A_1,B_1. Then, by local indistinguishability

    _BC[σ^BCE_A,BC]-_B[σ^BE_A,B]   = _B_1[(_B_2[σ^B_1B_2]-_B_2C[σ^B_1B_2C])E_A_1,B_1]
       ≤E_A_1,B_1d^|A_1| K  |∂ C|exp(-(B_1,C)/ξ)
    
        
           ≤  K_A,Bd^|∂_AB|K  |∂ C| exp(-l-r/ξ).
 

This concludes the proof of Claim 2.



Putting everything together, we get the desired result,

    (_BC[σ^ABC])(_B[σ^AB])^-1-
       ≤  K_A,B^4|∂ C|K̃exp(-l-r/ξ)+K_A,B^2d^|∂_A B| K^2K|∂ C|exp(-l-r/ξ) 
       = exp𝒪(β|∂_A,B|)𝒪(|∂ C|)exp(-l-r/ξ).


2) Assume l≥ 3r. To prove the mixing condition/strong tensorization (<ref>), following the steps above, or similarly those of <cit.>, we can rewrite

    σ_AC(σ_A⊗σ_C)^-1-   ≤ K_A,BK_B,C_B[σ^BE_A,BE_AB,C]|λ_ABC-1|
         + K_A,BK_B,C_BC[σ^BCE_A,BC]_AB[σ^ABE_AB,C]-_B[σ^BE_A,BE_AB,C]
       ≤ K_A,B^4K_B,C^2K|∂ C|exp(-l-r/ξ) 
       +K_A,BK_B,C_BC[σ^BCE_A,BC]_AB[σ^ABE_AB,C]-_B[σ^BE_A,BE_AB,C],


where the second inequality follows from Claim 1 as well as Lemma <ref>.



Claim 3: _BC[σ^BCE_A,BC]_AB[σ^ABE_AB,C]-_B[σ^BE_A,BE_AB,C] is exponentially decaying in l with correlation length ξ. 



Proof: Set B=B_1B_2B_3 with B_1:=∂ A, B_3:=∂ C, B_2:=B∖(B_1∪ B_3), then (B_1,B_3)≥ l-2r and E_A,BC=E_A,B=E_A,B_1 and E_AB,C=E_B,C=E_B_3,C and consequently

    _BC[σ^BCE_A,BC]_AB[σ^ABE_AB,C]-_B[σ^BE_A,BE_AB,C]
        = _BC[σ^BCE_A,B_1]_AB[σ^ABE_B_3,C]-_B[σ^BE_A,B_1E_B_3,C]
       ≤_B[σ^BE_A,B_1E_B_3,C]-_B[σ^BE_A,B_1]_B[σ^BE_B_3,C]_(I)
        +_B[σ^BE_A,B_1]_B[σ^BE_B_3,C]-_BC[σ^BCE_A,B_1]_AB[σ^ABE_B_3,C]_(II).

Next, we bound (I) and (II) separately. To bound (II) we use that, by the proof of Claim 2,

    _B[σ^BE_A,B_1]-_BC[σ^BCE_A,B_1]   ≤E_A,B_1_B_2B_3[σ^B]-_B_2B_3C[σ^BC]_1 
       ≤ K_A,Bd^|∂_AB||∂ C|Kexp(-1/ξ(B_1,C)), 
    _B[σ^BE_B_3,C]-_AB[σ^ABE_B_3,C]   ≤E_B_3,C_B_1B_2[σ^B]-_AB_1B_2[σ^AB]_1
       ≤ K_B,Cd^|∂_CB||∂ A|Kexp(-1/ξ(A,B_3)).

Then, putting both bounds together, 
    (II)   ≤_B[σ^BE_A,B_1] _B[σ^BE_B_3,C]-_AB[σ^ABE_B_3,C]
          + _B[σ^BE_B_3,C]_B[σ^BE_A,B_1]-_BC[σ^BCE_A,B_1]
       ≤ K_A,BK_B,Cd^|∂_CB||∂ A|K̃exp(-l-r/ξ)+K_B,CK_A,Bd^|∂_AB||∂ C|K̃exp(-l-r/ξ)
       =exp(𝒪(β|∂ A|+β|∂ C|))𝒪(|∂ A|+|∂ C|)exp(-l-r/ξ).


To bound (I), we use that exponential decay of covariance directly implies

    σ_AC-σ_A⊗σ_C_1≤K̃exp(-(A,C)/ξ)

for σ≡σ^ABC, by Hölder duality. Thus,

    (I)   = _B_1B_3[σ_B_1B_3(E_A,B_1⊗ E_B_3,C)]-_B_1[σ_B_1E_A,B_1]⊗_B_3[σ_B_3E_B_3,C]
       = _B_1B_3[(σ_B_1B_3-σ_B_1⊗σ_B_3)(E_A,B_1⊗ E_B_3,C)] 
       Proof of Claim 2≤E_A,B_1⊗ E_B_3,Cσ_B_1B_3-σ_B_1⊗σ_B_3_1d^|∂_AB|+|∂_CB|
       ≤ K_A,BK_B,CKexp(-B_1,B_3/ξ) = exp𝒪(β|∂ A|+|∂ C|)exp(-l-2r/ξ).

Therefore, we conclude the proof of Claim 3, since putting the bounds above together yields

    _BC[σ^BCE_A,BC]_AB[σ^ABE_AB,C]-_B[σ^BE_A,BE_AB,C]
        = exp(𝒪(β|∂ A|+β|∂ C|))𝒪(|∂ A|+|∂ C|)K̃exp(-l-2r/ξ)   .

 Finally, we can obtain the following bound for the mixing condition 

    σ_AC(σ_A⊗σ_C)^-1-≤exp𝒪(β(|∂ A|+|∂ C|))𝒪(|∂ A|+|∂ C|)exp(-l-2r/ξ).

The result on the mutual information follows directly from the one for the mixing condition by (<ref>). 

We can also get the mixing condition directly from strong local indistinguishability. 


Let  Λ be a graph with finite growth constant. Define S_Λ(l):=sup_x∈Λ|{v∈Λ|(x,v=l)}| the surface area of the maximal l-sphere in Λ. Note that we have

    ^D: D S_^D(l) = 𝒪(l^D-1), 
       Λ S_Λ(l) = exp(o(l)), 
       𝕋_b: b S_𝕋_b(l) = b^l.

Now if {σ^Γ}_Γ is a family of states which satisfy uniform strong local indistinguishability with decay function l↦ϵ_A,C(l), i.e. each σ≡σ^Γ satisfies _BCσ^ABC(_Bσ^AB)^-1-≤ϵ_A,C(l), then there exists a suitable function l↦η_A,C(l) such that this family also satisfies the uniform mixing condition as σ_AC(σ_A⊗σ_C)^-1-≤η_A,C(l) satisfying the following implications.

    ϵ_A,C(l) = exp(𝒪(|∂ A|))𝒪(|∂ C|)exp(-l/ξ)    η_A,C(l) = exp(𝒪(|∂ A|+|∂ C|))S_Λ(l/3)^3exp(-l/3ξ), 
    ϵ_A,C(l) = 𝒪(|∂ A|,|∂ C|)exp(-l/ξ)    η_A,C(l) = 𝒪((|∂ A|,|∂ C|))S_Λ(l/3)^3exp(-l/3ξ).
 

 
    In the case where strong local indistinguishability follows from decay of covariance, this derivation provides no improvement in scaling over <Ref>. However, whenever one can assume strong local indistinguishability with a polynomial prefactor on subexponential, respectively, polynomial graphs, then this directly gives us the mixing condition with a sub-exponential, respectively, polynomial prefactor.  


Fix some Γ⊂⊂Λ and some A,C⊂Γ such that (A,C)=l>3r. W.l.o.g. we assume l is divisible by 3, else use l-1 or l-2. Split B:=Γ∖(AC)=B_1B_2B_3 into 3 regions, where B_1:=∂_l/3A:={x∈ B|(x,A)≤l/3} and B_3:=∂_l/3C. Then (A,B_2),(B_1,B_3),(B_2,C)≥l/3. Now by the transitivity and tensor multiplicativity of the relation ∼ (see proposition <ref>) we have that

    σ_AC = _B_1B_2B_3σ^AB_1B_2B_3Cϵ_1∼_B_1B_3σ^AB_1⊗σ^B_3C =  ϵ_2ϵ_3+ϵ_2+ϵ_3∼_BCσ^ABC⊗_ABσ^ABC = σ_A⊗σ_C.

Using the crude bounds |B_1|≤ |∂ A|f_Λ(l/3) and |B_3|≤ |∂ C|f_Λ(l/3), and the assumptions on the functions ϵ_i arising from strong local indistinguishability directly yields the claim. Note that the A,C in the mixing condition are not necessarily the same regions A,C as in the strong local indistinguishability.





 §.§ q𝕃_1→𝕃_∞-clustering from decay of covariance or temperature

Here we consider a notion of clustering introduced in <cit.> as q𝕃_1→𝕃_∞-clustering, where it is instrumental in implying rapid thermalization of the Schmidt dynamics. As such, this notion of clustering will be key in our proofs of rapid mixing.

This is in principle a more abstract notion of clustering defined w.r.t the family of invariant states of a family of Lindbladians with certain properties. In this section we will only consider the family of Lindbladians {ℒ^S_Γ}_Γ⊂⊂Λ to be the one corresponding to the Schmidt conditional expectations {E^S_X}_X⊂Γ introduced in <ref>. In this case 

The uniform family of primitive, reversible, and frustration-free Schmidt generators {ℒ^S_Γ}_Γ⊂⊂Λ with fixed points {σ^Γ}_Γ⊂⊂Λ satisfies uniform q𝕃_1→𝕃_∞-clustering of correlations if 
there exists a function (C,D,l)↦η_C,D(l), exponentially decaying in l, with decay rate independent of C,D such that for two overlapping subregions C,D⊂Γ with l:=(C∖ D,D∖ C), it holds that


    max_α={α_i}_i∈ I_C∪ DE^S,(α)_C∘ E^S,(α)_D-E^S,(α)_C∪ D: 𝕃_1(τ_C∪ D^α)→𝕃_∞≤η_C,D(l)≡η(l),
 where {E^S_X}_X⊂⊂Λ are the Schmidt conditional expectations and (α) are the boundary conditions of the subset CD:=C∪ D⊂Λ. 


Recall that the Schmidt conditional expectations can only sensibly be defined for nearest neighbour interacting systems, hence this notion of clustering can only exist on systems with geometrically-2-local potentials.


In the following, we show that this notion of clustering is implied by uniform decay of covariance via strong local indistinguishability, 
which constitutes another result of independent interest from this work.  


Let Λ be a 
graph with finite growth constant and let Φ be a bounded, commuting, nearest-neighbour potential on Λ.  Then if, for some β>0, the family of Gibbs states {σ^Γ}_Γ⊂⊂Λ associated to (Λ,Φ,β) satisfies uniform exponential decay of covariance, then the Schmidt generators associated to (Λ,Φ,β) 
satisfy uniform q𝕃_1→𝕃_∞-clustering as in (<ref>), i.e.



   uniform ϵ(l)-decay of covariance  η_C,D(l)-q𝕃_1→𝕃_∞-clustering, 


 where η_C,D(l) = (𝒪(|∂ (C∖ D)|+|∂ (D∖ C)|))𝒪(1)ϵ(l).
The ⟸ implication and its proof can be found in <cit.>. That implication also follows from  <Ref> in the case Λ=ℤ.


An example of subsets C,D can be found in <Ref>. 


The exponential scaling of the prefactors of the decay in the q𝕃_1→𝕃_∞-clustering comes directly from Theorem <ref>, so if one had strong local indistinguishability with a linear (or polynomial) dependence on the boundary regions, then the decay in the q𝕃_1→𝕃_∞-clustering would also only be at worst polynomial in the boundaries of C∖ D and D∖ C.


Such a decay is already known to hold at high enough temperatures. Before proving <Ref>, we also present it in the context of our work here, as it will be a basis for our second main result.


Let Λ be a graph with finite growth constant ν and Φ a uniformly J-bounded, commuting, geometrically-2-local potential on it. Then if the temperature β^-1 is high enough, as 0≤β<1/10eJν, the family of Schmidt generators associated to (Λ,Φ,β) satisfies uniform q𝕃_1→𝕃_∞-clustering as in (<ref>) with decay function 
    η_C,D(l)≤ K|C∪ D|exp(-l/ξ^'),

where l=(C∖ D,D∖ C) and K,ξ^'>0 are some fixed constants independent of the local regions C,D, or Γ.    

Note that in this so-called the “high temperature" regime we have a linear dependence of the pre-factor on the size of the local regions, as compared to an exponential one in Theorem <ref>.
Although in <cit.> they only explicitly consider hyper-cubic lattice, the proofs there works equally for lattices with bounded degree, including trees. 



We show that, under the assumptions of the theorem, the strong local indistinguishability and mixing condition imply 

    max_(α)E^S,(α)_C∘ E^S,(α)_D-E^S,(α)_C∪ D: 𝕃_1,τ^C∪ D_α→𝕃_∞≤η(l)
 explicitly for the Schmidt conditional expectations. 
Since both of these are, by Theorem <ref>, implied by uniform decay of correlations, this proves the theorem. 
Here (α) is a boundary condition of the subset CD:=C∪ D⊂Λ and η(l) is exponentially decaying with decay length ξ. We split the proof into two steps: First, we establish that what we need to show is algebraically equivalent to a statement σ_1η(l)∼σ_2 for two states σ_1,σ_2, and we then employ results of <Ref> to prove this statement from the assumptions of the theorem.

Before we continue, let us first establish some nomenclature for the subregions we are considering. We split the region CD into the following disjoint subsets l̃_l̃_El_ l_ F r_ r_ G r̃_r̃_, where C=ElF, C_=l̃_ElFr_, D=FrG, D_ = l_ FrG r̃_. The projectors P^(α_l̃),P^(γ_l),P^(β_r),P^(α_r̃) act on the regions l̃,l,r,r̃ respectively. For a graphical representation of this see <Ref>.



Recall the notation for the Schmidt conditional expectation established in <Ref> and equation (<ref>).
Fix a boundary condition (α)=(α_l̃,α_r̃)={α_i}_i∈ I_CD for CD, where (α_l̃):={α_i}_i∈ I_∂ C∖ D labels the boundary of C not in D and (α_r̃):={α_i}_i∈ I_∂ D∖ C labels the boundary of D not in C. Similarly, denote with (β_r):={β_i}_i∈ I_∂ C∩ D and (γ_l):={γ_i}_i∈ I_∂ D∩ C the boundaries of C in D and of D in C respectively. For visualization, see also <Ref>.
Let 0≤ X≡ X^(α)∈𝕃_1,τ^(α)_CD_ such that X_1,τ^(α)_CD_=1. Set

    N≡ N^(α_r̃):=E_D^S,(α_r̃)(X)=⊕_(γ_l)E_D^S,(γ_l,α_r̃)(X) ∈ℬ(ℋ_D^(α_r̃)_)⊗_D^(α_r̃)_.
 By construction it holds that P^(α_r̃)NP^(α_r̃)=N. Recall that since D⊂ CD, it follows that E^S_CD∘ E_C^S=E^S_CD and hence


    (E_C^S,(α)∘ E_D^S,(α)-E_CD^S,(α))(X) 
       = (⊕_(β_r)E_C^S,(α_l̃,β_r)-E_CD^S,(α_l̃,α_r̃))(N) 
       = ⊕_(β_r)(P^(α_l̃)P^(β_r)(_C_,(α_l̃,β_r)[τ_C_^(α_l̃,β_r)N]⊗_C_)P^(α_l̃)P^(β_r)) 
       -  P^(α_l̃)P^(α_r̃)(_CD_,(α_l̃,α_r̃)[τ_CD_^(α_l̃,α_r̃)N]⊗_CD_)P^(α_l̃)P^(α_r̃)
       =P^(α_l̃)[⊕_(β_r)P^(β_r)(_C_,(α_l̃,β_r)[τ_C_^(α_l̃,β_r)P^(α_r̃)(N⊗_D_)P^(α_r̃)]⊗_C_)P^(β_r). 
       - . (⊕_(β_r)P^(β_r))P^(α_r̃)(
        _CD_,(α_l̃,α_r̃)[τ_CD_^(α_l̃,α_r̃)(N⊗_D_)]⊗_CD_)P^(α_r̃)] P^(α_l̃).

In the first line here we used the definition of X, in the second the explicit expressions for the Schmidt conditional expectation from <Ref>. Then in the third we factor out a common P^α_l̃ and introduce an identity =⊕_(β_r)P^(β_r), which commutes with the term just after. We also employ the fact that N≡ P^α_rN⊗_D_P^α_r. Therefore, we can write:

    (E_C^S,(α)∘ E_D^S,(α)-E_CD^S,(α))(X)
            =(P^(α_l̃)P^(α_r̃)⊕_(β_r)P^(β_r))_C_∖ D_,(α_l̃)[(_C_∩ D_,(β_r)[τ_C_^(α_l̃,β_r)]_=:σ_2^(α_l̃,β_r)≡σ_2-_D_,(α_r̃)[τ_CD_^(α_l̃,α_r̃)]_=:σ_1^α_l̃≡σ_1)N]×
               (P^(α_l̃)P^(α_r̃)⊕_(β_r)P^(β_r)) ≡⊕_(β_r)P^(α_l̃,β_r,α_r̃)[_C_∖ D_[(σ_2-σ_1)N]]P^(α_l̃,β_r,α_r̃),

where σ_2≡σ_2^(α_l̃,β_r):=_C_∩ D_,(β_r)[τ_C_^(α_l̃,β_r)] and σ_1≡σ_1^(α_l̃):=_D_,(α_r̃)[τ_CD_^(α_l̃,α_r̃)].
Here, we factored out the projections and rearrange the partial traces suitably. The last equality is then just introducing a simplifying notation. 
For simplicity we suppress the boundary conditions (α_l̃) index on the states. 
Hence

    (E_C^S(α)∘ E_D^S(α)-E_CD^S(α))(X)   ≤max_(β_r)_C_∖ D_,(α_l̃)[(σ_2-σ_1)N]
       = max_(β_r)|_C_∖ D_, (α_l̃)[(σ_2-σ_1)N]| 
       = max_(β_r)|_C_∖ D_, (α_l̃)[(-σ_1^-1/2σ_2σ_1^-1/2)(σ_1^1/2Nσ_1^1/2)]| 
       ≤max_(β_r)(-σ_1^-1/2σ_2σ_1^-1/2)(σ_1^1/2Nσ_1^1/2)]_1 
       ≤max_(β_r)σ_1^-1/2σ_2σ_1^-1/2-_∞N_𝕃_1,σ_1,
 
where the equality in the second line follows since N=E_D^S(γ_l,α_r̃)(X) is the identity on the complementary Hilbert space to (ℋ_C_∖ D_^(α_l̃)). In the last inequality we used Hölder and the definition of the 𝕃_1,σ_1 norm.
By definition of N, we have

    N_𝕃_1,σ_1   = σ_1^1/2Nσ_1^1/2_1 N≥0=[σ_1N] = [τ_CD_^(α_l̃,α_r̃)N] = [τ_CD_^(α)E_D^S(α)(X)] 
       = [E^S(α)_D*(τ_CD_^(α))X]=[τ_CD_^(α)X] X≥ 0=X_𝕃_1,τ_CD_^(α) = 1.

Hence, to prove the theorem, we need to establish that σ_1ϵ∼σ_2 for any boundary condition (α_l̃,β_r,α_r̃). We will do it with l↦ϵ(l) an exponentially decreasing function in l=(C∖ D,D∖ C) with decay length ξ. The two states can be written as

    σ_1    :=_D_,(α_r̃)[τ_CD_^(α_l̃,α_r̃)] = 1/[...]_∂_ (CD)∪ D_,(α_r̃)[P^(α_l̃)P^(α_r̃)σ^(CD∂)P^(α_l̃)P^(α_r̃)] 
       = 1/[...]_l̃_ l_ Dr̃[P^(α_r̃)P^(α_l̃)σ^(CD∂)P^(α_l̃)] ∈ℬ(ℋ_C_∖ D_^(α_l̃)), 
    σ_2    :=_F_,(β_r)[τ_C_^(α_l̃,β_r)] = 1/[...]_∂_ (C)∪ F_,(β_r)[P^(α_l̃)P^(β_r)σ^(C∂)P^(α_l̃)P^(β_r)] 
       = 1/[...]_l̃_ l_ Fr[P^(β_r)P^(α_l̃)σ^(C∂)P^(α_l̃)] ∈ℬ(ℋ_C_∖ D_^(α_l̃)).

Both are full-rank states on ℋ_C_∖ D_^(α_l̃) and thus have the same support.
The intuition now is as follows: since we have a Gibbs state that satisfies exponential decay of covariance, strong local indistinguishability, and the mixing condition (see <Ref>), these two states should be approximately the same in the bulk, where we compare them. This is because they might only differ significantly on D∖ C, but we look at them on C ∖ D. 


From the assumptions of the Theorem we have that the family of Gibbs states satisfy decay of covariance and hence by <Ref> also strong local indistinguishability and the mixing condition. We start by applying the mixing condition from (<ref>) to each of the states, which guarantees the existence of two exponentially decaying functions ϵ_1,ϵ_2 in (l,r̃)>(C∖ D,D∖ C) and (l,r)=(C∖ D,D∖ C), respectively, s.t.

    _l_ Dσ^CD∂ϵ_1∼_l_ Dr̃σ^CD∂⊗_l̃ElDσ^CD∂
       <ref>_r̃P^(α_r̃)_l_ Dσ^CD∂P^(α_r̃)ϵ_1∼_l_ Dr̃σ^CD∂[P^(α_r̃)σ^CD∂P^(α_r̃)], 
       _l_ Fσ^C∂ϵ_2∼_l_ Frσ^C∂⊗_l̃ElFσ^C∂
       <ref>_rP^(β_r)_l_ Fσ^C∂P^(β_r)ϵ_2∼_l_ Frσ^C∂[P^(β_r)σ^C∂P^(β_r)].
 Here the implication follows from Proposition <ref> 4) applied to the projections P^(α_r̃),P^(β_r) respectively. Now, by strong local indistinguishability (<ref>) there exists an exponentially decaying function ϵ_3 in (l,r)=(C∖ D,D∖ C), s.t.

    _l_ Dr̃σ^CD∂ = _l_ FrGr̃σ^CD∂ϵ_3∼_l_ Frσ^C∂

and from Theorem <ref> one gets that uniform decay of covariance with decay function, say ϵ()=Ke^-/ξ, implies that ϵ_1∝exp𝒪(|r̃|+|∂(El)|)ϵ(), ϵ_2∝exp𝒪(|r|+|∂(El)|)ϵ(), and ϵ_3∝exp𝒪(|∂(El)|)𝒪(|r|)ϵ().
Hence by transitivity and symmetry of the strong similarity relation, see <Ref> 1) and 2), it follows that 

    _l_ Dr̃[P^(α_r̃)σ^CD∂P^(α_r̃)]    ϵ_1∼_l_ Dr̃σ^CD∂[P^(α_r̃)σ^CD∂P^(α_r̃)] ϵ_3∼_l_ Frσ^C∂[P^(α_r̃)σ^CD∂P^(α_r̃)] 
       ϵ_2(1-ϵ_2)^-1∼_l_ Fr[P^(β_r)σ^C∂P^(β_r)][P^(α_r̃)σ^CD∂P^(α_r̃)]/[P^(β_r)σ^C∂P^(β_r)],
 
and hence

    σ_1:= _l_ Dr̃[P^(α_r̃)σ^CD∂P^(α_r̃)]/[P^(α_r̃)σ^CD∂P^(α_r̃)]η̃∼_l_ Fr[P^(β_r)σ^C∂P^(β_r)]/[P^(β_r)σ^C∂P^(β_r)] =: σ_2,
 where η̃ is exponentially decreasing in (C∖ D, D∖ C) with decay length ξ and prefactor scaling as exp(𝒪(|∂ (C∖ D)|+|∂ D∖ C)|)), since ϵ_1,ϵ_2,ϵ_3 are. By Proposition <ref> 4^') it follows that this strong similarity of the states also holds within each block (α_l̃):

    σ_1= 1/[...]_l̃_[P^α_l̃σ_1P^α_l̃] η:=η̃(2+η̃)∼1/[...]_l̃_[P^α_l̃σ_2P^α_l̃] = σ_2.
 This establishes the bound for any boundary condition (α_l̃,β_r,α_r̃), which concludes the proof.
Note that this also implies that the exponential decay rate of η is the same as the one of the decay of covariance we assumed, which is the correlation length of the Gibbs state ξ.




§ MAIN RESULTS


In this section we give sufficient conditions for rapid thermalization of Davies evolutions with unique fixed point the Gibbs state of a nearest-neighbour, commuting Hamiltonian. 

The results essentially show that a suitable q𝕃_1→𝕃_∞ decay (Definition <ref>), together with some geometric features of the lattice, implies the existence of a constant or log decreasing cMLSI constant α(ℒ^D_Γ), which directly implies rapid mixing. The main technical lemma is as follows. 



    Let Λ be a 2-colorable graph with finite growth constant, Φ a uniformly bounded, nearest-neighbour, commuting potential, and β>0 some inverse temperature. If the family of Schmidt generators associated to (Λ,Φ,β) satisfies an exponentially decaying q𝕃_1→𝕃_∞ decay function η_C,D(l) in l=(C∖ D,D∖ C) whenever picking the regions C,D convex and such that 
    
        
  a  Case 1: (C),(D)=𝒪(l^2), 
        
  a Case 2: (C),(D)=𝒪(l), 
    
 then a family of Davies generators associated to (Λ,Φ,β) satisfies the MLSI with
    
        
  a Case 1: a system size independent MLSI constant α(ℒ^D_Γ)=Ω(1)_|Γ|→∞,
        
  a Case 2: a linear in system diameter decaying MLSI constant α(ℒ^D_Γ)=Ω(((Γ))^-1)_|Γ|→∞.
    


The proof is deferred to  <Ref>. 
The intuition behind separating these two cases is as follows.  The bound on the MLSI requires certain choices of coarse-grainings of the lattice into overlapping regions C,D. The choice of how large those regions are as compared to their overlap is limited both by the geometry of the lattice, and the decay of η_C,D(l). Case 1 is the one where it is possible to choose the overlap to be a vanishing fraction of the regions, leading to a better scaling, while in Case 2 the overlap is a finite fraction of the region. The diameter is the relevant quantity here, because the decay of correlations is a function of the one dimensional distance between two subregions and not the number of sites in the overlap in general. 


In the next two subsections, Theorem <ref> is then applied to two types of decay of η_C,D(l), and for each of them we consider two different lattice geometries, corresponding to the two cases of the theorem. Case 1: will apply to the 1D quantum spin chain under gap and among others the D-dimensional hypercubic lattices at high temperature. On the other hand, Case 2: will apply to the 2D hypercubic lattice under the gap condition and b-ary trees at high enough temperature. In particular:


    
  * In <Ref> we consider the case where the q𝕃_1→𝕃_∞ decay is implied by the existence of a unformly bounded strictly positive gap λ(ℒ^D_Γ)=Ω(1)_|Γ|→∞ of the Davies Lindbladians established via the  clustering results <Ref> and <Ref>, and <cit.>.  This shows rapid thermalization with a constant decay rate of commuting 1D quantum systems here. This is visualised in <Ref>. We also show that the same argument can also be applied to 2D systems. There, we obtain a better bound on the thermalization time than one would naïvely get from the gap, but still too slow to be rapidly mixing.
    

  * In <Ref>, we consider the high temperature setting, where the q𝕃_1→𝕃_∞ is guaranteed to decay fast, see <Ref>. This setting notably includes D-dim hypercubic lattices and b-ary trees, for which we hence show rapid mixing at high enough temperatures. 








 §.§ Rapid Thermalization from Gap


Here we show two direct applications of <Ref>. The first is that for 1D quantum spin chains with commuting nearest-neighbour Hamiltonians, having a gap is a sufficient (and necessary) condition for rapid thermalization. We also show how in 2D we obtain an improvement over previous results on mixing. Before stating both results, we recall what was previously established in Section <ref>. 

Let Λ be a graph with finite growth constant, Φ a uniformly bounded, nearest-neighbour, commuting potential and β a suitable inverse temperature, such that the family of Davies Lindbladians associated to (Λ,Φ,β) has a positive spectral gap, i.e. inf_Γ⊂⊂Λλ(ℒ^D_Γ)>0 . Then, for any Γ⊂⊂ℤ or ℤ^2 and C,D ⊂Γ with C ∩ D ≠∅,  by <cit.>, <Ref>, and <Ref> we have q𝕃_1→𝕃_∞-decay with decay rate 

    η_gap,C,D(l)=exp𝒪(|∂(C∖ D)|+|∂(D∖ C)|)𝒪(exp(-(C∖ D,D∖ C)/ξ)),
where ξ is the thermal correlation length of the family of Gibbs states associated to (Λ,Φ,β). 
Together with this scaling, two judicious choices of regions C,D (one for each case in Theorem <ref>) and their corresponding overlap allows us to obtain the following proposition. 
 
    It holds that η_gap,C,D(l) is exponentially decaying in l when picking C,D convex with (C∖ D,D∖ C)=l and such that 
        
        
  a Case 1: For Λ=ℤ, we have (C),(D)=𝒪(l^2)  ,
        
  a Case 2: For Λ=ℤ^2, and the thermal correlation length ξ is ξ < ξ_max(Φ,β)= 𝒪(1), we have (C),(D)=𝒪(l)
    


    For Λ=ℤ, the boundary of convex sets is 2, since they are intervals of the chain. Since this is a constant number, the decay holds independently of the correlation length ξ. 
   For Λ=ℤ^2, the boundary of convex sets scales linearly in their diameter. Hence so do |∂(C∖ D)|,|∂(D∖ C)| and thus, if the correlation length ξ is small enough to dominate over the linear growth in the exponent, we get exponential decay of l↦η_gap,C,D(l).


Together with <Ref> this directly yields the main result of the section.

 
The Davies generator ^D_Λ:={ℒ^D_Γ}_Γ⊂⊂Λ corresponding to a uniform, nearest-neighbour, commuting family of Hamiltonians acting on the locally-finite dimensional quantum spin system ℋ_Λ  satisfies a MLSI with constant 

    
  1) α(ℒ^D_Γ) = Ω(1)_|Γ|→∞ independent of system size, when Λ= is the spin-chain, or
    
  2) α(ℒ^D_Γ) = Ω((√(|Γ|))^-1)_|Γ|→∞ square-root decreasing in system size, when Λ=ℤ^2 and the correlation length of the fixed point (Gibbs state) is ξ < ξ_max(Φ,β)= 𝒪(1),

whenever the family of thermal states satisfies uniform exponential decay of covariance or when the Davies Lindbladians are uniformly gapped, i.e.  inf_Γ⊂⊂Λλ(ℒ^D_Γ)>0, which implies the former.


The square root decay of the MLSI constant in the 2-dimensional setting in Theorem <ref> does not imply rapid mixing, since it only guarantees a polynomial thermalization time. However, it still improves the bound on the mixing time over the one we would get only from having a gap, i.e. we get t_mix(ϵ)=𝒪(√(|Γ|)log|Γ|,log1/ϵ), as opposed to an arbitrary t_mix(ϵ)=𝒪(poly(|Γ|),log1/ϵ).

Concerning the 1D result, we first of all note that
we do not require r=2, i.e. nearest-neighbour interactions assumption, since by coarse-graining, we can always map a geometrically-r-local 1D system to a nearest neighbour one.[This, however, only works in the 1D case. The problem in higher dimensions is that the coarse grained graph is no longer 2-colorable or 2-colorable under further coarse-graining.]
This result has important implications in the 1D setting, 
where Theorem <ref> not just gives rapid mixing of the Davies dynamics but also an optimal scaling of the cMLSI constant from either gap or exponential decay of covariance.

A priori, assuming gap is the stronger assumption, since in <cit.>, it was proved that existence of a system-size independent gap implies 𝕃_2-clustering, and thus exponential decay of covariance (or 𝕃_∞-clustering). 
However, for the 1D systems considered here,[In fact, this holds more generally for uniformly bounded, commuting, geometrically-local systems on any graph with a finite growth constant.] it is known that rapid thermalization implies existence of a striclty positive gap of the generator <cit.>. This means that in 1D we have shown equivalence of strictly positive gap of generators, decay of covariance of the invariant state, MLSI with uniformly strictly positive constant, and rapid thermalization. 
Therefore, at least in 1D quantum spin chains,
this answers in the affirmative an open question from <cit.>, namely whether for commuting systems on ℤ, existence of exponential decay of covariance is sufficient to prove existence of a spin-system size invariant strictly positive spectral gap for ℒ^D_Γ. This is due to the fact that, as a combination of <Ref> and <Ref>, exponential decay of covariance of a thermal state of a uniform geometrically-local, commuting Hamiltonian implies a strictly positive and system size invariant MLSI constant, which implies gap. We refer the reader to Figure <ref> for a more clear depiction of these implications. 
We thus highlight the strength of our result for the 1D setting in the following corollary. 






Let Λ=ℤ be the one-dimensional quantum spin chain endowed with a uniformly J-bounded, geometrically-r-local, commuting potential Φ for some r∈ℕ and let β>0 be any inverse temperature. 
There exists a strictly positive constant α independent of the spin-chain length that lower bounds the MLSI constant of any element of the Davies generators associated to (ℤ,Φ,β), i.e. α(^D_Γ)=Ω(1)_|Γ|→∞. Hence 
for any finite Γ⊂⊂Λ we have 

    D(ρ_tσ^Γ) ≤ e^-tα(ℒ_Γ^D)D(ρ_0σ^Γ),

and thus also rapid thermalization with mixing time t_mix(ϵ)=𝒪(log|Γ|,log1/ϵ).


This follows from the recent result in <cit.>, where it is shown that exponential decay of covariance holds for such systems with correlation length ξ=exp𝒪(β)  and any finite temperature β>0. Alternatively, for translation-invariant systems, a uniform lower bounded gap is known by <cit.>. 
We do not require r=2, i.e. nearest-neighbour interactions, since by coarse-graining, we can always map a geometrically-r-local 1D system to a nearest neighbour one.[This, however, only works in the 1D case. The problem in higher dimensions is that the coarse grained graph is no longer 2-colorable or 2-colorable under further coarse-graining.]

Corollary <ref> is a strict improvement over the best previous results <cit.>, where α(^D_Γ)=Ω(log|Γ|^-1) is logarithmically decreasing in the system size for translation-invariant Hamiltonians. With our result, we obtain the same system-size independence as with the so-called LSI constant in the classical setting, which is known to be optimal.










 §.§ Rapid Thermalization from High Temperature



In this section we consider rapid thermalization as a consequence of being at high temperature, as given in Theorem <ref>. Being at high temperature means that we can assume uniform-q𝕃_1→𝕃_∞-clustering with decay function 

    η_C,D;(l) = 𝒪(|C∪ D|)exp(-(C∖ D,D∖ C)/ξ^').

See <cit.>[Theorem 6,7, Proposition 2] for the proof.  The fact that we now have a linear dependence 𝒪(|C∪ D|) of the pre-factor as opposed to the exponential one in Eq. (<ref>) means that we can define more favourable choices of regions C,D in the proofs, thus yielding stronger results. This is of particular importance for D>1, where we prove rapid mixing under the high temperature assumption.

We now present the following condition on the decay length ξ^', which will feature in the main theorem below.
 
For an infinite graph Λ, recall N(l):=sup_x∈Λ|B_l(x)|, where B_l(x):={v∈Λ|(x,v)≤ l} is the ball around x of radius l. 
We require the decay length ξ^' of the uniform-q𝕃_1→𝕃_∞-clustering to satisfy 

    ξ^' < l/2log N(l)
 eventually in l, i.e. for l≥ l_0 for some l_0∈ℕ. 


The decay length ξ^' is not just a function of the graph alone, but also of the potential Φ and the inverse temperature β. If condition (<ref>) holds, then N(l)exp(-l/ξ^') (and in turn, η_C,D(l)) is at least exponentially decaying in l.

Recall that hypercubic lattices are sub-exponential graphs, whereas b-ary trees are exponential.  Hence for infinite hypercubic lattices of dimension D we have that N(l)∝ l^D and thus this condition is trivially fulfilled for any ξ^'>0. For b-ary trees we have that N(l)=∑_k=0^lb^k=b^l+1-1/b-1∝ b^l, and hence this condition becomes
ξ^'<1/2log b, which is an implicit condition on the temperature β^-1.  Since it holds trivially for β=0, it should also hold for a small enough constant β>0.

Having established this, we now have the following proposition, similarly to Proposition <ref>. 
 
    The function η_C,D;(l) is exponentially decaying in l when choosing C,D convex such that 
    
         
  a Case 1: For Λ=ℤ^D a D-dim hypercubic lattice, (C),(D)=𝒪(l^2).  [This also holds for any subexponential graph, when requiring that we still have exponential decay when on can choose (C),(D)=𝒪(l^1+δ) for any 0<δ≤ 1. The proof still goes through under this slightly weakend 'Case 2' condition, hence the main result directly extends from hypercubic to subexponential graphs.]  
           
  a Case 2:   For Λ exponential graph, (C),(D)=𝒪(l) and the the decay length ξ^' is small enough (see e.g. Condition (<ref>)).
    

 Let Λ be a D-dim hyper-cubic graph. Then |C∪ D| scales at worst as ((C)+(D))^D= 𝒪(l^2D), so η_C,D;(l)≤𝒪(l^2D)exp(-lξ^-1). This is eventually exponentially decaying.  

For exponential graphs there exists a finite growth constant, say b, such that we have that |C∪ D|≤ b^(C)+(D)=exp(𝒪(l)), and thus η_C,D;(l)≤exp(𝒪(l)-lξ^' -1) is exponentially decaying when ξ^' is small enough. In the proof of Lemma <ref> we will only be requiring exponential decay for sets C,D such that (C∪ D)=2l=2(C∖ D,D∖ C). In this case we have that condition (<ref>) is sufficient to guarantee the claimed exponential decay, as it holds that |C∪ D|≤ b^(C∪ D)≤exp(2llog b), and thus η_C,D;(l)≤exp(2llog b-lξ^' -1).



This allows us to apply Theorem <ref> directly. In this section, as opposed to Proposition <ref> above, Case 1 will involve generic sub-exponential lattices, and Case 2 will be exponential trees. With those geometries, and given the choice of regions that Proposition <ref> allows for, the arguments from Sec. <ref> below allow us to obtain the following main result in the high temperature setting.

 
Let Λ be a 2-colorable graph with finite growth constant ν and Φ a uniformly J-bounded, nearest-neighbour, commuting potential on it. If the temperature is high enough, i.e. β^-1>10eJν, the Davies generators ^D_Λ:={ℒ^D_Γ}_Γ⊂⊂Λ associated to (Λ,Φ,β) acting on the locally-uniformly-finite dimensional quantum spin system ℋ_Λ satisfy a MLSI with constant 

    
  1) α(ℒ^D_Γ) = Ω(1)_|Γ|→∞ independent of system size, when Λ is a sub-exponential graph, such as all ℤ^D, or
    
  2) α(ℒ^D_Γ) = Ω((log|Γ|)^-1)_|Γ|→∞ logarithmically decreasing in system size, when Λ is an exponential graph and the correlation length of the fixed point (Gibbs state) satisfies condition (<ref>). For tree graphs, this condition is ξ^'<(2log b)^-1.


This means that in both cases the dynamics generated by the Davies generators are rapidly mixing. Importantly, the result also holds when we have a polynomial dependence on |C∪ D| instead of the linear prefactor of η_C,D;(l). The requirement on the high temperature comes solely through <Ref>, hence if one could prove (<ref>) from some other starting point, such as having a gap, then one would equally obtain Theorem <ref>. 

The result constitutes a strict extension of the main Theorem of <cit.>, since we are able to prove rapid thermalization for Davies dynamics, as opposed to for the more artificial setting of Schmidt dynamics. For the sub-exponential graph such as hypercubic lattices, the constant bound on the MLSI yields an optimal scaling of the thermalization time of the semi-group. It should be possible to extend it to wider ranges of temperatures, as long as the right notion of decay of correlations holds.

The result for trees is, to the authors knowledge, the first of its kind in the quantum setting. Classically it is known, however, that the exponential decay rate of the relative entropy towards the equilibrium is tree-size independent <cit.>. We expect this to also hold in the quantum case, since the proof there hinges upon a very analogous condition on the temperature as the one here, which is implicit through ξ^'<(2log b)^-1.
























 §.§ Proof of main results


In this section we prove <Ref>. The structure is as follows. We first reproduce an important result from <cit.> that tells us that exponential q𝕃_1→𝕃_∞ decay implies an approximate tensorization statement in <Ref>. This result, the work in <cit.>, and Lemma <ref> are the ingredients we need to establish the main result <ref> for quantum systems on hypercubic latices of dimension D>1. 
The  main part of the proof in <Ref> then consists of a geometric argument where we apply the aforementioned approximate tensorization to show the main result separately for Case 1 and Case 2. This is done by proving the result for the two prototypical instances of both cases: quantum spin chains and b-ary trees, respectively. The derivations then generalise straightforwardly to all the instances of Cases 1 and 2. 

Denote with η(l)≡η_C,D((C∖ D,D∖ C)) the decay function of the q𝕃_1→𝕃_∞-clustering of the family of Schmidt generators associated to (Λ,Φ,β).
In what follows, it is important that, by assumption, in Case 1 we have that η(l) is exponentially decaying when picking C,D convex s.t. (C),(D)=𝒪(l^2) and in Case 2 that η(l) is exponentially decaying when picking C,D convex s.t. (C),(D)=𝒪(l). The two cases, as well as their consequences as MLSIs for the various models, are depicted in Figure <ref>. 





  §.§.§ Approximate tensorization for an almost classical state ω



In this step of the proof, we connect the exponential decay from uniform q𝕃_1→𝕃_∞-clustering from Definition <ref> with a decomposition of a relative entropy of a conditional expectation on some region with respect to that of smaller overlapping regions, which we refer to as “approximate tensorization".

Given a 2-colorable graph Λ and a fixed 2-coloring, we denote the set of vertices with labels 0 as Λ_0 and the set of vertices with labels 1 as Λ_1=Λ∖Λ_0.

 
Given the 2-colorable graph Λ with 2-coloring Λ = {Λ_0,Λ_1}, denote a finite subgraphs Γ⊂⊂Λ with induced 2-colorings {Γ_i:=Γ∩Λ_i}_i=0^1=:{Γ_0,Γ∖Γ_0}. For a quantum state ρ∈𝒟(ℋ_Γ), we define 

    ω := E_Γ_0*^S(ρ) = (◯_a∈Γ_0E_a*^S)(ρ),

where the second equality follows from Proposition <ref>. 

The importance of this “semiclassical" state is that in <cit.> it was shown to satisfy the following approximate tensorization statement from the q𝕃_1→𝕃_∞-decay we assume here. 




Let (Λ,Φ,β) be as assumed in this section. Fix some Γ⊂⊂Λ and given some ρ∈𝒟(ℋ_Γ) set ω:=E^S_A*(ρ) as defined above. Let C,D⊂Γ be two convex subsets, s.t. for #∈{R:=C∪ D, ∂ C, ∂ D, ∂ R} we have #∩Λ_0=∅, and l:=(C∖ D,D∖ C)>1. Then it holds that

    D(ωE_C∪ D*^S(ω)) ≤1/1-2η_C,D(l)[D(ωE_C*^S(ω))+D(ωE_D*^S(ω))],

where η(l) is the decay function of the uniform q𝕃_1→𝕃_∞-clustering. 


The proof of this approximate tensorization statement, given either <Ref> or <Ref>, is just an application of <cit.>. 
This theorem will allow us to execute what is sometimes called a “divide-and-conquer" strategy for the state ω. That is, approximate tensorization allows us to bound the relative entropy distance between ω and the projection onto its fixed point on some lattice region CD by the ones on the smaller regions C and D, up to a suitable factor that depends on the decay of correlations of the fixed point. We will then iterate this procedure to end up with finite size regions, on which we can bound the relative entropies by a local MLSI constant, and finally generalize this to arbitrary states ρ.


























From this we will show that, as long as we have exponential decay in this approximate tensorization argument for convex regions C,D with the right diameter, as per the two cases, we can establish a lower bound on the MLSI. In Case 1, with diameter 𝒪(l^2), we can establish a constant lower bound on the MLSI constant α(ℒ^D_Γ)=Ω(1)_|Γ|→∞ and for Case 2, where the convex regions C,D have diameters 𝒪(l), we can establish a logarithmic (for trees at high temperature) and square root (for 2D lattices also at high temperature) lower bound on the MLSI constant α(ℒ^D_Γ)=Ω((|Γ|)^-1)_|Γ|→∞. To get there, we will require a repeated application of Theorem <ref> with appropriate choices of regions C,D, as we show in the next subsection.












  §.§.§ Geometric argument


We now construct the aforementioned geometric divide-and-conquer-strategy employing a Cesaro averaging over different choices of regions C,D. 
The target is an upper bound on the relative entropy between the state ω and its image under the Schmidt conditional expectation on the whole Γ. This bound is based on Theorem <ref>, and will be expressed in terms of the sum of the relative entropies between ω and its image under the Schmidt conditional expectation on smaller coarse-grained sets of a fixed size, called {R_k}_k.

For the geometric argument, we will explicitly define the construction of the relevant coarse-grained-sets for the b-ary tree, for b∈ℕ. The 1-D case is then covered as the b=1 case in trees. Since the Cesaro averaging and clustering works analogously for b-ary trees and D-dim hypercubic lattices, we will for simplicity only conduct it for the former. For the definition of the subsets and further details on the D-dimensional hypercubic construction see Section <ref> and also <cit.>.

Denote with K:={k|x_k∈Γ_0} the index set of Γ_0=Γ∩Λ_0. 

The {R_k}_k will be of a sufficiently large but finite minimal size and s.t. E^S_R_k*∘ E^S_Γ_0*=E^S_Γ_0*∘ E^S_R_k* holds, which is always the case if their boundaries satisfy ∂ R_k∩Γ_0=∅. As Γ_0 is the union of single vertices each with distance 2 from each other, each x_j∈Γ_0 is either an element of R_k, or (x_j,R_k)=2, so the commutation of the conditional expectations follows from Proposition <ref>. For trees, we define the subsets as follows [Note that the constructions are not unique.]. 


Denote the infinite b-ary tree with 𝕋_b. Denote with B_x,l the subtree rooted at site x∈𝕋_b of height l. 
We define the following set of subsets

    {R_k}_k∈ K:={B_x_k,l_0∩Γ}_k∈ K={B_x,l_0∩Γ}_x∈Γ_0,
where l_0∈2ℕ is a a suitably large constant to be fixed later on. The {R_k}_k form a coarse-graining into subtrees based on each vertex of the same label (i.e. in Γ_0) of finite fixed height l_0. We consider the cMLSI constant of our evolution on these sets, and extend these via a Cesaro averaging argument to the whole lattice.


Through these sets, we define the following quantity with respect to the state ω from (<ref>)

    D_R(ω):=∑_R_k⊂ RD(ωE^S_R_k*(ω)),
 where the sets {R_k}_k∈ K are the coarse grained sets as specified above. Observe that D_R(ω) is monotonically increasing in R, i.e. if A⊂ B⊂Γ are two subregions, then D_A(ω)≤ D_B(ω) by positivity of the relative entropy. It is also additive up to boundary terms, i.e. if A,B⊂Γ are two disjoint subregions, then 

    D_AB(ω)=D_A(ω)+D_B(ω)+∑_R_k∩ A≠∅R_k∩ B≠∅D(ωE^S_R_k*(ω)).

Next we define a function L↦ C(L):ℕ→ℝ, such that C(L) is the smallest number for which 

    D(ωE^S_Γ*(ω)) ≤ C(L)D_Γ(ω),
 where L:=(Γ). It follows from this definition that C(L) is monotonically non-decreasing. The main target of this section is thus to upper bound C(L). These bounds are given in the following key lemma, which is proven by inductively applying the approximate tensorization and averaging suitably over the choices of partitions C,D. It is the main ingredient of the geometric part of the proof. 

 For sub-trees of 𝕋_b of the form Γ=B_x_j,L for some j∈ K, (<ref>) holds with  

       
  a Case 1:  C(L)=𝒪(1)_|Γ|→∞ is uniformly upper bounded by C(∞)<∞.
      
  a Case 2:  C(L)=𝒪(L)_L→∞=𝒪(|Γ|)_|Γ|→∞=𝒪(log|Γ|)_|Γ|→∞.



This lemma can be viewed as an approximate tensorization statement on many regions for a state of the form of ω. That is, it upper bounds the relative entropy between it and its Schmidt conditional expectation on the whole lattice Γ in terms of the relative entropies of it and the Schmidt conditional expectations of the fixed finite size regions {R_k}_k∈ K. 

    






In order to prove it, we will make a repeated use of the approximate tensorization result Theorem <ref>. First breaking up a region B_x_k,L into the following regions, then breaking these up in a recursive way until we reach a decomposition in terms of the initial defined {R_k}_k. To apply it, we now proceed with the choice of regions C,D to be used in the result. 
For a set B_x_k,L define 

    C_k^l̃ := B_x_k,l̃  ,  D_k^l̃,l := ⋃_(x_m,x_k)=l̃-lm∈ K B_x_m,L+l-l̃  .

Hence we cover the subtree B_x_k,L of height L by a subtree with the same root of height l̃, called C_k^l̃, and a union of disjoint subtrees of height L+l-l̃, called D_k^l̃,l, s.t. their overlap has height l and is s.t. we can apply the approximate tensorization result <Ref> with the function η_C_k^l̃,D_k^l̃,l(l).  

Importantly we require that each of these sets begins and ends with some vertices of the same index 0, i.e. in Γ_0.
In formulae, this is C_k^l̃∪ D_k^l̃,l=B_x_k,L and (C_k^l̃∖ D_k^l̃,l,D_k^l̃,l∖ C_k^l̃)=l for all 0≤l̃≤ L.
Hence each set B_x_k,L has the family of non-trivial partitions {C_k^l̃,D_k^l̃,l}_l̃=1^L-1 and for each of these it holds, due to <Ref>, that

    D(ωE^S_B_x_k,L*(ω)) ≤1/1-2η_C_k^l̃,D_k^l̃,l(l)[D(ωE^S_C_k^l̃*(ω))+D(ωE^S_D_k^l̃,l*(ω))],

For an example of these regions see <Ref>. 

































In the 1D setting, these can be written as C_k^l̃=[x_k,x_k+l̃] and D_k^l̃,l=[x_k+l̃-l,x_k+L], where the  becomes , which is just the length of the intervals.

Now, recalling the definition of the two cases we are considering, we have the following key observation.

       
  a Case 1:  We can pick the overlap l=√(L)= 𝒪(√(L)). Then η(l)≡η_C_k^l̃,D_k^l̃,l(l) 
    is exponentially decaying in L for any ξ>0, by assumption. Set l_min,1 to be the smallest L, s.t. η(l)<1/2 for all l≥ l_min,1.
      
  a Case 2:  We have to pick the overlap l=L/N=𝒪(L) for some N∈ℕ∖{1} and require the decay length (ξ,ξ^') to be small enough such that we have eventual exponential decay in η(l)≡η_C_k^l̃,D_k^l̃,l(l) in l. This suffices for N=2, but we will keep it general in the following derivation. 
    
    We set l_0 to be the smallest l such that η(l)<1/2 for all l≥ l_0.

We now write the proof for each of these cases separately.

 
The proof idea is to average the approximate tensorization result, <Ref>, over all the above defined coverings {C_k^l̃,D_k^l̃,l}_l̃, for different l̃, to get the relative entropy between ω and its Schmidt conditional expectation on the whole of B_x_k,L in terms of the relative entropy between it and the Schmidt conditional expectation on subregions of height ϵ L, where ϵ<1.  
This allows us to bound C(L) in terms of the smaller C(ϵ L) times a multiplicative factor, which allows us to ultimately bound C(L) by a constant through a recursive process. Hence to bound C(L) by C(l_0), where l_0 is a fixed finite size, requires 𝒪(log(L)) steps. However, since in Case 1 the decay is assumed strong enough, we will be able to upper bound this product uniformly by a constant. 


Fix x_j∈Γ_0 and let 1/2<ϵ<1. 
We enumerate a maximal set of partitions of B_x_j,L into {C_j^l̃,D_j^l̃,√(L)}, s.t. (C_j^l̃)=l̃,(D_j^l̃,√(L))=L+√(L)-l̃≤ϵ L and s.t. different partitions have disjoint overlaps, i.e. (C_j^l̃_1∩ D_j^l̃_1,√(L))∩(C_j^l̃_2∩ D_j^l̃_2,√(L)) = ∅, whenever l̃_1≠l̃_2.
This works as long as √(L)≲(2ϵ-1)L which gives another condition on the minimal size of l=√(L)≥: l_min,2. There exist L/√(L)=𝒪(√(L)) of these partitions, since their overlap is of height √(L)=𝒪(√(L)). To simplify notation, we refer to these partitions {C_i,D_i}^𝒪(√(L))_i=1, where the partition index i is not to be confused with the fixed root j. We now average over all the approximate tensorization results of these partitions to get

    D(ωE^S_B_x_j,L*(ω))    ≤1/𝒪(√(L))∑_i=1^𝒪(√(L))1/1-2η_C_i,D_i(√(L))[D(ωE^S_C_i*(ω))+D(ωE^S_D_i*(ω)) ] 
       ≤1/1-2η(√(L))1/𝒪(√(L))∑_i=1^𝒪(√(L)) C((C_i))D_C_i(ω)+C((D_i))D_D_i(ω) 
       ≤ C(ϵ L)1/1-2η(√(L))1/𝒪(√(L))(∑_i=1^𝒪(√(L))(2D_C_i∩ D_i(ω)+D_C_i∖ D_i ∪ D_i∖ C_i(ω)).
            +∑_R_k∩ (C_i∖ D_i)≠∅R_k∩ (C_i∩ D_i)≠∅D(ωE^S_R_k*(ω))+∑_R_k∩ (D_i∖ C_i)≠∅R_k∩ (C_i∩ D_i)≠∅D(ωE^S_R_k*(ω)) ) 
       ≤ C(ϵ L)1/1-2η(√(L))1/𝒪(√(L))(2D_⋃_i=1^𝒪(√(L))(C_i∩ D_i)(ω)+∑_i=1^𝒪(√(L))D_(C_i∖ D_i)∂∪ (D_i∖ C_i)∂(ω)) 
       ≤ C(ϵ L)1/1-2η(√(L))√(L)/L(2+L/√(L))D_B_x_j,L(ω),
 
where in the second line we used that η_C_i,D_i(√(L)) does not depend on i, and then the definition of C((·)). In the third line we used that C((C_i)),C((D_i))≤ C(ϵ L) since (C_i)=l̃,(D_i)=L+√(L)-l̃≤ϵ L, and also used Eq. (<ref>).
Hence it follows that C(L)≤ C(ϵ L)1/1-2η(√(L))(1+2/√(L))=:C(ϵ L)f(L). This allows us to bound C(L) through an iterative process, since by definition in Eq. (<ref>) and the sets {R_k}_k∈ K, C(l_0)=1. Using C(L)≤ C(ϵ L)f(L),   M=𝒪(log L) times, s.t. ϵ^M L=l_0=:max{l_min,1,l_min,2} then gives

    C(L)≤ C(l_0)∏_k=1^Mf(ϵ^kL)≤ C(l_0)∏_k=0^∞ f(l_0ϵ^-k)<∞,

where it can be checked by inspection that the infinite product converges to a constant since η(√(L)) 
is exponentially decaying in √(L) and (1+2/√(L))→ 1 fast enough. Note that C(l_0)=1, independent of the x_j which we fixed initially. Hence the result follows.


The proof follows exactly the same idea and techniques as the one above. The main difference is that we need to choose the overlap of the coverings C,D to scale as 𝒪(L), where L=B_x_j,L, to still get decay in the approximate tensorization. Hence, the number of partitions has to be constant in system size, which yields a constant multiplicative factor in each inductive step. Since we again need 𝒪(log(L)) steps in the iteration to bound C(L), this gives the 𝒪(L), where L=(B_x_k,L) of the original set, scaling of the bound on C(L). The details of this proof can be found in <Ref> for completeness.


After this key lemma on approximate tensorization, in Sec. <ref> below we will generalise this statement to arbitrary states ρ and the Davies maps instead of the Schmidt conditional expectation. This will then allow us to extend the existence of a local cMLSI constant, see <Ref>, to the whole lattice with the cost of C(L)^-1. 






  §.§.§ Extension to hypercubic lattices


The main idea behind the proof of Lemma <ref>, where we  average over all suitable partitions of a given set to obtain the desired upper bound, naturally extends to hypercubic lattices (D>1). To adapt it, we need to adap the definitions of both the coarse-grained sets {R_k }_k and the `suitable partitions'. This then yields the second case of Theorem <ref> and the first case of Theorem <ref>.

First, for the {R_k }_k, instead of the definition in equation (<ref>) we have the following definition.

Denote with x+A the set A⊂ℤ^D shifted by x∈ℤ^D. Then, define the sets 

    {R_k}_k∈ K:=   {(x_k+[-l_0-1,l_0-1]^D∪(Γ_0∩(x_k+[l_0,l_0]^D)))∩Γ}_k∈ K
    =   {(x+[-l_0-1,l_0-1]^D∪(Γ_0∩(x+[l_0,l_0]^D)))∩Γ}_x∈Γ_0,
 where again l_0∈ℕ is a suitably large constant to be fixed later on. These are essentially jagged hyper-cubes of side-length 2l_0+1 around a center x_k∈Γ_0.  An example for one of these jagged hyper-cubes in 2D is shown in <Ref>.





For the suitable partitions C_k,D_k, the idea is to take jagged hyper-rectangles with sufficient overlap as the sets `C_k,D_k'. Then, we apply the approximate tensorization statement  D times to get (<ref>), but with 2^D terms on the right hand side, where the conditioned relative entropy is on sub-regions with a strictly smaller diameter. The approximate tensorization statement we obtain then yields, through iterative application as in the proof of Lemma <ref>, the analogous following Lemma.
 For sub-lattices of ℤ^D of form x_j+[-L,L]^D for some j∈ K, (<ref>) holds, with the  coarse-grained sets defined in (<ref>), with  

     
  a Case 1:  C(L)=𝒪(1)_|Γ|→∞ is uniformly upper bounded by a C(∞)<∞.
     
  a Case 2:  C(L)=𝒪(L)_L→∞=𝒪(|Γ|)_|Γ|→∞=𝒪(|Γ|^1/D)_|Γ|→∞.


For the 2-D graph in Case 2, which is the only use of this Case 2 in our main results, this gives us
C(L)=𝒪(L)_L→∞=𝒪(√(|Γ|))_|Γ|→∞. Case 1 instead yields the constant MLSI at high temperature. 



  §.§.§ Putting everything together


In the last steps, we generalise the result above for the state ω=E_Γ_0^S(ρ) and the Schmidt conditional expectations to arbitrary states ρ and to the Davies conditional expectations. This finalises the bounds on the MLSI for the Davies generators.



Let (Λ,Φ,β) be as assumed in <Ref> 
, Λ = {Λ_0,Λ_1} be a two colouring, and {E^S_Γ}_Γ⊂⊂Λ, {E^D_Γ}_Γ⊂⊂Λ the families of Schmidt and Davies conditional expectations. 
Let Γ⊂⊂Λ[e.g. for the b-ary tree, b∈ℕ, wlog Γ=B_0,L.] be a complete connected finite subgraph, where we set {x_k}_k∈ K=Γ_0:=Γ∩Λ_0, and set ω:=E^S_Γ_0(ρ) for a state ρ∈𝒟(ℋ_Γ).  
We first apply the chain rule for the relative entropy (<ref>), with σ≡σ^Γ=E^D_Γ*(ρ)=E^S_Γ_0*(σ), so that

    D(ρE^D_Γ*(σ))=D(ρσ) = D(ρE^S_Γ_0*(ρ)) + D(E^S_Γ_0*(ρ)σ) = D(ρω)+ D(ωσ).

The first summand D(ρω) satisfies exact tensorization (a form of strong subadditivity), since the Schmidt conditional expectations of two sets with distance two between each other commute, E^S_{x_k}*∘ E^S_{x_j}*= E^S_{x_j}*∘ E^S_{x_k}* = E^S_{x_k}∪{x_j}* for all x_k,x_j∈Γ_0. See Proposition <ref>  and <cit.>. This means that

    D(ρω)=D(ρE^S_Γ_0*(ρ)) ≤∑_x_k∈Γ_0D(ρE^S_{x_k}*(ρ)) Lemma <ref>≤∑_x_k∈Γ_0D(ρE^D_{x_k}∂ *(ρ)).

We can also bound the second summand D(ωσ) as follows, using Lemma <ref>, the DPI for the relative entropy, and finally Lemma <ref>.

    D(ωσ) = D(ωE^S_Γ*(ω))    Lemma <ref>≤ C∑_x_k∈Γ_0D(ωE^S_R_k*(ω)) = C∑_x_k∈Γ_0D(E^S_Γ_0*(ρ)(E^S_R_k*∘ E^S_Γ_0*)(ρ)) 
       DPI≤ C∑_x_k∈Γ_0D(ρE^S_R_k*(ρ)) Lemma <ref>≤ C∑_x_k∈Γ_0D(ρE^D_R_k∂*(ρ)),
 where in the second line we used that E^S_R_k*∘ E^S_Γ_0*=E^S_Γ_0*∘ E^S_R_k*, which holds by the construction of the sets R_k in (<ref>).[This is since ∂ R_k∩Γ_0=∅. Recall that Γ_0 is the union of single vertices each with distance 2 from each other. Hence the claim follows from Proposition <ref>.]
Recall that by Lemma <ref> in Case 1  the constant C is independent of system size, whereas in Case 2 for trees it scales logarithmically with system size.
Importantly, the regions {{x_k}∂,R_j∂}_k,j are of fixed finite size. Hence by <Ref> there exists cMLSI constants α_0,α_1>0, s.t. for any j,k

    α_0D(ρE^D_{x_k}∂*(ρ)) ≤_ℒ^D_{x_k}∂(ρ), α_1D(ρE^D_R_j∂*(ρ)) ≤_ℒ^D_R_j∂(ρ).

Putting everything above together, we have 

    D(ρE^D_Γ*(ρ) = D(ρσ)    ≤1/min{α_0,α_1}(∑_x_k∈Γ_0_ℒ^D_{x_k}∂(ρ) +C∑_k|x_k∈Γ_0_ℒ^D_R_k∂(ρ)) 
       ≤2mC/min{α_0,α_1}_ℒ^D_Γ(ρ),
 where in the last inequality we used the positivity and additivity of the entropy production and the fact that, by construction, each site x∈Γ is contained in at most a constant number, say 2m, of regions R_k∂, since they are of fixed finite size. The same holds for {x_k}∂ with ≤2m regions. Thus, it follows that

    α(ℒ^D_Γ)≥min{α_0,α_1}/2mC > 0,

so that the final scaling is Ω(C^-1).
Therefore in Case 1 from Lemma <ref> we get α(ℒ^D_Γ)= 𝒪(1)_|Γ|→∞ whereas in Case 2 we get α(ℒ^D_Γ)= Ω((|Γ|)^-1)_|Γ|→∞. Note that for trees (B_0,L)=𝒪(log|B_0,L|).


 






§ FURTHER APPLICATIONS


We have established a constant lower bound on the cMLSI as our main result for subexponential graphs. Here we explore the consequences of the system size independence of this bound for physical and statistical properties of such systems under dissipative evolution and of its steady state σ. 
Assume (Λ,Φ,β) suitable, such that the system admits a system size independent lower bound on the Davies cMLSI constant α(ℒ^D_Γ) as in <ref>.





 §.§ Exponential convergence to Gibbs states in the thermodynamic limit

A direct consequence of the spin-system size |Γ| independence of the cMLSI constant α(^D_Γ) is that in the thermodynamic limit we have an exponential decay of the relative entropy density between the initial state at time t and the thermal state, with a fixed decay rate α. This illustrates how quantum materials in the thermodynamic limit thermalize when weakly coupled to an external environment. 

For local Hamiltonians D(ρ_tσ^Γ)=𝒪(|Γ|), and hence

    lim_Γ↑Λ1/|Γ|D(ρ_tσ^Γ) ≤ e^-α tlim_Γ↑Λ1/|Γ|D(ρσ^Γ).


For a local Hamiltonian we have H_Γ≤∑_diam(X)≤ rX⊂ΓΦ_X = 𝒪(|Γ|) and hence

    d^|Γ|e^-β𝒪(|Γ|) = [e^-βH]≤[e^-β H]≤[e^βH] = d^|Γ|e^β𝒪(|Γ|),
 hence taking logarithms gives log Z_Γ= log[e^-β H_Γ]=𝒪(|Γ|) and we can bound

    D(ρσ^Γ) ≤ -[ρloge^-β H/Z_Γ] =log Z_Γ+β[ρ H_Γ] =𝒪(|Γ|).

The result follows directly from MLSI with system size independent constant  α by dividing through |Γ| and taking the limit.





 §.§ Local Mixing


The relative entropy D(e^tℒ(ρ) σ ), as it appears in the rapid mixing results, is rarely an adequate figure of merit in many-body experiments, since it is associated with expectation values of observables with support across the whole system. Even if two states are very different in relative entropy, that difference may be invisible to reasonable read-out capabilities.

Instead, the measurements that typically take place there are of observables with support on a small number of subsystems, such as local magnetization or currents. These, however, are always expected to thermalize in 𝒪(1) time irrespective of the initial conditions, much faster than global observables. 

That this is the case can be shown to follow already from the relative entropy decay. The proof only requires rapid mixing with a constant rate and the Lieb-Robinson bound for local Lindbladians which, under the assumption that N(l) = e^o(l), reads as follows <cit.>.

Let ℒ be the Lindbladian defined on the whole lattice Λ. Also, let Y be a region of the lattice, and V_l a larger region such that Y ⊂ V_l ⊂Λ and l ≡(Y, Λ∖ V_l). For any observable O_Y with (O_Y)⊂ Y,

    e^t ℒ^† (O_Y)- e^t ℒ_V_l^† (O_Y)  ≤ O_Y  e^-Ω (l-vt),

where ℒ_V_l=∑_(ℒ_X) ⊂ V_lℒ_X.

This means that the time evolution of O_Y is to a good approximation generated by the jump operators around its vicinity. Notice that in <cit.> it is assumed that N(l)=poly(l), but sub-exponential is also enough. With this, the result on local thermalization is as follows.


    Assuming that the system has rapid mixing with constant decay rate α, as in Corollary <ref>, we have that, for any region A, 

    D( _A^c[e^t ℒ(ρ)] σ_A ) ≤𝒪 ( | A | + t^κ+1) × e^-α t,

    where σ_A=_A^c[σ] is the local marginal of the Gibbs state.



Our starting point is a local Lindbladian of the form of Eq. (<ref>), as a sum of local terms. 

We now define three regions A,B,C such that A is our subsystem of interest, and B shields A from C with l_0=dist(A,C), and also such that A ∪ B ∪ C=Λ. We also define the Gibbs state on these regions as τ_ν=e^-β H_ν/Z_ν, with H_ν the subset of Hamiltonian terms with support in the respective ν∈{A,B,C } and Z_ν= Tr[e^-β H_ν]. Notice that these states are different from the respective marginals of the global Gibbs state σ_ν=_ν^c [σ].  Also τ_AB,Z_AB are those defined on A ∪ B.

Consider an arbitrary initial state on the lattice ρ, with ρ_AB=_C[ρ]. A direct calculation shows that

    D(ρ_AB⊗τ_C σ)= D(ρ_ABτ_AB)+logZ/Z_ABZ_C+Tr[(ρ_AB⊗τ_C)H_I],

where H_I=H-H_AB-H_C, the interaction between C and AB. Notice that D(ρ_ABτ_AB)= 𝒪(| A | + | B |), so that

    D(ρ_AB⊗τ_C σ) ≤ D(ρ_ABτ_AB) + 𝒪(|∂_BC|) ≤𝒪(| A | + | B |),

where |∂_BC| is the size of the boundary between B and C, which by definition is smaller than | B |.

We now consider the time evolution of this state. By data processing, we have

    D( e^t ℒ(ρ_AB⊗τ_C) σ) ≥ D( _BC[e^t ℒ(ρ_AB⊗τ_C)] σ_A).

We can compare this state with the one evolved under ℒ_AB,

    _BC[e^t ℒ(ρ_AB⊗τ_C)]  - _B[e^t ℒ_AB(ρ_AB )] _1    = max_ O_A =1|[O_A (e^t ℒ(ρ_AB⊗τ_C)  - e^t ℒ_AB(ρ_AB ) ⊗τ_C)]  |
        = max_ O_A =1|Tr[(e^t ℒ^†(O_A)- e^t ℒ_AB^†(O_A))ρ_AB⊗τ_C]  |
       ≤max_ O_A =1 e^t ℒ^†(O_A)- e^t ℒ_AB^†(O_A) ≤ e^-Ω (l_0-vt),

where we have used the definition of the 1-norm and the Lieb-Robinson bound. Similarly, we also obtain

    _BC[e^t ℒ(ρ)]- _BC[e^t ℒ_AB(ρ_AB⊗τ_C)] _1 ≤ e^-Ω (l_0-vt),

so that by the triangle inequality,

    _BC[e^t ℒ(ρ_AB⊗τ_C)]  - _BC[e^t ℒ(ρ)]  _1  ≤ e^-Ω (l_0-vt).

From continuity, this translates into a bound on relative entropy

    | D( _BC[e^t ℒ(ρ_AB⊗τ_C)] σ_A ) - D( _BC[e^t ℒ(ρ)] σ_A )  |≤εlogσ_A^-1 + 2h (ε/1+ε),

where ε= e^-Ω (l_0-vt) and h(p) is the binary entropy. Notice that logσ_A^-1=𝒪( | A |).

Additionally, by assumption, ℒ is such that it obeys rapid mixing with constant α independent of system size.
Hence, applied to the initial state ρ_AB⊗τ_C, it implies that 

    D(e^t ℒ(ρ_AB⊗τ_C) σ ) ≤ D( ρ_AB⊗τ_C σ) e^-α t.

Let us choose l_0 such that e^-Ω (l_0-vt)≤ e^-α t, which is possible as long as the radius of B increases linearly as l_0 ∝ t. Data processing, together with Eq. (<ref>) and Eq. (<ref>), then imply that

    D( _BC[e^t ℒ(ρ)] σ_A ) ≤𝒪 ( | A | + t^κ+1) × e^-α t,

which finishes the proof.


Eq. (<ref>) captures the most easily measurable character of the thermalization process, which occurs even in the thermodynamic limit. If we look at an individual subsystem, both the environment and the remainder of the system A^c can be seen as the entire bath that the subsystem A is coupled to. There is a crucial difference, however, between the two parts of this extended bath: while A is weakly coupled to the external environment, it is strongly coupled to A^c. This means that, as Eq. (<ref>) shows, the thermalization happens towards the state σ_A, often called the mean force Gibbs state <cit.>, instead of to the bare Gibbs state σ^A=e^-β H_A/Z_A with H_A=∑_supp(h_X) ⊂ A h_X. A similar global-to-local reduction of rapid mixing was already proven in <cit.> for the 1-norm.



 §.§ Wasserstein distance and transportation cost inequalities




A distance on 𝒟(ℋ_Γ), which is important in what follows is the quantum Wasserstein distance of order 1 <cit.> between two finite dimensional quantum states ρ,σ∈𝒟(ℋ_Γ). It is defined as

    W_1(ρ,σ)   ≡ρ-σ_W_1:= 
       1/2min{∑_i∈ΛX^(i)_1|[X^(i)]=0, X^(i)*=X^(i), _iX^(i)=0  ∀ i∈Γ, ρ-σ=∑_i∈ΛX^(i)}.
 
Its dual norm with respect to the Hilbert-Schmidt inner product distance is the Lipschitz distance <cit.>, i.e. for any self-adjoint observable A∈ℬ(ℋ_Γ)

    A_L:=max{[AX]|[X]=0, X=X^*, X_W_1≤1} = 2max_i∈Γmin_A^(i)∈ℬ(ℋ_Γ∖{i})A-_i⊗ A^(i),
where _i∈ℬ(ℋ_i) is the identity on system i and A^(i) does not act on system i.
Thus by definition it holds that |[AX]|≤X_W_1A_L for suitable X,A. For a thorough overview and some properties see <cit.>.
From the existence of an MLSI constant α independent of the spin-system size |Γ|, one can show that the following transport cost inequality holds with a transport cost c^'=c|Γ|/α linear in spin-system size |Γ| 
<cit.>.

    ρ-σ_W_1≤√(c^' D(ρσ)).
 

For a proof of this see <cit.> and <cit.>. The fact that the transport cost scales linearly in |Γ| has the following important consequences.



  §.§.§ Tighter bounds on the entropy difference and convergence via relative entropy

The first application of this concerns bounding the von-Neumann entropy difference between two quantum states ρ,σ by their relative entropy.

Let ρ∈𝒟(ℋ_Γ) be arbitrary and σ≡σ^Γ the invariant Gibbs state in Γ. The following two bounds hold

    i)      |S(ρ)-S(σ)|≤ g(c√(|Γ|/α)√(D(ρσ)))+c√(|Γ|/α)log(d^2|Γ|)√(D(ρσ))
       = 𝒪(√(|Γ|)log |Γ|)_|Γ|→∞√(D(ρσ)),
        
    
        ii)      |S(ρ_t)-S(σ)| = 𝒪(|Γ|log |Γ|,e^-α t/2/√(α)),
 where g(t)=(t+1)log(t+1)-tlog t=o(t)_t→∞,  ρ_t:=e^tℒ_Λ*(ρ), and c is some constant independent of |Γ| depending only on the locality of the Lindbladian ℒ^D_Λ.


These inequalities  represent a 𝒪(√(|Γ|)/log |Γ|) improvement compared to Pinsker's inequality (see <cit.>) as 

    ρ-σ_W_1≤|Γ|/2ρ-σ_1≤|Γ|/√(2)√(D(ρσ))  ,

or to the following inequality from <cit.>

    |S(ρ)-S(σ)|≤√(3)log(d^|Γ|)√(D(ρσ))=𝒪(|Γ|)√(D(ρσ))  .

The difference is that these apply to arbitrary quantum states ρ,σ, whereas our result in Eq.  (<ref>) requires σ to be the Gibbs state of a suitable Hamiltonian. 
Note that these entropy difference bounds are optimal in their scaling in |Γ| up to logarithmic correction, since the entropy difference is an extensive quantity, i.e. it scales as 𝒪(|Γ|).


We first use the following continuity bound from <cit.>, that states that for any two density matrices ρ,σ∈𝒟(ℋ_Γ)

    |S(ρ)-S(σ)|≤ g(ρ-σ_W_1)+ρ-σ_W_1log(d^2|Γ|),
 where d is the local Hilbert space dimension, ·_W_1 the quantum Wasserstein distance of order 1 (see (<ref>) for the definition), and g(t)=(t+1)log(t+1)-tlog t. By the transport cost inequality from <cit.> and <cit.>[Note that in that paper a different normalization convention for the quantum Wasserstein distance was used, as compared to here or the other references in this section], which holds under the assumptions of the Corollary, we get Eq. (<ref>).
Combining this with the inequality just above gives i). For ii) use i), the bound D(ρσ^Γ)≤𝒪(|Γ|) from above, and the MLSI in its integrated form  D(ρ_tσ)≤ e^-α tD(ρσ).




 


  §.§.§ Gaussian concentration bound
 
A system size independent cMLSI constant also yields the following Gaussian concentration bound.
 
Let O∈ℬ(ℋ_Λ) be a k-local observable, i.e. 

    O=∑_|X|≤ kX⊂Λo_X,

such that for all i∈Λ,  ∑_X⊂Λ:X∋ io_X≤ g, where each o_X acts only non trivially on sites X⊂Λ. Let σ be the Gibbs state of some geometrically-2-local, uniformly bounded, commuting Hamiltonian with uniform exponential decay of correlations, at any fixed inverse temperature β>0. Then for r≥ 0 it holds that

    ℙ_σ(|O-[Oσ]|≥ s)≤ 2exp[-α s^2/𝒪(|Γ|)].


This means that quantum spin systems with geometrically-local, commuting Hamiltonians with uniform exponential decay of correlations give rise to a sub-Gaussian random variable in their thermal equilibrium states for any observables of the above form. 
This bound reproduces the scaling of independent random variables from Hoeffding's inequality, and is thus optimal. Eq. (<ref>) constitutes a tightening in terms of its |Γ|-dependence and generalization to a larger class of observables and s-values of the previous best known Gaussian concentration bound in <cit.>[The statement there <cit.> is equivalent to ℙ_σ(|O-[Oσ]|≥ s)≤ 2exp[-α s/𝒪(√(|Γ|))].].


This follows from <Ref> and <cit.>, when using the transport cost c^'=c|Γ|/α in Eq. (<ref>) and the fact that Δ^1/2_σ(O)_L≤ 4gC=𝒪(1)_|Γ|→∞, since

    Δ^1/2_σ(O)_L    ≤ 2max_i∈ΛΔ_σ^1/2(O)-^(i)_d⊗t̂r̂_iΔ_σ^1/2(O)
       =2max_i∈Λ∑_X:|X|≤ kΔ_σ^1/2(o_X)-^(i)_d⊗t̂r̂_i∑_X:|X|≤ kΔ_σ^1/2(o_X)
       =2max_i∈Λ∑_X:X∂∋ i, |X|≤ k(Δ_σ^1/2(o_x)-^(i)_d⊗t̂r̂_iΔ_σ^1/2(o_x))
       ≤ 2max_i∈Λ∑_X:X∂∋ i, |X|≤ k2Δ_σ^1/2(o_x)≤ 4max_i∈Λ∑_X:|X|≤ k, X∂∋ io_Xexp(β/2∑_B:B∩ X≠∅diam(B)≤ rΦ_B)
       ≤ 4gexp(β Jc_r,k,ν/2) ≡ 4gC,
 where t̂r̂_i is the normalized partial trace. The last two inequalities follow from

    Δ_σ^1/2(o_X)   =σ^1/2o_Xσ^-1/2 = e^-β/2∑_B∩ X≠∅Φ_Bo_Xe^β/2∑_B∩ X≠∅Φ_B
       ≤o_Xexp(β/2∑_B:B∩ X≠∅|X|≤ k, diam(B)≤ rΦ_B)≤o_Xexp(β J c_r,k,ν/2).

Here c_r,k,ν is a constant which depends only on the locality of O, the geometric locality r of the Hamiltonian, and the growth constant ν of the graph. 





  §.§.§ Ensemble equivalence under (long-range) Lipschitz observables

The canonical ensemble with inverse temperature β is given by the Gibbs state σ_β, where we now write the temperature dependence explicitly. 

Let σ_E,δ be the microcanonical ensemble with average energy E defined on an energy-shell width δ, so that if P is the spectral measure of the Hamiltonian H=∑_EEP(E)≡∑_mE_m|E_m⟩⟨E_m| then

    σ_E,δ:=P((E-δ,E])/[P((E-δ,E])]=1/𝒩_E,δ∑_E_m∈(E-δ,E]|E_m⟩⟨E_m|.

Now, let 
E=argmax_E∈ℝ(e^-β E𝒩_E,δ), where 𝒩_E,δ:=[P((E-δ,E])] is the number of eigenstates in the energy interval (E-δ,E]. This defines the average energy of the microcanonical ensemble corresponding to the energy of σ_β <cit.>.


Two ensembles represented respectively by the families of states {σ_1^Γ,σ_2^Γ}_Γ⊂⊂Λ are said to be equivalent if, in the thermodynamic limit, they produce the same expectation values on averaged geometrically-local observable O/|Γ|=1/|Γ|∑_i=1^|Γ| O_i, with O_i≤ g <cit.>. That is, if for any such observable

    |[σ_1^ΓO/|Γ|]-[σ_2^ΓO/|Γ|]|= 1/|Γ||[(σ_1^Γ-σ_2^Γ) O]||Γ|→∞⟶ 0 .
 
In <cit.> it was shown that the microcanonical and canonical ensembles are equivalent in this sense when the system satisfies suitable concentration bounds, such as Eq. (<ref>). 
We extend this notion of equivalence in the 1D case to the more general class of Lipschitz observables, defined as O∈ℬ(ℋ_Γ), s.t. O_L<∞. These notably include long-range locally bounded, k-local observables of the form of Eq.  (<ref>).

For any Lipschitz observable O∈𝒜_Λ, i.e. O∈𝒜_Λ s.t. O_L<∞ and for σ_E,δ, σ_β the micro- and canonical ensemble states with the same energy E=max_E∈ℝ(e^-β E𝒩_E,δ), respectively,  it holds that 

    1/|Γ||[σ_E,δO]-[σ_β O]|≤O_Lo(1)_|Γ|→∞,

for all σ_E,δ such that δ=e^-𝒪(|Γ|).


This follows directly from <cit.>  when employing the linearity of the transport cost in the system size. The proof follows from the transport cost inequality, which is used to bound the relative entropy between the microcanonical and canonical ensembles, which yields the result.




§ OUTLOOK


In this paper, we have addressed the problem of estimating the speed of convergence of certain quantum dissipative evolutions governed by Lindbladians to their steady states, given by Gibbs states of local, commuting Hamiltonians. We have particularly considered a large class of Davies generators with nearest neighbour, commuting Hamiltonians, and derived for them a positive (and, in some cases, system-size independent) MLSI from either a positive spectral gap in the Lindbladian, or at high enough temperature. 

After the completion of this work, the knowledge about existence of positive MLSIs, and therefore rapid mixing, for various physically relevant systems stands as follows:


    
  * For 1D systems, for Davies generators with k-local, commuting, translation-invariant Hamiltonians, at any finite, inverse temperature β, a log-decreasing MLSI α(ℒ_Γ^D) = Ω ((log |Γ| )^-1)_|Γ| →∞ was derived in <cit.>. Here, we have lifted that to a constant MLSI  α(ℒ_Γ^D) = Ω (1)_|Γ| →∞ and extended it to non-translation-invariant Hamiltonians. Both results yield rapid mixing.  
    
  * For 2D systems, for Davies generators with k-local, commuting, Hamiltonians, at any finite, inverse temperature β, a positive spectral gap was shown to be equivalent to 𝕃_2-clustering in <cit.>. Here, we have shown that, whenever the correlation length is small enough, this is further equivalent to a square-root decreasing MLSI  α(ℒ_Γ^D) = Ω (√(|Γ|)^-1)_|Γ| →∞, yielding a 𝒪(√(|Γ|)log |Γ|)-mixing time.  
    
  * For high D systems and their Schmidt generators with nearest neighbour, commuting, Hamiltonians, a positive, constant MLSI α(ℒ_Γ^D) = Ω (1)_|Γ| →∞ was shown to hold at high-enough temperature in <cit.>. Here, we have extended this to Davies generators of nearest neighbour, commuting, Hamiltonians at high-enough temperature, concluding for them rapid mixing. 
    
  * For trees, we have provided the first results in the direction of rapid mixing, showing a log-decreasing MLSI α(ℒ_Γ^D) = Ω ((log |Γ| )^-1)_|Γ| →∞ whenever the correlation length is small enough. 

This shows definite progress in our understanding of the mixing times of Davies generators with commuting Hamiltonians, for which we aim to eventually have a complete picture. In this respect, there are still unsolved natural questions, such as:

    
  * Can we extend the results for high D systems to geometrically-k-local Hamiltonians? This would require a totally new approach, as the Schmidt generators cannot be defined beyond nearest neighbour interactions.
    
  * Can we provide a more accurate result for specific models? For instance, for Davies generators associated to quantum double models, a positive spectral gap was shown to hold at any positive temperature in <cit.>, extending the prior <cit.>. Therefore, it might be possible that a similar behaviour translates to the mixing time, yielding rapid mixing at every positive temperature.    

The natural extension of this problem to the context of non-commuting Hamiltonians is much less understood. As the Davies generator loses its desirable properties whenever the Hamiltonian considered is non-commuting, specially the locality, an alternative, physically-relevant Lindbladian needs to be considered in this case.  The recent <cit.> introduced a possible quasi-local Lindbladian that by-passes the problems induced by the Davies, which some of the authors of this manuscript recently showed to be gapped in <cit.>. The question whether this can be lifted to a positive MLSI, or any other means to derive rapid mixing, even for any specific case (e.g. 1D systems) remains open. An alternative approach to estimate mixing times of Lindbladians has just been introduced in <cit.>, but it is unclear whether it could be employed in this context.

The problem of estimating mixing times of Lindbladians has been well explored in the quantum community in the past few years, with a notable increase of interest very recently. The reason for this partly stems from an upraise in the tools at our disposal to tackle this problem, and the appearance of more applications derived from it. In particular, rapid mixing constitutes a fundamental approach for quantum Gibbs sampling, as stated in <cit.>, which nevertheless can also be explored with alternative approaches as in <cit.>. It has also been recently shown <cit.> that systems with rapid mixing are promising candidates for quantum advantage experiments in the context of analogue open quantum simulation. 













Acknowledgments: A.C.  acknowledges the support of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 470903074 – TRR 352. A.M.A. acknowledges support from the Spanish Agencia Estatal de Investigacion through the grants “IFT Centro de
Excelencia Severo Ochoa CEX2020-001007-S" and “Ramón
y Cajal RyC2021-031610-I", financed by
MCIN/AEI/10.13039/501100011033 and the European
Union NextGenerationEU/PRTR. This project was funded within the QuantERA II Programme that has received funding from
the EU’s H2020 research and innovation programme under the GA No 101017733.

abbrv




§ PROOF OF PROPERTIES OF THE RELATION FROM DEFINITION <REF>


This section contains the proofs of Proposition <ref>, Corollary <ref>, and Proposition <ref>.


 
    
  1) The reflexivity is trivial.
    
  2) For symmetry, notice that the spectra and thus the spectral radii of B^-1/2AB^-1/2- and A^1/2B^-1A^1/2- are the same.  Since they are both normal, we have B^-1/2AB^-1/2-≤ϵ which proves the second implication. Now the first implication follows via 

    B^1/2A^-1B^1/2- = (B^-1/2AB^-1/2)^-1-    = ∑_k=1^∞(-1)^k(B^-1/2AB^-1/2-)^k
       ≤∑_k=1^∞B^-1/2AB^-1/2-^k 
       ≤ϵ/1-ϵ,

where in the last line we used that by assumption B^-1/2AB^-1/2-≤ϵ.
 
  3) For transitivity, notice that 

    A^1/2C^-1A^1/2-   = A^1/2B^-1/2(B^1/2C^-1B^1/2-+)B^-1/2A^1/2-
       ≤A^1/2B^-1/2(B^1/2C^-1B^1/2-)B^-1/2A^1/2 + A^1/2B^-1A^1/2-
       ≤(B^1/2C^-1B^1/2-)(B^-1/2AB^-1/2) + A^1/2B^-1A^1/2-
       ≤ϵ_2(1+ϵ_1) + ϵ_1 = η.

For the second inequality, note that XYX^*=|spec(XYX^*)|=|spec(X^*XY)|≤X^*XY holds, for X,Y some operators with Y self-adjoint, where |spec(X)| denotes the spectral radius of X. Using this with X=A^1/2B^-1/2 and Y=B^1/2C^-1B^1/2- then gives the second inequality. The third inequality follows from the assumptions and the fact that B^-1/2AB^-1/2≤ 1+ϵ_1 by the second implication of symmetry.

  4)  For the tensor multiplicativity, notice that

    (A⊗ F)^1/2(A⊗F)^-1(A⊗ F)^1/2-   = A^1/2A^-1A^1/2⊗ F^1/2F^-1F^1/2-⊗
       ≤(A^1/2A^-1A^1/2-)⊗(F^1/2F^-1F^1/2-)
           +(A^1/2A^-1A^1/2-)⊗+⊗(F^1/2F^-1F^1/2-)
       ≤ϵ_1ϵ_2+ϵ_1+ϵ_2 = η.


  5)  The following chain of implications holds

    D^1/2E^-1D^1/2-≤ϵ   ⇔ (1-ϵ) ≤ D^1/2E^-1D^1/2≤ (1+ϵ) 
       ⇔1/1+ϵ ≤ D^-1/2ED^-1/2≤1/1-ϵ 
       ⇔D/1+ϵ≤ E ≤D/1-ϵ  .

Since P is self-adjoint the above implies that the following holds:

    PDP/1+ϵ≤ PEP ≤PDP/1-ϵ_ℋ^'PDP/1+ϵ≤_ℋ^'PEP ≤_ℋ^' PDP/1-ϵ  ,

where we identified P with ⊗ P for sake of clarity. The above holds also in reverse on _ℋ^'(PDP)=_ℋ^'(PEP). The fact that the (operator algebraic) supports of these two are the same is directly implied by the previous sandwich inequality and positivity. So it implies that 

    (_ℋ^'PDP)^1/2(_ℋ^'PEP)^-1(_ℋ^'PDP)^1/2-_(_ℋ^'(PDP))≤ϵ  ,

where (·)^-1 represents the generalized inverse. 

  6) First note that if Dϵ∼ E, then Dμ∼λ E for λ>0, where μ=ϵ/λ+|1-1/λ|, since

    D^1/2(λ E)^-1D^1/2-≤λ^-1 D^1/2E^-1D^1/2-+λ^-1-.

Now since Dϵ∼ E _ℋ^'PDP ϵ∼_ℋ^'PEP ⇔_ℋ^'PDP/1+ϵ≤_ℋ^'PEP ≤_ℋ^'PDP/1-ϵ 1+ϵ≥[PDP]/[PEP]≥ 1-ϵ by the proof of property 5). It follows that

    𝒳   ≡(_ℋ^'PDP/[PDP])^1/2(_ℋ^'PEP/[PEP])^-1(_ℋ^'PDP/[PDP])^1/2-
       =  (_ℋ^'PDP)^1/2(_ℋ^'PEP)^-1(_ℋ^'PDP)^1/2λ-,

with 1-ϵ≤λ^-1=[PDP]/[PEP]≤ 1+ϵ.
Hence 𝒳≤ϵλ^-1+|1-λ^-1|≤ϵ(1+ϵ)+ϵ=ϵ(2+ϵ).





The corollary is easily proved by induction using the transitivity of the relation, spelled out here for convenience. Assume
A_iϵ∼A_i+1 for all i and set A_0η_k∼A_k, then
by transitivity we have the recursion 

    η_k+1≤η_k(1+ϵ)+ϵ.

For k=1, noting that η_1=ϵ, this gives η_2=ϵ(1+ϵ)+ϵ=ϵ^2+2ϵ=(1+ϵ)^2-1.
For any k we have by induction that 

    η_k+1≤η_k(1+ϵ)+ϵ = ((1+ϵ)^k-1)(1+ϵ)+ϵ = (1+ϵ)^k+1-1.




Assume that (ω)⊂(τ), else the statement clearly holds. We may now WLOG assume the supports to be equal.
    Set ϵ = ω^1/2τ^-1ω^1/2, then the lower bound follows from
    
    D_max(ωτ)=logω^1/2τ^-1ω^1/2≤logω^1/2τ^-1ω^1/2-+≤log(1+ϵ)≤ϵ.

    For the upper bound see that for X:=ω^1/2τ^-1ω^1/2≥ 0 we have
    
    -X   = -explog X =-∑_n=0^∞(log X)^n/n! = ∑_n=1^∞(log X)^n/n! = ∑_n=1^∞(log X)^n/n!
       = log X∑_n=0^∞1/n(log X)^n/n!≤log Xexp(log X),
 where the last inequality follows from subadditivity aubmultiplicativity of the norm and 1/n≤ 1. This is the desired upper bound, hence we are done.




§ PROOF OF LEMMA <REF>



Let us prove each item separately:


  1)  If Λ and Φ are chosen as in the statement of the lemma, then there exists a constant c_r,ν depending only on r,ν such that

    E_A,B^±1   =e^∓β H_ABe^±β H_A±β H_B
       = exp{∓β∑_X ∩ A≠∅, X ∩ B≠∅Φ_X }
       ≤exp{(β J ∑_X ∩ A≠∅, X ∩ B≠∅diam(X)≤ r1 ) }
       ≤exp(β J c_r,νmin{|∂_B A|,|∂_A B|})=:K_A,B=exp(𝒪(β|∂_A,B|)),

where we are denoting ∂_A B := (∂ B),  ∩ A, ∂_B A := (∂ A),  ∩ B, and ∂_A,B:=min{|∂_B A|,|∂_A B|}. Hence it also holds

    E_A,B^±1E_AB,C^±1≤E_A,B^±1E_AB,C^±1≤ K_A,BK_B,C.

    
  2)  Consider the map

    [                                 ℬ(ℋ_Γ)                                      →                              ℬ(ℋ_Γ∖ B)                                       ;                                      Q                                      ↦                               _B[σ^BQ] =1/[e^-H_B]_B[e^-1/2H_BQe^-1/2H_B]   , ]
     
   which is positive and unital. Note that if Q>0 is strictly positive, then so is Q^-1>0 and λ_min(Q)=Q^-1^-1≤ Q≤Q=λ_max(Q). Applying the aforementioned map to this inequality immediately gives that 

    Q^-1^-1_B^c≤_B[σ^BQ]≤Q_B^c, 
       Q^-1_B^c≤_B[σ^BQ]^-1≤Q^-1_B^c,
 since inversion of two commuting operators is order reversing. We conclude by taking norms.

  3) This is just a special case of 2) given 1), since each of the E_A,B are strictly positive, as the potential Φ is commuting, e.g. by self-adjointness and the spectral theorem.






§ PROOF OF STRONG LOCAL INDISTINGUISHABILITY IN 1D


This section is devoted to the proof of <Ref>. 


The proof follows a similar path as the proof of <cit.> and uses local indistinguishability (see e.g. <cit.>, <cit.>), which follows from the assumptions of the theorem. Thus, we know that

    _BCσ^ABC-_Bσ^AB_1≤ K^' e^-a^' l  ,

 with K^',a^'>0 depending only on r,Jβ.
 
Using the notation E_X,Y:=e^-H_XYe^H_X+H_Y for X,Y⊂ I disjoint, we rewrite:

    (_BC[σ^ABC])(_B[σ^AB])^-1
       = _BC[e^-H_ABC]_B[e^-H_AB]^-1[e^-H_ABC]^-1[e^-H_AB] 
        = _BC[e^-H_ABC]e^H_Ae^-H_A_B[e^-H_AB]^-1[e^-H_ABC]^-1[e^-H_AB] 
        = _BC[e^-H_BCe^-H_ABCe^H_Ae^H_BC]_B[e^-H_Be^-H_ABe^H_Ae^H_B]^-1[e^-H_ABC]^-1[e^-H_AB] 
        = _BC[σ^BCE_A,BC]_B[σ^BE_A,B]^-1[e^-H_AB][e^-H_BC]/[e^-H_ABC][e^-H_B]
       ≡_BC[σ^BCE_A,BC]_B[σ^BE_A,B]^-1·λ_ABC^-1.

In the second line we multiplied by =e^H_Ae^-H_A, which in the third line we can separate and pull into the partial traces, since neither of them trace out the region A. 
Thus we may rewrite

    (_BCσ^ABC)(_Bσ^AB)^-1-   
    ≤   (_BCσ^ABC)(_Bσ^AB)^-1- (_BC[σ^BCE_A,BC])(_B[σ^BE_A,B])^-1
       + (_BC[σ^BCE_A,BC])(_B[σ^BE_A,B])^-1-
    ≤   _BC[σ^BCE_A,BC](_B[σ^BE_A,B])^-1|λ_ABC^-1-1| 
       + (_B[σ^BE_A,B])^-1_BC[σ^BCE_A,BC]-_B[σ^BE_A,B]  .

By  <cit.>, it holds that (_B[σ^BE_A,B])^-1≤ C and _BC[σ^BCE_A,BC]≤ C, for the same constant C depending only on r,J, β. Furthermore in <cit.>, it is proven that λ_ABC-1 decays exponentially, thus by the geometric series so does λ_ABC^-1-1, i.e. there exist K^'',a^''>0, depending only on r,Jβ s.t. 

    |λ_ABC^-1-1|≤ K^''e^-a^''l  .

Hence, it remains to bound (_BC[σ^BCE_A,BC])-(_B[σ^BE_A,B])^-1 exponentially in l. To do this we adopt a similar strategy to <cit.>. Let us split B = B_1B_2 into to halves, such that |B_1|=|B_2|=l, and write

    _BC[σ^BCE_A,BC]-_B[σ^BE_A,B]≤   _BC[σ^BCE_A,BC]-_BC[σ^BCE_A,B_1]
        + _BC[σ^BCE_A,B_1]-_B[σ^BE_A,B_1]
        + _B[σ^BE_A,B_1]- _B[σ^BE_A,B]  .

Here we just used the triangle inequality of the operator norm twice.
For the first and third summand in Equation (<ref>) use that the map Q↦_X[σ^XQ] is a contraction in B(ℋ_Γ)→ B(ℋ_Γ∖ X) by the Russo Dye theorem (see e.g lemma <ref> or <cit.>), and thus 

    _BC[σ^BCE_A,BC]-_BC[σ^BCE_A,B_1]≤E_A,BC-E_A,B_1  ,

where the right-hand side is exponentially decaying in |B_1|=l by <cit.>. The same holds true for 

    _B[σ^BE_A,B_1]-_B[σ^BE_A,B]≤E_A,B_1-E_A,B  .

 For the second summand in Equation (<ref>) we use  <cit.>, which gives 

    _BC[σ^BCE^*_A,B_1]-_B[σ^BE^*_A,B_1]≤Ke^-al

for some K,a>0, depending only on r,J,β. The same holds true for the adjoints, which is exactly the second summand. In total we have that there exist K^''',a^'''>0 depending only on r,J, β such that 

    _BC[σ^BCE_A,BC]-_B[σ^BE_A,B]≤ K^'''e^-a^'''l.

Now putting all of the above together we have our desired result.

    (_BC[σ^ABC])(_B[σ^AB])^-1-   ≤_BC[σ^BCE_A,BC](_B[σ^BE_A,B])^-1|λ_ABC^-1-1| 
          + (_B[σ^BE_A,B])^-1_BC[σ^BCE_A,BC]-_B[σ^BE_A,B]
       ≤ C^2K^''e^-a^''l+CK^'''e^-a^'''l  .





§ PROOF OF PROPERTIES OF SCHMIDT CONDITIONAL EXPECTATIONS



If (A_1,A_2)≥2, then by definition of the algebras 𝒩_A_1 and 𝒩_A_2 have only = 𝒩_∅=𝒩_A_1∩ A_2 in common.
Since it holds, that ∂ A_1 ∪∂ A_2 = ∂(A_1∪ A_2) their union is given by 

    𝒩_A_1∪𝒩_A_2   =ℬ(ℋ_A_1)⊗ℬ(ℋ_A_2)⊗_ℋ_(A_1)^c⊗_ℋ_(A_2)^c⊗_i∈ I_A_1⊗_j∈ J^(i)∖{0}𝒜^j_b_j⊗⊗_i∈ I_A_2⊗_j∈ J^(i)∖{0}𝒜^j_b_j
       = ℬ(ℋ_A_1∪ A_2)⊗_ℋ_(A_1∪ A_2)^c⊗⊗_i∈ I_A_1∪ A_2⊗_j∈ J^(i)∖{0}𝒜^j_b_j
       = 𝒩_A_1∪ A_2

For the other case we can WLOG assume A_1⊂ A_2, hence A_1∪ A_2=A_2 and A_1∩ A_2=A_1. Then clearly 𝒩_A_1∪𝒩_A_2=𝒩_A_2=𝒩_A_1∪ A_2. Similarly 𝒩_A_1∩𝒩_A_2=𝒩_A_1=𝒩_A_1∩ A_2.




§ PROOF OF CASE 2 IN <REF>



Let N∈ℕ∖{1} and 1/2<ϵ<1, s.t. 1/2+1/2N≥ϵ.
We enumerate a maximal set of partitions of B_x_j,L into {C_j^l̃,D_j^l̃,L/N}, s.t.  (C_j^l̃)=l̃,(D_j^l̃,L/N)=L+L/N-l̃≤ϵ L and s.t. different partitions have disjoint overlaps, i.e. (C_j^l̃_1∩ D_j^l̃_1,L/N)∩(C_j^l̃_2∩ D_j^l̃_2,L/N) = ∅, whenever l̃_1≠l̃_2.
There exist L/L/N=N of these partitions, since their overlap is of 'width' L/N. Call these partitions {C_i,D_i}^N_i=1. Now we average over all the approximate tensorization results of these partitions to get 

    D(ωE^S_B_x_j,L*(ω))   ≤1/N∑_i=1^N1/1-2η_C_i,D_i(L/N)[D(ωE^S_C_i*(ω))+D(ωE^S_D_i*(ω)) ] 
       ≤1/1-2η(L/N)1/N∑_i=1^N C((C_i))D_C_i(ω)+C((D_i))D_D_i(ω) 
       ≤ C(ϵ L)1/1-2η(L/N)1/N(∑_i=1^N(2D_C_i∩ D_i(ω)+D_C_i∖ D_i ∪ D_i∖ C_i(ω) . 
          +∑_R_k∩ (C_i∖ D_i)≠∅R_k∩ (C_i∩ D_i)≠∅D(ωE^S_R_k*(ω))+∑_R_k∩ (D_i∖ C_i)≠∅R_k∩ (C_i∩ D_i)≠∅D(ωE^S_R_k*(ω)) ) 
       ≤ C(ϵ L)1/1-2η(L/N)1/N(2D_⋃_i=1^N(C_i∩ D_i)(ω)+∑_i=1^ND_(C_i∖ D_i)∂∪ (D_i∖ C_i)∂(ω)) 
       ≤ C(ϵ L)1/1-2η(L/N)1/N(2+N)D_B_x_j,L(ω),
 
where in the second line we used the definition of C((·)) and
 in the third line we used that (C_i),(D_i)≤ C(ϵ L) since (C_i)=l̃,(D_i)=L+L/N-l̃≤ϵ L.
 
Hence it follows that C(L)≤ C(ϵ L)1/1-2η(L/N)(1+2/N)=:C(ϵ L)f̃(L)(1+2/N).
Repeating this M=𝒪(log L) times s.t. ϵ^M L=l_0=:L_min,1 gives

    C(L)   ≤ C(l_0)(1+2/N)^M∏_k=1^Mf̃(ϵ^kL)≤ C(l_0)(1+2/N)^M∏_k=0^∞f̃(l_0ϵ^-k)
       = 𝒪((1+2/N)^𝒪(log L))=𝒪(L),

where the infinite product converges since η(L/N) 
is exponentially decaying in L under the condition on the decay length ξ,ξ^'. Finally, notice that for exponential graphs (i.e. trees) we have that |B_x_j,L|=b^L and hence 𝒪(L)=𝒪(log|B_x_j,L|). Again, note that it is independent of the x_j which we fixed, so the result follows.
The optimal N for which this works is N=2, hence it suffices if we have decay when (C∪ D)= B_x_j,L=L=2l=2(C∖ D,D∖ C), which is the one we have in Case 2.


