The mixing time of Markovian dissipative evolutions of open quantum many-body systems can be bounded using optimal constants of certain quantum functional inequalities, such as the modified logarithmic Sobolev constant. For classical spin systems, the positivity of such constants follows from a mixing condition for the Gibbs measure via quasi-factorization results for the entropy. Inspired by the classical case, we present a strategy to derive the positivity of the modified logarithmic Sobolev constant associated with the dynamics of certain quantum systems from some clustering conditions on the Gibbs state of a local, commuting Hamiltonian. In particular, we show that for the heat-bath dynamics of 1D systems, the modified logarithmic Sobolev constant is positive under the assumptions of a mixing condition on the Gibbs state and a strong quasi-factorization of the relative entropy.

The fields of quantum information theory and quantum many-body systems have strong ties, as do their classical analogs. In the last few years, we have come to see that the results and tools developed in quantum information theory have helped to solve fundamental problems in condensed matter physics, whereas some new models created for many-body systems have been used for the storage and transmission of quantum information. There are numerous connections between these two fields, as well as interesting problems lying in their intersection.

One of these problems is the thermalizalization of a quantum system. It has recently generated great interest in both communities for several reasons, one of them being the increasing number of tools available from quantum information theory35,31 to address both the conditions under which a system thermalizes in the infinite time limit and how fast this thermalization occurs. Another one is the possible use of such systems for the implementation of quantum memory devices, as was suggested in the theoretical proposal of dissipative state engineering,37,23 where the authors proposed that quantum dissipative evolutions might constitute a robust way of constructing interesting quantum systems that preserve coherence for relatively long periods. Some experimental results have confirmed this idea, which has also raised the interest in this kind of system and, thus, in the problem of thermalization.

This paper concerns the question mentioned above, namely, how fast a dissipative system thermalizes. This “velocity” of thermalization will be studied by means of the mixing time, i.e., the time that it takes for every initial state undergoing a dissipative evolution to be almost indistinguishable from the thermal equilibrium state. In particular, we will be interested in physical systems for which this convergence is fast enough in a regime that is called rapid mixing. Several bounds for the mixing time, and thus, conditions for rapid mixing to hold, can be found via the optimal constants associated with some quantum functional inequalities, as it is the case for the spectral gap (optimal constant for the Poincaré inequality)36 or the modified logarithmic Sobolev constant (associated with the modified logarithmic Sobolev inequality).22 We will focus on the latter.

This problem was previously addressed in the classical setting. In Ref. 13, the authors showed that a classical spin system in a lattice, for a certain dynamics and a clustering condition in the Gibbs measure associated with this dynamics, satisfies a modified logarithmic Sobolev inequality. In Ref. 11, the usual logarithmic Sobolev inequality was studied via another similar condition of clustering in the Gibbs measure. Both results were inspired by the seminal work of Martinelli and Olivieri29,28 and aimed to notably simplify their proof via a result of quasi-factorization of the entropy39 in terms of some conditional entropies. Previously, a result of quasi-factorization of the variance4 had been used to prove positivity of the spectral gap for certain dynamics under certain conditions in the Gibbs measure.

The latter found its quantum analog in Ref. 21, where the authors introduced the notion of conditional spectral gap and proved positivity of the spectral gap for the Davies and heat-bath dynamics associated with a local commuting Hamiltonian, via a result of quasi-factorization of the variance, under a condition of strong clustering in the Gibbs state. In this paper, we aim to study the quantum analog of the aforementioned strategy to obtain positivity of a modified logarithmic Sobolev inequality via a result of quasi-factorization of the relative entropy, thus obtaining an exponential improvement on the mixing time with respect to the spectral gap case.

The main purpose of this paper is to present a strategy to obtain a positive modified logarithmic Sobolev constant and apply it to the particular case of the heat-bath dynamics in 1D. Our strategy is based on the following five points:

  1. Definition of a conditional modified logarithmic Sobolev constant (conditional MLSI constant, for short).

  2. Definition of the clustering condition on the Gibbs state.

  3. Quasi-factorization of the relative entropy in terms of a conditional relative entropy (introduced in Ref. 8).

  4. Recursive geometric argument to provide a lower bound for the global modified logarithmic Sobolev constant (MLSI constant, for short) in terms of the conditional MLSI constant in a fixed-sized region.

  5. Positivity of the conditional MLSI constant.

We remark that point (1) is not needed in the classical case, i.e., whenever all the involved density matrices are diagonal in a product basis of the Hilbert space of states. Indeed, there, the Dobrushin–Lanford–Ruelle (DLR) condition15,24 allows us to reduce the study of the MLSI on a domain and its boundary. In the quantum case, the DLR condition fails18 and the definition of the conditional MLSI is here to compensate this feature by taking into account the whole exterior of the domain, not only its boundary. This leads to several new difficulties, as illustrated by point (5), which is completely trivial in the classical case.

These five points together will imply the positivity of the modified logarithmic Sobolev constant independently of the system size. Note that the first two consist of careful definitions of some notions, the first of which will be used during the proof, whereas the second one will constitute the initial assumptions to impose on the Gibbs state. The third point has been previously addressed in Ref. 8. It constitutes the quantum analog of the results of quasi-factorization of the entropy in the classical case and provides an upper bound for the relative entropy between two states in terms of the sum of two conditional relative entropies and a multiplicative error term that measures how far the second state is from a tensor product. This result allows us to reduce from the global MLSI constant to a conditional one in the fourth point. Finally, in the last point, we prove that the conditional MLSI constant is positive and independent of the system size, yielding thus the same fact for the global one. In general, this will be the trickiest part of the strategy.

This strategy of five steps constitutes a generalization of the one used in Ref. 8 to prove that the heat-bath dynamics with the tensor product fixed point has a positive MLSI constant, lower bounded by 1/2 (see also Ref. 3). This result showed, in particular, that local depolarizing on-site noise destroys very fast any quantum memory. Then, it is only natural that in this paper, we weaken the conditions imposed on the fixed point and address the analogous problem for a more general class of quantum systems evolving under this dynamics. Indeed, in the main result of this paper, we apply the strategy above to the heat-bath dynamics in dimension 1 to obtain positivity of the MLSI constant under two additional assumptions on the Gibbs state.

Before proceeding to the informal exposition of the main result of this paper, let us emphasize the importance of the strategy to prove MLSI presented in this paper, which is also at the core of the thesis.7 Indeed, after the completion of the first version of this paper, a modification of this strategy for a new conditional relative entropy has been used in Refs. 2 and 10 to obtain the first examples of positivity of the MLSI for quantum lattice spin systems independently of the system size, thus solving a long-standing open problem. However, those examples concern a different class of semigroups, called Schmidt generators, whose main feature is that they present better properties regarding the decomposition of the algebras over different regions of the lattice. Therefore, while this paper does not solve the problem of positivity of MLSI for the heat-bath dynamics, the strategy we present here has already proven to be successful in analyzing MLSI, even beyond the specific generators considered here.

In this article, we consider the quantum heat-bath dynamics in 1D, whose generator is constructed following the same idea than for the classical heat-bath Monte-Carlo algorithm. More specifically, given a finite lattice ΛZ and a state ρΛSΛ, it is defined as

LΛ*(ρΛ)=xΛσΛ1/2σxc1/2ρxcσxc1/2σΛ1/2ρΛ,

where the first term in the sum of the RHS coincides with the Petz recovery map for the partial trace at every site x ∈ Λ, comprising the partial trace in x, and σΛ is the Gibbs state of a commuting k-local Hamiltonian.

Analogous to the classical case and the quantum spectral gap case, as part of the previous strategy, we need to assume that a couple of clustering conditions on the Gibbs state hold. The first one is related to the exponential decay of correlations in the Gibbs state of the given commuting Hamiltonian and is satisfied, for example, by Gibbs states at high-enough temperature depending logarithmically of the system size.

Assumption 1
(mixing condition). LetC, D ⊂ Λ be the union of non-overlapping finite-sized segments of Λ. The following inequality holds for positive constantsK1, K2independent of Λ:
σC1/2σD1/2σCDσC1/2σD1/21CDK1eK2d(C,D),
where d(C, D) is the distance betweenCandD, i.e., the minimum distance between two segments ofCandD.40 

The second assumption constitutes a stronger form of quasi-factorization of the relative entropy than the ones appearing in Ref. 8. An example where it holds is for Gibbs states verifying σΛ=xΛσx.

Assumption 2
(strong quasi-factorization). GivenX ⊂ Λ, for everyρΛSΛthe following inequality holds
DX(ρΛσΛ)fX(σΛ)xXDx(ρΛσΛ),
where 1 ≤ fX(σΛ) < depends only onσΛand does not depend on the size of Λ, whereasDX(ρΛσΛ), resp.Dx(ρΛσΛ), is the conditional relative entropy inX, resp.x, ofρΛandσΛ, whose explicit form will be recalled inSubsection 2BII B.

Then, the main result of this paper is the positivity of the modified logarithmic Sobolev constant for the heat-bath dynamics in 1D.

Theorem.

Considering that the two previous assumptions hold, the modified logarithmic Sobolev constant of the generator associated with the heat-bath dynamics in 1D systems with invariant state, the Gibbs state of a local commuting Hamiltonian is strictly positive and independent of |Λ|.

The rigorous statement of this theorem will be given in Theorem 7.

In Sec. II, we introduce the necessary notation, preliminary notions, and basic properties to follow the rest of this paper. In Sec. III, we prove several technical tools (which do not depend on the necessary assumptions for the main result), of independent interest, that will be of use in the proof of the main result, which we subsequently address in Sec. IV, providing a complete and self-contained proof. In Sec. V, we discuss the assumptions imposed on the Gibbs state, providing examples and situations in which they hold. We finally conclude in Sec. VI with some open problems.

In this paper, we consider finite dimensional Hilbert spaces. For Λ a set of |Λ| parties, we denote the multipartite finite dimensional Hilbert space of |Λ| parties by HΛ=xΛHx, whose dimension is dΛ. Throughout this text, Λ will often consist of three parties, and we will denote by HABC=HAHBHC the corresponding tripartite Hilbert space. Furthermore, most of this paper concerns quantum spin lattice systems, and we often assume that ΛZd is a finite subset. In general, we use uppercase Latin letters to denote systems or sets.

For every finite dimensional HΛ, we denote the associated set of bounded linear operators by BΛB(HΛ) and by AΛA(HΛ) its subset of observables, i.e., Hermitian operators, which we denote by lowercase Latin letters. We further denote by SΛS(HΛ)={fΛAΛ:fΛ0 and tr[fΛ]=1} the set of density matrices, or states, and denote its elements by lowercase Greek letters. In particular, whenever they appear in the text, Gibbs states are denoted by σΛ. We usually denote the space where each operator is defined using the same subindex as for the space, but we might drop it when it is unnecessary.

In this paper, we often consider quantum channels, i.e., completely positive and trace-preserving maps. In general, a linear map T:BΛBΛ is called a superoperator. We write 1 for the identity matrix and id for the identity superoperator. For bipartite spaces HAB=HAHB, we consider the natural inclusion AAAAB by identifying each operator fAAA with fA1B. In this way, we define the modified partial trace in A of fABAAB by trB[fAB]1B, but we denote it by trB[fAB] in a slight abuse of notation. Moreover, we say that an operator gABAAB has support in A if it can be written as gA1B for some operator gAAA. Note that given fABAAB, we write fA ≔ trB[fAB].

Finally, given x,yΛZd, we denote by d(x, y) the Euclidean distance between x and y in Zd. Hence, the distance between two subsets of Λ, A and B, is given by d(A, B) ≔ min{d(x, y): xA, yB}. Furthermore, we denote by ‖·‖ the usual operator norm and by ‖·‖1 = tr[|·|] the trace-norm.

1. Von Neumann entropy

Let HΛ be a finite dimensional Hilbert space, and consider ρΛSΛ. The von Neumann entropy of ρΛ is defined as

S(ρΛ)tr[ρΛlogρΛ].

The applications of this notion to quantum statistical mechanics and quantum information theory are numerous. Here, we focus on one of its most fundamental properties, which will appear often throughout the text.

Proposition 1
(strong subadditivity, Ref. 25). LetHABC=HAHBHCbe a tripartite Hilbert space, and considerρABCSABC. Then, the following inequality holds:
S(ρABC)+S(ρB)S(ρAB)+S(ρBC).

2. Relative entropy

A measure of distinguishability between two states that will appear often throughout this text is the relative entropy. Let HΛ be a finite dimensional Hilbert space, and consider ρΛ,σΛSΛ. We define the relative entropy of ρΛ and σΛ as

D(ρΛσΛ)tr[ρΛ(logρΛlogσΛ)].

Some fundamental properties of the relative entropy that will be of use are the following:

Proposition 2

(properties of the relative entropy, Ref. 38). LetHAB=HAHBbe a bipartite Hilbert space, and considerρAB,σABSAB. Then, the following properties hold:

  1. Non-negativity.D(ρABσAB) ≥ 0 andD(ρABσAB) = 0 if, and only if,ρAB = σAB.

  2. Additivity.D(ρAρBσAσB) = D(ρAσA) + D(ρBσB).

  3. Superadditivity.D(ρABσAσB) ≥ D(ρAσA) + D(ρBσB).

  4. Data processing inequality.For every quantum channelT:SABSAB,D(ρABσAB)D(T(ρAB)T(σAB)).

3. Conditional relative entropy

The conditional relative entropy provides the value of the distinguishability between two states in a certain system, given the value of their distinguishability in a subsystem. Let HAB=HAHB be a bipartite finite dimensional Hilbert space, and consider ρAB,σABSAB. The conditional relative entropy in A of ρAB and σAB is given by

DA(ρABσAB)D(ρABσAB)D(ρBσB).

We recall in the next proposition some properties of the conditional relative entropy that will be of use in Secs. II EV.

Proposition 3

(some properties of the conditional relative entropy, Ref. 8). LetHAB=HAHBbe a bipartite finite dimensional Hilbert space, and considerρAB,σABSAB. Then, the following properties hold:

  1. Non-negativity.DA(ρABσAB) ≥ 0.

  2. DA(ρABσAσB) = Iρ(A: B) + D(ρAσA), whereIρ(A: B)≔D(ρABρAρB) is the mutual information.

Given a state ρΛ in an open quantum many-body system under the Markov approximation, its time evolution is described by a one-parameter semigroup of completely positive trace-preserving maps Tt*etLΛ*, also known as quantum Markov semigroup (QMS), where LΛ*:SΛSΛ denotes the generator of the semigroup, which is called Liouvillian or Lindbladian, since its dual version in the Heisenberg picture satisfies the Lindblad (or GKLS) form26,19 for every XΛBΛ,

LΛ(XΛ)=i[H,XΛ]+12k=1l2Lk*XΛLk(Lk*LkXΛ+XΛLk*Lk),

where HAΛ, LkBΛ are the Lindblad operators, and [·,·] denotes the commutator.

We say that the QMS is primitive if there is a unique full-rank σΛSΛ, which is invariant for the generator, i.e., such that LΛ*(σΛ)=0. Furthermore, we say that the Lindbladian is reversible or satisfies the detailed balance condition, with respect to a state σΛSΛ if its version for observables verifies

fΛ,LΛ(gΛ)σΛ=LΛ(fΛ),gΛσΛ

for every fΛ,gΛAΛ, where this weighted scalar product is defined for every fΛ,gΛAΛ by

fΛ,gΛσΛtr[fΛσΛ1/2gΛσΛ1/2].

We recall now the notion of entropy production as the derivative of the relative entropy in the following form:

Definition 1.
LetΛZdbe a finite lattice, and letHΛbe the associated Hilbert space. LetLΛ*:SΛSΛbe a primitive reversible Lindbladian with fixed pointσΛSΛ. Then, for everyρΛSΛ, the entropy production is defined as
EP(ρΛ)ddtt=0D(ρtσΛ)=tr[LΛ*(ρΛ)(logρΛlogσΛ)],
where we are writingρtTt*(ρΛ).

Note that the entropy production of a primitive QMS only vanishes on σΛ. The fact that both the negative derivative of the relative entropy between the elements of the semigroup and the fixed point and the relative entropy between the same states have the same kernel and converge to zero in the long time limit, for every possible initial state for the semigroup, allows us to consider the possibility of bounding one in terms of the other. This is the reason to define a modified logarithmic Sobolev inequality and its optimal constant.

Definition 2.
LetΛZdbe a finite lattice,HΛbe its associated Hilbert space, andLΛ*:SΛSΛbe a primitive reversible Lindbladian with fixed pointσΛSΛ. Then, the modified logarithmic Sobolev constant (MLSI constant) is defined as
α(LΛ*)infρΛSΛtr[LΛ*(ρΛ)(logρΛlogσΛ)]2D(ρΛσΛ).

A family of quantum logarithmic Sobolev inequalities was introduced in Ref. 22, where the modified logarithmic Sobolev inequality, whose optimal constant we have just recalled, is identified with the one-logarithmic Sobolev inequality. In the same paper, it is shown that the existence of a positive MLSI constant implies a bound in the mixing time of an evolution, i.e., the time that it takes for every initial state to be almost indistinguishable of the fixed point, which constitutes an exponential improvement in terms of the system size to the bound provided by the existence of a positive spectral gap. Indeed, if α(LΛ*)>0, then for every ρΛSΛ,

ρtσΛ12log(σΛ1)eα(LΛ*)t.

This constitutes a way to obtain sufficient conditions for a QMS to satisfy rapid mixing, a property that has profound implications in the system, such as stability against external perturbations12 and the fact that its fixed point satisfies an area law for the mutual information.5 

Given a finite lattice ΛZd, let us define a k-local bounded potential as Φ:ΛAΛ such that for any x ∈ Λ, Φ(x) is a Hermitian matrix supported in a ball of radius k centered at x and there exists a constant C < such that ‖Φ(x)‖ < C for every x ∈ Λ.

We define the Hamiltonian from this potential in the following way: For every subset A ⊂ Λ, the Hamiltonian in A, HA, is given by

HAxAΦ(x).

We further say that this potential is commuting if [Φ(x), Φ(y)] = 0 for every x, y ∈ Λ.

Consider now A ⊂ Λ and Φ a bounded k-local potential. Since the potential is local, we can define the boundary of A as

A{xΛ\A|d(x,A)<k},

and we denote by A∂ the union of A and its boundary. Note that HA clearly has support in A∂. Since in this paper we only focus on 1D systems, for every bounded connected subset A ⊂ Λ, the boundary will be composed of two parts, which we will intuitively denote by (∂A)Left and (∂A)Right, respectively.

In the full lattice ΛZd, the Gibbs state is defined as

σΛeβHΛtr[eβHΛ].

Note that, by a slight abuse of notations, we will denote by σA, for A ⊂ Λ, the state given by trAc[σΛ], which should not be confused with the restricted Gibbs state corresponding to the terms of the Hamiltonian HA.

Let ΛZd be a finite lattice and Φ:ΛAΛ be a k-local bounded commuting potential. Consider σΛ to be the associated Gibbs state. Given A ⊆ Λ, we define the heat-bath conditional expectation as follows: for every ρΛSΛ,

EA*(ρΛ)σΛ1/2σAc1/2ρAcσAc1/2σΛ1/2.

Note that it is a quantum channel, and moreover, it coincides with the Petz recovery map for the partial trace in A with respect to σΛ, comprising the partial trace in A,33 i.e., EA*()PtrAσΛtrA[] for

PtrAσΛ()σΛ1/2σAc1/2()σAc1/2σΛ1/2.

Furthermore, it is the dual map of the minimal conditional expectation that appears in Ref. 21. As opposed to what its name suggests, it is not a usual conditional expectation, but a quasi-conditional expectation,32 since it lacks some of the basic properties in the definition of conditional expectation.

We can now define the heat-bath generator on Λ by

LΛ*(ρΛ)xΛEx*(ρΛ)ρΛ

for every ρΛSΛ. Analogously, for every A ⊂ Λ, we denote by LA* the generator where the summation is only over elements xA. Note that the Lindbladian is defined as the sum of terms containing conditional expectations considered over single sites. Some basic properties concerning the heat-bath generator are collected in the following proposition:

Proposition 4
(Ref. 21). LetΛZdbe a finite lattice andΦ:ΛAΛbe ak-local bounded commuting potential. Then, the following properties hold:
  1. For anyA ⊂ Λ,LA*is the generator of a semigroup of CPTP maps of the formetLA*.

  2. LΛ*isk-local in the sense that each individual composing term acts non-trivially only on balls of radiusk.

  3. For anyA, B ⊂ Λ, we have

LA*+LB*=LAB*+LAB*.

To conclude this subsection, let us introduce two concepts that will be of use in the proof of the main result. They are conditional versions of notions defined on the whole system, in the same spirit as the conditional relative entropy.

Definition 3.
LetΛZdbe a finite lattice, and letLΛ*:SΛSΛbe the heat-bath generator with fixed pointσΛSΛ. GivenA ⊂ Λ, we define the entropy production inAfor everyρΛSΛby
EPA(ρΛ)tr[LA*(ρΛ)(logρΛlogσΛ)].

Considering the notions of entropy production in a subsystem and conditional relative entropy, one can address again the problem of relating both of them via an inequality, thus obtaining a conditional version of the aforementioned MLSI constant.

Definition 4.
LetΛZdbe a finite lattice, and letLΛ*:SΛSΛbe the heat-bath generator with fixed pointσΛSΛ. GivenA ⊂ Λ, we define the conditional MLSI constant by
αΛ(LA*)infρΛSΛtr[LA*(ρΛ)(logρΛlogσΛ)]2DA(ρΛσΛ),
whereDA(ρΛσΛ) is the conditional relative entropy introduced inSubsection 2BII B.

In the classical setting, there is no need to define a conditional MLSI constant, since it coincides with the MLSI constant due to the DLR condition.13 Not only this last property fails in general in the quantum case18 but also the study of the conditional MLSI constant is essential in our case, as it is part of our strategy to prove the positivity of the MLSI constant.

Consider a tripartite space HABC=HAHBHC. We define a recovery mapRBBC from B to BC as a completely positive trace-preserving map that reconstructs the C-part of a state σABCSABC from its B-part only. If that reconstruction is possible, i.e., if for a certain σABCSABC, there exists such RBBC verifying

σABC=RBBC(σAB),

we say that σABC is a quantum Markov chain (QMC) between ABC. When this is the case, the recovery map can be taken to be the Petz recovery map. More specifically, σABC is a QMC (ABC) if, and only if, it is a fixed point of the composition of the Petz recovery map for the partial trace in C, with respect to the state σBC, with the partial trace in C, i.e.,

σABC=PtrCσBCtrC[σABC]=σBC1/2σB1/2σABσB1/2σBC1/2.

This class of states has been deeply studied in the last few years. In the next proposition, we collect an equivalent condition for a state to be a QMC.

Theorem 1

(Refs. 33 and 34). LetHABC=HAHBHCbe a tripartite Hilbert space andσABCSABC. Then,σABCis a quantum Markov chain, if, and only if,Iσ(A: C|B) = 0, forIσ(A: C|B) = S(σAB) + S(σBC) − S(σABC) − S(σB) the quantum conditional mutual information.

Another important equivalent condition for a state to be a quantum Markov chain, concerning its structure as a direct sum of tensor products, appears in the next result.

Theorem 2
(Theorem 6 of Ref. 20). A tripartite stateσABCofHAHBHCsatisfiesIσ(A: C|B) = 0 if and only if there exists a decomposition of systemBasHB=jHbjLHbjRinto a direct sum of tensor products such that
σABC=jqjσAbjLσbjRC,
with the stateσAbjL(respectively, the stateσbjRC) being onHAHbjL(respectively, onHbjRHC) and a probability distribution {qj}.

Turning now to Gibbs states, as they were introduced in Subsection II D, we recall an important result about their Markovian structure.

Theorem 3

(Theorem 3 of Ref. 6). Given ak-local commuting potential on Λ, its associated Gibbs stateσΛis a quantum Markov network, that is, for all disjoint subsetsA, B, C ⊂ Λ such thatBshieldsAfromCwithd(A, C) > k,Iσ(A: C|B) = 0.

The notion of “shield” used in the statement of the theorem denotes that systems A and B are not adjacent, i.e., no site of A is at distance 1 from C and vice versa. Moreover, the condition d(A, C) > k implies that there are at least k sites of B between any site of A and another one of C. Therefore, combining the results of Theorems 3 and 2, we obtain the following essential result for the structure of Gibbs states:

Corollary 1.
LetΛZdbe a finite lattice andσΛbe the Gibbs state of a commuting Hamiltonian. Then, for any tripartitionÃB̃C̃of Λ such thatB̃shieldsÃfromC̃, the stateσΛcan be decomposed as
σΛ=jqjσÃb̃jLσb̃jRC̃.
(1)

Using the previous properties for quantum Markov chains, we can easily show the identity of the next proposition.

Proposition 5.
LetHABC=HAHBHCbe a tripartite Hilbert space andσABC be a quantum Markov chain betweenABC. Then, the following identity holds:
logσABC+logσB=logσBC+logσAB.
(2)

Proof.
Since σABC is a quantum Markov chain between ABC, by Theorem 2, we can write it as
σΛ=jqjσAbjLσbjRC.
(3)
Hence,
logσABC+logσBC+logσABlogσB=jlogσAbjLσbjRC+logσbjLσbjRC+logσAbjLσbjRlogσbjLσbjR=0,
where we have used the fact that the logarithm of a tensor product splits as a sum of logarithms.□

As a consequence of this identity, we have the following result:

Corollary 2.
LetHABC=HAHBHCbe a tripartite Hilbert space andσABCbe a quantum Markov chain betweenABC. Then, for anyρABCSABC, the following identity holds:
DA(ρABCσABC)=DA(ρABσAB)+Iρ(A:C|B),
(4)
whereIρ(A: C|B) denotes the conditional mutual information ofρABC.
In particular,
DA(ρABCσABC)DA(ρABσAB).

Proof.
Since σABC is a quantum Markov chain between ABC, by Proposition 5, we have
DA(ρABCσABC)DA(ρABσAB)=D(ρABCσABC)D(ρBCσBC)D(ρABσAB)+D(ρBσB)=S(ρABC)+S(ρBC)+S(ρAB)S(ρB)Iρ(A:C|B)+tr[ρABClogσABC+logσBC+logσABlogσB]=Iρ(A:C|B).
In particular, since Iρ(A: C|B) ≥ 0 for every state ρABCSABC,
DA(ρABCσABC)DA(ρABσAB).

This section aims at presenting a collection of technical results, which will be necessary in the proof of the main result of this paper in Sec. IV. Some of them, as we will see below, are of independent interest to quantum information theory. Note that all the results that appear in this section hold independently of Assumptions 1 and 2 and do not depend on the geometry of Λ.

The main technical result of this section is Theorem 6. In its proof, we will make use of the following lemma, which provides a lower bound for a conditional entropy production in a single site (see Definition 3) in terms of a conditional relative entropy in the same single site:

Lemma 4.
For a single sitex ∈ Λ and for everyρΛ,σΛSΛ, the following holds:
EPx(ρΛ)Dx(ρΛσΛ),
(5)
whereEPx(ρΛ) is defined with respect toσΛ. Therefore,EPA(ρΛ) ≥ 0 for anyA ⊂ Λ andρSΛ.

Proof.
The proof is a direct consequence of the data processing inequality and the fact that Ex*() is the Petz recovery map for the partial trace in x, comprising the partial trace (and, in particular, a quantum channel). Indeed, let us recall that EPx(ρΛ) is given by
EPx(ρΛ)=tr[Lx*(ρΛ)(logρΛlogσΛ)]=tr[ρΛEx*(ρΛ)(logρΛlogσΛ)]=D(ρΛσΛ)tr[Ex*(ρΛ)(logρΛlogσΛ)].
(6)
In the second term of (6), let us add and subtract logEx*(ρΛ). Then,
tr[Ex*(ρΛ)(logρΛlogσΛ)]=tr[Ex*(ρΛ)(logρΛlogσΛ+logEx*(ρΛ)logEx*(ρΛ))]=D(Ex*(ρΛ)ρΛ)+D(Ex*(ρΛ)σΛ)D(Ex*(ρΛ)σΛ),
(7)
where we have used the fact that the relative entropy of two states is always non-negative.
Finally, since Ex*() is the Petz recovery map for the partial trace in x comprising the partial trace [denote Ex*()=Ptrxσtrx[]], note that σΛ is a fixed point. Then,
D(Ex*(ρΛ)σΛ)=D(Ptrxσtrx[ρΛ]Ptrxσtrx[σΛ])D(ρxcσxc),
and thus,
EPx(ρΛ)D(ρΛσΛ)D(ρxcσxc)=Dx(ρΛσΛ).

Remark 1.
If we recall the definition for the conditional MLSI constant introduced in Sec. II, Lemma 4 can clearly be seen as a lower bound for the conditional MLSI constant at a single site x ∈ Λ for the heat-bath dynamics, i.e.,
αΛ(Lx*)12.

This inequality, in particular, can be used to prove positivity of the MLSI constant for the heat-bath dynamics when σΛ is a tensor product, as it appears in Ref. 8 (see also Refs. 3 and 1).

Remark 2.
Note that in the previous lemma, we have only used the fact that the partial trace is a quantum channel and Ex*() its Petz recovery map comprising it. Hence, in more generality, Lemma 4 could be stated as follows: Let T be a quantum channel, and denote by T̂ its Petz recovery map with respect to σΛ. Then, for any ρΛSΛ, the following holds:
tr[(ρΛT̂T(ρΛ))logρΛlogσΛ]D(ρΛσΛ)D(T(ρΛ)T(σΛ)).
(8)

Another tool that will be of use in the main result of this section is the following lemma, which appeared first in Ref. 27. It can be seen as an equivalence between blocks of spins and allows us to prove an equivalence between the usual conditional Lindbladian associated with the heat-bath dynamics in A ⊆ Λ, given as a sum of local terms, and a modified one, given as a unique term. Note that it is stated in the Heisenberg picture.

Lemma 5
(Ref. 27). LetA ⊆ Λ, and letσΛbe the Gibbs state of thek-local commuting Hamiltonian mentioned above. There exist constants 0 < cA, CA < , possibly depending onAbut not on Λ such that for anyfΛAΛ, the following holds:
cAxAfΛ,fΛEx(fΛ)σΛfΛ,fΛEA(fΛ)σΛCAxAfΛ,fΛEx(fΛ)σΛ,
(9)
whereEx(respectively,EA) is the dual ofEx*(respectively, ofEA*) and is given by
Ex(fΛ)σxc1/2trx[σΛ1/2fΛσΛ1/2]σxc1/2,
for everyfΛAΛand analogously forEA.

Let us now state and prove the main technical result of this section, which will be essential for the proof of Theorem 7 but has independent interest on its own.

Theorem 6.
LetΛZdbe a finite lattice, and letσΛSΛbe the Gibbs state of a commuting Hamiltonian over Λ. For anyA ⊆ Λ andρΛSΛ, the following equivalence holds:
ρΛ=EA*(ρΛ)ρΛ=Ex*(ρΛ)xA.
(10)

Proof.
Let us first recall that for every ρΛSΛ, the local Lindbladian in A ⊆ Λ is given by
LA*(ρΛ)=xAEx*(ρΛ)ρΛ,
and define
L̃A*(ρΛ)=EA*(ρΛ)ρΛ.
Analogously, defining the superoperator ΓσΛ:fΛσΛ1/2fΛσΛ1/2, we can write every observable fΛAΛ as
fΛ=ΓσΛ1(ρΛ)=σΛ1/2ρΛσΛ1/2,
(11)
and thus, we have
LA(fΛ)=xAEx(fΛ)fΛ,
L̃A(fΛ)=EA(fΛ)fΛ.
With this notation, inequality (9) in Lemma 5 can be rewritten as
cAfΛ,LA(fΛ)σΛfΛ,L̃A(fΛ)σΛCAfΛ,LA(fΛ)σΛ,
and thus,
fΛ,LA(fΛ)σΛ=0xA,fΛ,Lx(fΛ)σΛ=0fΛ,L̃A(fΛ)σΛ=0,
which, thanks to the detailed-balance condition, and the subsequent positivity of the generators, leads to
LA(fΛ)=0xA,Lx(fΛ)=0L̃A(fΛ)=0.
(12)
Now, because of (11), one can easily see that Ex*=ΓσΛExΓσΛ1 and the same holds for EA*. Hence, (12) is equivalent to
LA*(ρΛ)=0xA,Lx*(ρΛ)=0L̃A*(ρΛ)=0.
(13)
Recalling the expressions for LA*(ρΛ) and L̃A*(ρΛ), we obtain
ρΛ=EA*(ρΛ)ρΛ=Ex*(ρΛ)xA.

This result can also be stated in terms of conditional relative entropies. Indeed, note that as a consequence of Petz’s characterization for conditions of equality in the data processing inequality, all the conditions above can be seen as necessary and sufficient conditions for vanishing conditional relative entropies. We have then the following corollary:

Corollary 3.
LetΛZdbe a finite quantum lattice, and letσΛSΛbe the Gibbs state of a commuting Hamiltonian. For anyA ⊆ Λ andρΛSΛ, the following equivalence holds:
DA(ρΛσΛ)=0Dx(ρΛσΛ)=0xA.
(14)

Another consequence of the previous result is that a state is recoverable from a certain region whenever it is recoverable from several components of that region that cover it completely, no matter the size of those components. More specifically, we have the following corollary:

Corollary 4.
Given a finite lattice Λ, a partition of it into three subregionsA, B, C, andσABCthe Gibbs state of a commuting Hamiltonian, if we denote byEA*()the conditional expectation onAassociated with the heat-bath dynamics (with respect to the Gibbs state), we have for anyρABCSABC,
EAB*(ρABC)=ρABCEA*(ρABC)=ρABC   andEB*(ρABC)=ρABC.
(15)
In particular,
DAB(ρABCσABC)=0DA(ρABCσABC)=0   andDB(ρABCσABC)=0.
(16)

Proof.
By virtue of Theorem 6, it is clear that
EAB*(ρABC)=ρABCEx*(ρABC)=ρABCxABEx*(ρABC)=ρABCxAEA*(ρABC)=ρABC   and    andEx*(ρABC)=ρABCxBEB*(ρABC)=ρABC
The second part is a direct consequence of Ref. 33 and Corollary 3.□

In this section, we state and prove the main result of this paper, namely, a static sufficient condition on the Gibbs state of a k-local commuting Hamiltonian for the heat-bath dynamics in 1D to have a positive modified logarithmic-Sobolev constant (MLSI constant in short). For that, we first need to introduce two assumptions that need to be considered in order to prove the result, which will be discussed in further detail in Sec. V.

The first condition can be interpreted as an exponential decay of correlations in the Gibbs state of the commuting Hamiltonian. In Sec. V A, we will see that only a weaker assumption is necessary, although this form is preferable here for its close connections to its classical analog.13 

Assumption 1
(mixing condition). LetΛZbe a finite chain, and letC, D ⊂ Λ be the union of non-overlapping finite-sized segments of Λ. LetσΛbe the Gibbs state of a commuting Hamiltonian. The following inequality holds for certain positive constantsK1, K2independent of Λ, C, D:
σC1/2σD1/2σCDσC1/2σD1/21CDK1eK2d(C,D),
whered(C, D) is the distance betweenCandD, i.e., the minimum distance between two segments ofCandD.

The second condition that needs to be assumed constitutes a strong form of quasi-factorization of the relative entropy.

Assumption 2
(strong quasi-factorization). LetΛZbe a finite chain andX ⊂ Λ. LetσΛbe the Gibbs state of ak-local commuting Hamiltonian. For everyρΛSΛ, the following inequality holds:
DX(ρΛσΛ)fX(σΛ)xXDx(ρΛσΛ),
(17)
where 1 ≤ fX(σΛ) < depends only onσΛonX∂(in particular, it is independent of |Λ|).

This form of quasi-factorization is stronger than the one that appeared in Ref. 8, since another conditional relative entropy appears in the LHS of the inequality, instead of a relative entropy as in the main results of quasi-factorization of the aforementioned paper. Moreover, the error term depends only on the second state, as in usual quasi-factorization results, but only on its value in the regions where the relative entropies are being conditioned and their boundaries. In particular, it is independent of the size of the chain.

As in the case of Assumption 1, we will see in Subsection V B that only a weaker condition is necessary for Theorem 7 to hold true, since this condition will only appear in the proof concerning sets X of small size.

Let us now state and prove the main result of this paper, namely, the positivity of the MLSI constant for the heat-bath dynamics in 1D.

Theorem 7.

LetΛZbe a finite chain. LetΦ:ΛAΛbe ak-local commuting potential andHΛ=xΛΦ(x)be its corresponding Hamiltonian, and denote byσΛits Gibbs state. LetLΛ*be the generator of the heat-bath dynamics. Then, if Assumptions 1 and 2 hold, the MLSI constant ofLΛ*is strictly positive and independent of |Λ|.

The proof of this result will be split into four parts. First, we need to define a splitting of the chain into two (not connected) subsets A, B ⊂ Λ, with a certain geometry so that (1) they cover the whole chain, (2) their intersection is large enough, and (3) each one of them is composed of smaller segments of fixed size, but large enough to contain two non-overlapping half-boundaries of two other segments, respectively.

More specifically, fix lN so that K1eK2l<12 for K1 and K2 the constants appearing in the mixing condition, and consider the splitting of Λ given in terms of A and B, verifying the following conditions (see Fig. 1):

  1. Λ = AB.

  2. A=i=1nAi and B=j=1nBj.

  3. |AiBi| = |BiAi+1| = l for every i = 1, …, n − 1.

  4. |Ai| = |Bj| = 2(k + l) − 1 for all i, j = 1, …, n, where k comes from the k-locality of the Hamiltonian.

FIG. 1.

Splitting of Λ in fixed-sized subsets Ai and Bi, of which we just show the first four terms. We reduce for simplicity to the case k = 2, l = 1.

FIG. 1.

Splitting of Λ in fixed-sized subsets Ai and Bi, of which we just show the first four terms. We reduce for simplicity to the case k = 2, l = 1.

Close modal

Note that the total size of Λ is then n(4k + 2l − 2) + l sites. Hence, fixing l and k as already mentioned, we can restrict our study here to lattices of size n(4k + 2l − 2) + l for every nN, as we will be interested in the scaling properties in the limit.

In the first step, considering this decomposition of the chain, we show an upper bound for the relative entropy of two states on Λ (the second of them being the Gibbs state) in terms of the sum of two conditional relative entropies in A and B, respectively, and a multiplicative error term that measures how far the reduced state σAcBc is from a tensor product between Ac and Bc, where Ac ≔ Λ\A and Bc ≔ Λ\B.

Step 1.
For the regionsAandBdefined above and for anyρΛSΛ, we have
D(ρΛσΛ)112h(σAcBc)DA(ρΛσΛ)+DB(ρΛσΛ),
(18)
where
h(σAcBc)=σAc1/2σBc1/2σAcBcσAc1/2σBc1/21AcBc.

Proof.
In Ref. 9, the authors showed that given a bipartite space HXY=HXHY, for every ρXY,σXYSXY, one has
(1+2h(σXY))D(ρXYσXY)D(ρXσX)+D(ρYσY)
(19)
for h(σXY) given by
h(σXY)=σX1/2σY1/2σXYσX1/2σY1/21XY.
Considering a tripartite space HXZY and ρXZY,σXZYSXZY, inequality (19) was shown to be equivalent to
(1+2h(σXY))D(ρXZYσXZY)D(ρXσX)+D(ρYσY)
in Ref. 8, and recalling the definition for the conditional relative entropy in XZ and ZY, this inequality can be rewritten as
D(ρXZYσXZY)112h(σXY)DXZ(ρXZYσXZY)+DZY(ρXZYσXZY).

Finally, inequality (18) follows just by replacing in this expression Λ = XZY, AB = Z, Ac = Y, and Bc = X.□

For the second step of the proof, we focus on one of the two components of Λ, e.g., A, and upper bound the conditional relative entropy of two states in the whole A in terms of the sum of the conditional relative entropies in its fixed-size small components. In this case, there is no multiplicative error term due to the structure of quantum Markov chain of the Gibbs state between one component, its boundary, and the complement and the fact that the boundaries of these components do not overlap.

Step 2.
ForA=i=1nAidefined as above (seeFig. 2 ) and for everyρΛSΛ, the following holds:
DA(ρΛσΛ)i=1nDAi(ρΛσΛ).
(20)

FIG. 2.

Splitting of A in fixed-sized subsets Ai so that their boundaries do not overlap. For simplicity, we restrict to the case k = 2, l = 1.

FIG. 2.

Splitting of A in fixed-sized subsets Ai so that their boundaries do not overlap. For simplicity, we restrict to the case k = 2, l = 1.

Close modal

Proof.

Without loss of generality, we assume that A = A1A2 (the general result follows by induction in the number of subsets Ai). For convenience, we denote D(ρAσA), respectively, DA(ρΛσΛ), by D(A), respectively, DA(Λ), since we are considering the same states ρΛ and σΛ in every (conditional) relative entropy.

With this notation, it is enough to show that
DA(Λ)DA1(Λ)DA2(Λ)0.
(21)
In this step, we are going to consider three different regions: A1, ∂A1, and A1c, but we could argue analogously for A2. We write C1A1Left and C2A1Right, the left and right components of ∂A1, respectively, and C3 all the sites that appear left to C1, between C2 and A2, and right to A2, i.e., C3(A1A2)c. We further write Cm=13Cm. Note that A1, A2, and C are disjoint and that A1A2C = Λ. Then,
DA(Λ)DA1(Λ)DA2(Λ)=D(Λ)D(C)D(Λ)+D(A2C)D(Λ)+D(A1C)=D(C)D(Λ)+D(A2C)+D(A1C)=trρΛ(logρΛlogρC+logρA1C+logρA2C)+trρΛ(logσClogσA2C+logσΛlogσA1C)=S(ρΛ)+S(ρC)S(ρA1C)S(ρA2C)+trρΛ(logσClogσA2C+logσΛlogσA1C)trρΛ(logσClogσA2C+logσΛlogσA1C),
(22)
where the last inequality follows from strong subadditivity of the von Neumann entropy. Now, from the structure of the quantum Markov chain of the Gibbs state and by Proposition 5, the sum of logarithms vanishes.□

Combining expressions (18) and (20) from Steps 1 and 2, respectively, we get

D(ρΛσΛ)112h(σAcBc)i=1nDAi(ρΛσΛ)+DBi(ρΛσΛ).
(23)

In the third step of the proof, using the first two, we get a lower bound for the global MLSI constant of the whole chain in terms of the conditional MLSI constants on the aforementioned fixed-sized regions Ai and Bi. For that, we need to consider that Assumption 1 holds true.

Step 3.
If Assumption 1 holds, we have
α(LΛ*)K̃mini{1,n}{αΛ(LAi*),αΛ(LBi*)},
whereK̃=12K1eK2l2andαΛ(LAi*), respectively,αΛ(LBi*), denotes the conditional MLSI constant ofLΛ*onAi, respectively,Bi, as introduced in Definition 4.

Proof.
By Eq. (23) and Assumption 1, we have
D(ρΛσΛ)112K1eK2li=1nDAi(ρΛσΛ)+DBi(ρΛσΛ).
(24)
Now, by virtue of the definition of conditional MLSI constants on each Ai and Bi, it is clear that
D(ρΛσΛ)112K1eK2li=1ntr[LAi*(ρΛ)logρΛlogσΛ]2αΛ(LAi*)+tr[LBi*(ρΛ)logρΛlogσΛ]2αΛ(LBi*)112K1eK2l12mini{1,,n}{αΛ(LAi*),αΛ(LBi*)}i=1nEPAi(ρΛ)+EPBi(ρΛ).
Therefore,
2mini{1,,n}{αΛ(LAi*),αΛ(LBi*)}D(ρΛσΛ)112K1eK2ltrLΛ*(ρΛ)+LAnBn*(ρΛ)logρΛlogσΛ+i=1n1LAiBi*(ρΛ)+LAi+1Bi*(ρΛ)logρΛlogσΛ212K1eK2ltr[LΛ*(ρΛ)logρΛlogσΛ],
(25)
where we have used the locality of the Lindbladian and the positivity of the entropy productions.
Finally, note that the last term of expression (25) is the entropy production of ρΛ. Hence, considering the quotient of this term over the relative entropy of the LHS and taking infimum over ρΛSΛ, we get
α(LΛ*)=infρΛSΛEP(ρΛ)2D(ρΛσΛ)K̃mini{1,n}{αΛ(LAi*),αΛ(LBi*)},
where K̃12K1eK2l2>0.□

Finally, in the last step of the proof, we show that the conditional MLSI constants on every Ai and Bi are strictly positive and, additionally, independent of the size of Λ. For that, we need to suppose that Assumption 2 holds true. We also make use of some technical results from Sec. III.

Step 4.
If Assumption 2 holds, for anyAidefined as above, we have
αΛLAi*CAi(σΛ)>0,
withCAi(σΛ)independent of the size of Λ, and analogously for anyBi.

Proof.
Consider X ∈ {Ai, Bi: 1 ≤ in}. Let us first recall that the conditional MLSI constant in X is given by
αΛ(LX*)=infρΛSΛEPX(ρΛ)2DX(ρΛσΛ)=infρΛSΛxXtr[Lx*(ρΛ)logρΛlogσΛ]2DX(ρΛσΛ).
By virtue of Lemma 4, we have
EPx(ρΛ)Dx(ρΛσΛ)
for every xX, and thus,
αΛ(LX*)infρΛSΛxXDx(ρΛσΛ)2DX(ρΛσΛ).
(26)
Note that the quotient in the RHS of (26) is well-defined, since we have seen in Corollary 3 that the kernel of DX(ρΛσΛ) coincides with the intersection of the kernels of Dx(ρΛσΛ) for every xX. Furthermore, because of Assumption 2, we obtain the following lower bound for the conditional MLSI constant:
αΛ(LX*)12fX(σΛ),
(27)
which is strictly positive, only depends on σΛ, and does depend on the size of Λ.□

Finally, putting together Steps 1, 2, 3, and 4, we conclude the proof of Theorem 7.

In this subsection, we will elaborate on the mixing condition introduced in Assumption 1 and provide sufficient conditions for it to hold. Consider ΛZ as a finite chain and A, B ⊂ Λ as in the splitting of Λ in the proof of Theorem 7 (see Fig. 1). Denote CBc and DAc so that they can be expressed as the union of disjoint segments, C=i=1nCi and D=j=1nDj, respectively. For every i = 1, …, n, respectively, i = 1, …, n − 1, denote by Ei, respectively, Fi, the connected set that separate Ci from Di, respectively, Di from Ci+1 (see Fig. 3). Note that because of the construction of A and B described in Sec. IV, every Ei and Fi are composed of l sites and every Ci and Di of, at least, 2k − 1 sites.

FIG. 3.

Notation introduced in the splitting of Λ into size-fixed Ai and Bi for the discussion in Assumption 1. For simplicity, we restrict to the case k = 2, l = 1.

FIG. 3.

Notation introduced in the splitting of Λ into size-fixed Ai and Bi for the discussion in Assumption 1. For simplicity, we restrict to the case k = 2, l = 1.

Close modal

Let σΛ be the Gibbs state of a k-local commuting Hamiltonian. Then, with this construction, Assumption 1 can be read as the existence of positive constants K1, K2 independent of Λ for which the following holds:

σC1/2σD1/2σCDσC1/2σD1/21CDK1eK2l,
(A1a)

where l = d(C, D).

This exponential decay of correlations on the Gibbs state is similar to certain forms of decay of correlations of states that frequently appear in the literature of both classical and quantum spin systems. In the latter, this is closely related, for instance, to the concept of LTQO (Local Topological Quantum Order)30 or the local indistinguishability that was introduced in Ref. 21.

The main difference with the (strong) mixing condition of the classical case13 lies in the fact that they considered a decay of correlations with the distance between two connected regions (in particular, rectangles), whereas in our case, we have a finite union of regions of that kind. The fact that the regions are connected is essential for some properties that can be derived from the Dobrushin condition (Refs. 17 and 28, and Condition III.d of Ref. 16).

Nevertheless, the mixing condition that we need to assume for the proof of Theorem 1 to hold is actually a bit weaker. Indeed, the only necessary thing is that we can bound the LHS of (A1a) by something that is strictly smaller than 1/2, i.e.,

σC1/2σD1/2σCDσC1/2σD1/21CD<12,
(A1b)

It is clear that (A1a) implies (A1b), as one can always choose l big enough. This new condition is a bit more approachable and we will show below a couple of examples of systems satisfying it. First, we consider the situation in which the fixed point is close enough to the normalized identity (in other words, we consider high-enough temperature, depending logarithmically of the system size). Then, we can prove the following result:

Proposition 6.
Let Λ be a finite chain, and consider a splitting on it as the one ofFig. 3 . Let us assume thatσΛSΛis close to the normalized identity in the following sense:
σΛ1ΛdΛ<ε
(28)
for a certain smallɛ ≥ 0. Then,(A1b)holds.

Proof.
Note that Eq. (28) is equivalent to
(1ε)1ΛdΛ<σΛ<(1+ε)1ΛdΛ,
and furthermore, given C, D ⊂ Λ with CD = ∅ (independently of the geometry), the following holds:
(1ε)1CdC<σC<(1+ε)1CdC,
as well as analogously for D and CD, respectively. Recalling now that for every A, B positive definite matrices, the following equivalence holds: ABA−1B−1, and we have
dCD(1ε)21CD>σC1/2σD1/2>dCD(1+ε)21CD,
and thus,
(1ε)(1+ε)21CD<σC1/2σD1/2σCDσC1/2σD1/2<(1+ε)(1ε)21CD.
Therefore, it is easy to see that
σC1/2σD1/2σCDσC1/2σD1/21CD<max3ε+ε2(1+ε)2,3εε2(1ε)2,
where indeed this maximum is always attained by the second element, which constitutes a strictly decreasing function with ɛ. Hence, to conclude, it is enough to find the values of ɛ ≥ 0 for which
3εε2(1ε)2<12,
which holds for ɛ < 0.13.□

Remark 3.

The previous result holds, in particular, for classical systems at high-enough temperature. Furthermore, note that the same proof allows to show that (A1b) holds for systems whose fixed point is close enough to a tensor product between C and D (with a distance scaling logarithmically with the system size).

Next, with a much more elaborate but similar in spirit proof, we can show that states with a defect at site i so that the interaction is bigger there, but interactions decay away from that site, also satisfy (A1b).

FIG. 4.

Decomposition of σΛ into the product of commuting terms for k = 3 and l = 5, assuming that Λ is decomposed only into A1, B1, and A2 for simplification.

FIG. 4.

Decomposition of σΛ into the product of commuting terms for k = 3 and l = 5, assuming that Λ is decomposed only into A1, B1, and A2 for simplification.

Close modal

Proposition 7.
Let Λ be a finite chain, and consider a splitting on it as the one ofFig. 3 . If we assume the following condition:
i=1nγi2>23i=1nδi2>13,
where we are writing
  • γiγCE(i)γED(i)γDF(i)γFC(i), fori = 1, …, n − 1,

  • δiδCE(i)δED(i)δDF(i)δFC(i), fori = 1, …, n − 1,

  • γnγCE(n)γED(n),

  • δnδCE(n)δED(n),

and for which eachγGH(i), respectively,δGH(i), is the minimum, respectively, maximum, eigenvalue ofσ(Gi)Hi, then(A1b)holds.

Proof.
First, note that condition (A1b) is equivalent to the following:
12σCσD<σCD<32σCσD.
(29)
Now, the state σΛ on the full chain can be decomposed into the following product of commuting terms (see Fig. 4):
ZσΛi=1n1χiσ̃C̊nσ̃(Cn)Enσ̃En̊σ̃(Dn)Enσ̃Dn̊,
(30)
with
χiσ̃C̊iσ̃(Ci)Eiσ̃Ei̊σ̃(Di)Eiσ̃Di̊σ̃(Di)Fiσ̃Fi̊σ̃(Ci+1)Fi,
(31)
where Z is the normalization factor and G̊ denotes the interior of G that is the set of sites in G whose corresponding interaction is fully supported in G. We use the notation σ̃G to remark that this term does not coincide, in general, with trGc[σΛ]. We will bound the boundary terms as follows: For any consecutive Gj, Hi ∈ {Cj, Di, Ei, Fi} so that Hi = Ei or Fi (and thus Gj = Ci, Ci+1, or Di), we have
γGH(i)1(Gj)Hiσ̃(Gj)HiδGH(i)1(Gj)Hi.
(32)
Note that in a slight abuse of notation, we are denoting by γFC(i) and δFC(i) the coefficients corresponding to the term σ̃Fi(Ci+1).Then, since (∂Gj) ∩ Hi consists of 2(k − 1) sites, half of which belong to Gj and the other half to Hi, we can write
γGH(i)σ̃G̊iσ̃H̊iσ̃G̊iσ̃(Gi)Hiσ̃H̊iδGH(i)σ̃G̊iσ̃H̊i,
and thus, replacing (32) in (30) after tracing out E and F, it is easy to show that
i=1n1γiσ̃C̊iD̊itr(σ̃E̊iF̊i)γCE(n)γED(n)σ̃C̊nD̊ntr(σ̃E̊n)ZσCDi=1n1δiσ̃C̊iD̊itr(σ̃E̊iF̊i)δCE(n)δED(n)σ̃C̊nD̊ntr(σ̃E̊n),
where γiγCE(i)γED(i)γDF(i)γFC(i) and δiδCE(i)δED(i)δDF(i)δFC(i), and d is the dimension of the local Hilbert space associated with each site.
On the other hand, if we proceed analogously to get a bound for σCσD to compare it with σCD, we obtain
i=1n1γi2σ̃C̊iσ̃D̊itr(σ̃E̊iF̊i)2γCE(n)γED(n)2σ̃C̊nσ̃D̊ntr(σ̃C̊D̊)Z2σCσDi=1n1δi2σ̃C̊iσ̃D̊itr(σ̃E̊iF̊i)2δCE(n)δED(n)2σ̃C̊nσ̃D̊ntr(σ̃C̊D̊).
Therefore, a sufficient condition for (A1b) is that
12tr(σC̊D̊)i=1n1γitr(σE̊iF̊i)δi2γnδn2<Z<32tr(σC̊D̊)i=1n1γi2tr(σE̊iF̊i)δiγn2δn,
(33)
with γnγCE(n)γED(n) and δnδCE(n)δED(n). Note that when β → 0, Zd|Λ|, where the number |Λ| of sites is equal to |E̊|+|F̊|+8(k1)(n1)+|C̊|+|D̊|. Moreover, δi = γi = 1 in the limit. Therefore, (33) holds trivially, since it reduces to 12<1<32. It is reasonable then to think that, close to infinite temperature (in a distance depending logrithmically with the system size), (33) holds.
Indeed, let us assume the following inequality between the γi and δi:
i=1nγi2>23i=1nδi2>13.
(34)

To conclude the proof that Eq. (34) implies Eq. (A1b), it is enough to bound Z, the normalization factor, in the same way that we have bounded σCD and σCσD. Introducing those bounds in the inequalities appearing in (33), it is easy to see that this expression reduces to (34).□

In this subsection, we will discuss Assumption 2, which can be seen as a strong quasi-factorization of the relative entropy, and provide some sufficient conditions on σΛ for it to hold.

Given Λ a finite chain and A a subset of Λ, if we denote by σΛ the Gibbs state of a k-local commuting Hamiltonian, Assumption 2 reads as

DA(ρΛσΛ)fA(σΛ)xADx(ρΛσΛ)ρΛSΛ,
(35)

where 1 ≤ fA(σΛ) < depends only on σΛ and is independent of |Λ|.

Let us first recall that A has a fixed size of 2(k + l) − 1 sites, so |A∂| = 2(2k + l − 1) − 1 and is, in particular, fixed. Moreover, if we separate one site from the rest in each step, i.e., for every 2 ≤ m ≤ |A|, if we consider the only connected B(m)A of size m that contains the first site of A, and we split B(m) into two connected regions B1(m) and B2(m) so that |B1(m)|=1, it is clear that the following inequality

DB(m)(ρΛσΛ)fB(m)(σΛ)DB1(m)(ρΛσΛ)+DB2(m)(ρΛσΛ)ρΛSΛ
(36)

implies inequality (35) by induction, taking

fA(σΛ)sup2m|A|fB(m)(σΛ).

Therefore, we can pose the following natural question:

Question 1.
Given two adjacent subsetsA, B ⊂Λ, under which condition on the Gibbs stateσΛdoes there exist a boundedfAB(σΛ) only depending onσΛand independent of the size of Λ such that the following inequality holds for everyρΛSΛ:
DAB(ρΛσΛ)fAB(σΛ)DA(ρΛσΛ)+DB(ρΛσΛ)?
(37)

We remark that we only need to answer this question for |A|, |B| < 2(k + l). Although we cannot give a general answer to this question, we can provide some motivation for situations in which it might hold. For that, we prove the following lemma, which shows that a conditional relative entropy in a certain region can be upper bounded by a quantity depending only on the reduced states in that region independently of the cardinality of the whole lattice.

Lemma 8.
LetA ⊂Λ. For anyρΛSΛ,
DA(ρΛσΛ)DA(ρΛσAσAc)+D(ρAσA).

Proof.
A simple use of the definition of the conditional relative entropy leads to the following identity:
DA(ρΛσΛ)DA(ρΛσAσAc)=D(ρΛσΛ)D(ρΛσAσAc)=tr[ρΛlogσΛ+logσAσAc].
(38)
By the quantum Markov chain property of the state σΛ between AAAc and by Proposition 5, we have
logσΛ=logσAc+logσAlogσA.
Plugging this in Eq. (38), we arrive at
DA(ρΛσΛ)DA(ρΛσAσAc)=tr[ρΛlogσA+logσAσA]=D(ρAσA)D(ρAσAσA)D(ρAσA).

Note that when ρ is classical, inequality (37) holds true for any Gibbs state of a classical k-local commuting Hamiltonian in 1D, and under some further assumptions, it also does in more general dimensions, since (37) coincides in the classical setting with a usual result of quasi-factorization of the entropy due to the DLR conditions. More specifically, this inequality holds classically whenever the Dobrushin–Shlosman complete analiticity condition holds. Moreover, in that setting, one can see that fAB(σΛ) actually depends only on σ(AB).

It is then reasonable to believe that this inequality might also hold true for Gibbs states of quantum k-local commuting Hamiltonians in 1D, although fAB could possibly depend on σ on the whole lattice Λ (without depending on its size). The intuition behind this is that σΛ is also a quantum Markov chain, and Lemma 8 shows that the conditional relative entropy in a certain region can be approximated by its analog for σΛ a tensor product obtaining an additive error term that can be bounded by something that only depends on the region and its boundary.

However, if we define

fAB(σΛ)supρΛSΛDAB(ρΛσΛ)DA(ρΛσΛ)+DB(ρΛσΛ),

we lack a proof that, in general, it satisfies the necessary conditions for (37) to hold. The study of examples of Hamiltonians whose Gibbs state satisfies the aforementioned inequality is left for future work.

Nevertheless, let us recall here some situations for which we already know that inequality (37) holds. First, if σΛ is a tensor product, this inequality holds with f = 1,9 as a consequence of strong subadditivity. Moreover, for a more general σΛ, if A and B are separated enough, we have seen in Step 2 that it also holds with f = 1 due to the structure of the quantum Markov chain of σΛ. Since in (37), we are assuming that A and B are adjacent, we cannot use this property to “separate” A from B, i.e., write σΛ as a direct sum of tensor products that separate A from B, and thus, the proof of Step 2 cannot be used here.

Additionally, let us mention that the idea used in Proposition 6 to show that Assumption 1 holds for systems at high-enough temperature cannot be used for Assumption 2. Indeed, by assuming Eq. (28), we get

log(1ε)+log1XdX<logσX<log(1+ε)+log1XdX

for X = A, B, Λ, and thus,

DABρΛ1ΛdΛ+log1ε1+ε<DAB(ρΛσΛ)<DABρΛ1ΛdΛ+log1+ε1ε,

and analogously for DA(ρΛσΛ) and DB(ρΛσΛ). Therefore, this allows us to reduce the expression above for fAB(σΛ) to

fAB(σΛ)>supρΛSΛDABρΛ1ΛdΛDAρΛ1ΛdΛ+DBρΛ1ΛdΛ+g(ε)DAρΛ1ΛdΛ+DBρΛ1ΛdΛ,

where g(ɛ) is a function of ɛ, and even though the first summand in the right-hand side above is known to be lower bounded by 1, the second term can be, in principle, arbitrarily large. Therefore, this method does not help to prove Assumption 2, with a multiplicative error term, for high-enough temperature. However, it would if we allowed for an additive term to appear. Indeed, using the previous comparisons for the logarithms, it is easy to show that the following inequality holds:

DAB(ρΛσΛ)<DA(ρΛσΛ)+DB(ρΛσΛ)+3log1+ε1ε

using the fact that fAB(1Λ/dΛ)=1. This inequality, though, cannot be used to show the positivity of a MLSI constant, although some generalizations of quasi-factorization inequalities with additive error terms for a different notion of conditional relative entropy have found different applications in quantum information theory, including new entropic uncertainty relations, as shown in Ref. 2.

In this paper, we have addressed the problem of finding conditions on the Gibbs state of a local commuting Hamiltonian so that the generator of a certain dissipative evolution has a positive modified logarithmic Sobolev constant. Building on the results from classical spin systems13,11 and following the steps of Ref. 21, where the authors addressed the analogous problem for the spectral gap, we have developed a strategy based on five points that provides positivity of the modified logarithmic Sobolev constant. Moreover, we have used this strategy to present two conditions on a Gibbs state so that its corresponding heat-bath dynamics in 1D satisfies this positivity.

This strategy opens a new way to obtain positivity of MLSI constants, and thus, it will probably be of use to prove this condition not only for the dynamics studied in this paper but also for some other dynamics, such as the one of the Davies semigroups (for instance, Ref. 14). However, for the time being, some natural questions arise from this work, such as the existence of examples of non-trivial Gibbs states for which these static conditions, and thus positivity of the modified logarithmic Sobolev constant, hold.

  • Question 1.Are there any easy examples ofσΛfor which the strong quasi-factorization of Assumption 2 holds withfdifferent fromAssumption 1?

    So far, the only example we have for this condition to hold is for σΛ a tensor product everywhere, for which the value of f is always 1. It is reasonable to think that this condition holds, for instance, when σΛ is a classical state, since in this case, one could expect to recover the classical case in which this inequality would agree with the usual quasi-factorization, thanks to the DLR condition. However, this is left for future work.

  • Question 2.Are there any more examples ofσΛfor which the mixing condition of Assumption 1 holds?

    Even though we have mentioned that this condition holds for classical states and we have shown a more complicated example of Gibbs state verifying this in Proposition 7, most of the tools available in the setting of quantum many-body systems to address the problem of decay of correlations on the Gibbs state depend strongly on the geometry used to split the lattice and, more specifically, on the number of boundaries between the different regions A and B. Since, in our case, this number scales linearly with Λ, there is no hope to use any of those tools to obtain more examples of Gibbs states satisfying Assumption 1. However, it is possible that a different approach allows for more freedom in this sense.

  • Question 3.Could this result be extended to more dimensions?

    Following the same approach from this paper, we would need to cover an n-dimensional lattice with small rectangles overlapping pairwise in an analogous way to the construction described here for dimension 1. It is easy to realize that even in dimension 2, one would need at least three systems to classify the small rectangles so that two belonging to the same class would not overlap. Thus, for our strategy to hold in dimension, at least, 2, we would need a result of quasi-factorization that provides an upper bound for the relative entropy of two states in terms of the sum of three conditional relative entropies, instead of two, and a multiplicative error term. Since we are lacking a result of this kind so far, this question constitutes an open problem.

  • Question 4.Can we change the geometry used to split the lattice?

    Another possible approach to tackle this problem could be based on the geometry presented in the classical papers13,11 and the quantum case for the spectral gap.21 In this approach, in each step, one splits the rectangle into two connected regions and carries out a more evolved geometric recursive argument. One of the main benefits from this approach would be a weakening in the mixing condition assumed in the Gibbs state. However, the main counterpart would be the necessity of a strong result of quasi-factorization for the relative entropy, even stronger than the one appearing in (37) since the multiplicative error term should converge to 1 exponentially fast, in which both sides of the inequality would contain conditional relative entropies.

The authors would like to thank Nilanjana Datta for fruitful discussions and for her comments on an earlier version of the draft. I.B. was supported by French A.N.R. (Grant No. ANR-14-CE25-0003 “StoQ”). A.C. was partially supported by a La Caixa-Severo Ochoa grant (ICMAT Severo Ochoa Project No. SEV-2011-0087, MINECO) and the MCQST Distinguished PostDoc fellowship from the Munich Center for Quantum Science and Technology. A.C. and D.P.-G. acknowledge support from MINECO (Grant No. MTM2017-88385-P) and from Comunidad de Madrid (Grant Nos. QUITEMAD-CM and ref. P2018/TCS-4342). A.L. acknowledges support from the Walter Burke Institute for Theoretical Physics in the form of the Sherman Fairchild Fellowship as well as support from the Institute for Quantum Information and Matter (IQIM), an NSF Physics Frontiers Center (NFS Grant No. PHY-1733907), from the BBVA Fundation, and from the Spanish Ramón y Cajal Programme (RYC2019-026475-I / AEI / 10.13039/501100011033). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No 648913). C.R. acknowledges financial support from the TUM University Foundation Fellowship and by the DFG Cluster of Excellence 2111 (Munich Center for Quantum Science and Technology).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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This and other terms in italics used in the Introduction will be defined precisely later in the paper, in Sec. II.

40.

The precise choice of the distance measure will be introduced later.