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.
I. INTRODUCTION
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 theory^{35,31} 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 Olivieri^{29,28} and aimed to notably simplify their proof via a result of quasi-factorization of the entropy^{39} in terms of some conditional entropies. Previously, a result of quasi-factorization of the variance^{4} 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:
Definition of a conditional modified logarithmic Sobolev constant (conditional MLSI constant, for short).
Definition of the clustering condition on the Gibbs state.
Quasi-factorization of the relative entropy in terms of a conditional relative entropy (introduced in Ref. 8).
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.
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) condition^{15,24} allows us to reduce the study of the MLSI on a domain and its boundary. In the quantum case, the DLR condition fails^{18} 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.
A. Informal exposition of the main result
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 $\Lambda \u2282Z$ and a state $\rho \Lambda \u2208S\Lambda $, it is defined as
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 σ_{Λ} 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.
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 $\sigma \Lambda =\u2a02x\u2208\Lambda \sigma x$.
Then, the main result of this paper is the positivity of the modified logarithmic Sobolev constant for the heat-bath dynamics in 1D.
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.
B. Layout of the paper
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.
II. PRELIMINARIES AND NOTATION
A. Notation
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 $H\Lambda =\u2297x\u2208\Lambda Hx$, whose dimension is d_{Λ}. Throughout this text, Λ will often consist of three parties, and we will denote by $HABC=HA\u2297HB\u2297HC$ the corresponding tripartite Hilbert space. Furthermore, most of this paper concerns quantum spin lattice systems, and we often assume that $\Lambda \u2282\u2282Zd$ is a finite subset. In general, we use uppercase Latin letters to denote systems or sets.
For every finite dimensional $H\Lambda $, we denote the associated set of bounded linear operators by $B\Lambda \u2254B(H\Lambda )$ and by $A\Lambda \u2254A(H\Lambda )$ its subset of observables, i.e., Hermitian operators, which we denote by lowercase Latin letters. We further denote by $S\Lambda \u2254S(H\Lambda )={f\Lambda \u2208A\Lambda :f\Lambda \u22650\u2009and\u2009tr[f\Lambda ]=1}$ 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 σ_{Λ}. 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:B\Lambda \u2192B\Lambda $ is called a superoperator. We write $1$ for the identity matrix and id for the identity superoperator. For bipartite spaces $HAB=HA\u2297HB$, we consider the natural inclusion $AA\u21aaAAB$ by identifying each operator $fA\u2208AA$ with $fA\u22971B$. In this way, we define the modified partial trace in A of $fAB\u2208AAB$ by $trB[fAB]\u22971B$, but we denote it by tr_{B}[f_{AB}] in a slight abuse of notation. Moreover, we say that an operator $gAB\u2208AAB$ has support in A if it can be written as $gA\u22971B$ for some operator $gA\u2208AA$. Note that given $fAB\u2208AAB$, we write f_{A} ≔ tr_{B}[f_{AB}].
Finally, given $x,y\u2208\Lambda \u2282\u2282Zd$, 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): x ∈ A, y ∈ B}. Furthermore, we denote by ‖·‖_{∞} the usual operator norm and by ‖·‖_{1} = tr[|·|] the trace-norm.
B. Entropies
1. Von Neumann entropy
Let $H\Lambda $ be a finite dimensional Hilbert space, and consider $\rho \Lambda \u2208S\Lambda $. The von Neumann entropy of ρ_{Λ} is defined as
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.
2. Relative entropy
A measure of distinguishability between two states that will appear often throughout this text is the relative entropy. Let $H\Lambda $ be a finite dimensional Hilbert space, and consider $\rho \Lambda ,\sigma \Lambda \u2208S\Lambda $. We define the relative entropy of ρ_{Λ} and σ_{Λ} as
Some fundamental properties of the relative entropy that will be of use are the following:
(properties of the relative entropy, Ref. 38). Let $HAB=HA\u2297HB$ be a bipartite Hilbert space, and consider $\rho AB,\sigma AB\u2208SAB$. Then, the following properties hold:
Non-negativity. D(ρ_{AB}‖σ_{AB}) ≥ 0 and D(ρ_{AB}‖σ_{AB}) = 0 if, and only if, ρ_{AB} = σ_{AB}.
Additivity. D(ρ_{A} ⊗ ρ_{B}‖σ_{A} ⊗ σ_{B}) = D(ρ_{A}‖σ_{A}) + D(ρ_{B}‖σ_{B}).
Superadditivity. D(ρ_{AB}‖σ_{A} ⊗ σ_{B}) ≥ D(ρ_{A}‖σ_{A}) + D(ρ_{B}‖σ_{B}).
Data processing inequality. For every quantum channel $T:SAB\u2192SAB$, $D(\rho AB\Vert \sigma AB)\u2265D(T(\rho AB)\Vert T(\sigma AB))$.
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 $HAB=HA\u2297HB$ be a bipartite finite dimensional Hilbert space, and consider $\rho AB,\sigma AB\u2208SAB$. The conditional relative entropy in A of ρ_{AB} and σ_{AB} is given by
We recall in the next proposition some properties of the conditional relative entropy that will be of use in Secs. II E–V.
(some properties of the conditional relative entropy, Ref. 8). Let $HAB=HA\u2297HB$ be a bipartite finite dimensional Hilbert space, and consider $\rho AB,\sigma AB\u2208SAB$. Then, the following properties hold:
Non-negativity. D_{A}(ρ_{AB}‖σ_{AB}) ≥ 0.
D_{A}(ρ_{AB}‖σ_{A} ⊗ σ_{B}) = I_{ρ}(A: B) + D(ρ_{A}‖σ_{A}), where I_{ρ}(A: B)≔D(ρ_{AB}‖ρ_{A} ⊗ ρ_{B}) is the mutual information.
C. Modified logarithmic Sobolev constant
Given a state ρ_{Λ} 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 $Tt*\u2254etL\Lambda *$, also known as quantum Markov semigroup (QMS), where $L\Lambda *:S\Lambda \u2192S\Lambda $ 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) form^{26,19} for every $X\Lambda \u2208B\Lambda $,
where $H\u2208A\Lambda $, $Lk\u2208B\Lambda $ are the Lindblad operators, and [·,·] denotes the commutator.
We say that the QMS is primitive if there is a unique full-rank $\sigma \Lambda \u2208S\Lambda $, which is invariant for the generator, i.e., such that $L\Lambda *(\sigma \Lambda )=0$. Furthermore, we say that the Lindbladian is reversible or satisfies the detailed balance condition, with respect to a state $\sigma \Lambda \u2208S\Lambda $ if its version for observables verifies
for every $f\Lambda ,g\Lambda \u2208A\Lambda $, where this weighted scalar product is defined for every $f\Lambda ,g\Lambda \u2208A\Lambda $ by
We recall now the notion of entropy production as the derivative of the relative entropy in the following form:
Note that the entropy production of a primitive QMS only vanishes on σ_{Λ}. 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.
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 $\alpha (L\Lambda *)>0$, then for every $\rho \Lambda \u2208S\Lambda $,
D. Gibbs states
Given a finite lattice $\Lambda \u2282\u2282Zd$, let us define a k-local bounded potential as $\Phi :\Lambda \u2192A\Lambda $ 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, H_{A}, is given by
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
and we denote by A∂ the union of A and its boundary. Note that H_{A} 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 $\Lambda \u2282\u2282Zd$, the Gibbs state is defined as
Note that, by a slight abuse of notations, we will denote by σ_{A}, for A ⊂ Λ, the state given by $trAc[\sigma \Lambda ]$, which should not be confused with the restricted Gibbs state corresponding to the terms of the Hamiltonian H_{A}.
E. Heat-bath generator
Let $\Lambda \u2282\u2282Zd$ be a finite lattice and $\Phi :\Lambda \u2192A\Lambda $ be a k-local bounded commuting potential. Consider σ_{Λ} to be the associated Gibbs state. Given A ⊆ Λ, we define the heat-bath conditional expectation as follows: for every $\rho \Lambda \u2208S\Lambda $,
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 σ_{Λ}, comprising the partial trace in A,^{33} i.e., $EA*(\u22c5)\u2254PtrA\sigma \Lambda \u25e6trA[\u22c5]$ for
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
for every $\rho \Lambda \u2208S\Lambda $. Analogously, for every A ⊂ Λ, we denote by $LA*$ the generator where the summation is only over elements x ∈ A. 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:
For any A ⊂ Λ, $LA*$ is the generator of a semigroup of CPTP maps of the form $etLA*$.
$L\Lambda *$ is k-local in the sense that each individual composing term acts non-trivially only on balls of radius k.
For any A, B ⊂ Λ, we have
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.
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.
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 case^{18} 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.
F. Quantum Markov chains
Consider a tripartite space $HABC=HA\u2297HB\u2297HC$. We define a recovery map $RB\u2192BC$ from B to BC as a completely positive trace-preserving map that reconstructs the C-part of a state $\sigma ABC\u2208SABC$ from its B-part only. If that reconstruction is possible, i.e., if for a certain $\sigma ABC\u2208SABC$, there exists such $RB\u2192BC$ verifying
we say that σ_{ABC} is a quantum Markov chain (QMC) between A ↔ B ↔ C. When this is the case, the recovery map can be taken to be the Petz recovery map. More specifically, σ_{ABC} is a QMC (A ↔ B ↔ C) 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 σ_{BC}, with the partial trace in C, i.e.,
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.
(Refs. 33 and 34). Let $HABC=HA\u2297HB\u2297HC$ be a tripartite Hilbert space and $\sigma ABC\u2208SABC$. Then, σ_{ABC} is a quantum Markov chain, if, and only if, I_{σ}(A: C|B) = 0, for I_{σ}(A: C|B) = S(σ_{AB}) + S(σ_{BC}) − S(σ_{ABC}) − S(σ_{B}) 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.
Turning now to Gibbs states, as they were introduced in Subsection II D, we recall an important result about their Markovian structure.
(Theorem 3 of Ref. 6). Given a k-local commuting potential on Λ, its associated Gibbs state σ_{Λ} is a quantum Markov network, that is, for all disjoint subsets A, B, C ⊂ Λ such that B shields A from C with d(A, C) > k, I_{σ}(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:
Using the previous properties for quantum Markov chains, we can easily show the identity of the next proposition.
As a consequence of this identity, we have the following result:
III. TECHNICAL TOOLS
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:
This inequality, in particular, can be used to prove positivity of the MLSI constant for the heat-bath dynamics when σ_{Λ} is a tensor product, as it appears in Ref. 8 (see also Refs. 3 and 1).
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.
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.
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:
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:
IV. MAIN RESULT
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}
The second condition that needs to be assumed constitutes a strong form of quasi-factorization of the relative entropy.
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.
Let $\Lambda \u2282\u2282Z$ be a finite chain. Let $\Phi :\Lambda \u2192A\Lambda $ be a k-local commuting potential and $H\Lambda =\u2211x\u2208\Lambda \Phi (x)$ be its corresponding Hamiltonian, and denote by σ_{Λ} its Gibbs state. Let $L\Lambda *$ be the generator of the heat-bath dynamics. Then, if Assumptions 1 and 2 hold, the MLSI constant of $L\Lambda *$ 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 $l\u2208N$ so that $K1e\u2212K2l<12$ for K_{1} and K_{2} 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):
Λ = A ∪ B.
$A=\u22c3i=1nAi$ and $B=\u22c3j=1nBj$.
|A_{i} ∩ B_{i}| = |B_{i} ∩ A_{i+1}| = l for every i = 1, …, n − 1.
|A_{i}| = |B_{j}| = 2(k + l) − 1 for all i, j = 1, …, n, where k comes from the k-locality of the Hamiltonian.
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 $n\u2208N$, 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 $\sigma AcBc$ is from a tensor product between A^{c} and B^{c}, where A^{c} ≔ Λ\A and B^{c} ≔ Λ\B.
Finally, inequality (18) follows just by replacing in this expression Λ = XZY, A ∩ B = Z, A^{c} = Y, and B^{c} = 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.
Without loss of generality, we assume that A = A_{1} ∪ A_{2} (the general result follows by induction in the number of subsets A_{i}). For convenience, we denote D(ρ_{A}‖σ_{A}), respectively, D_{A}(ρ_{Λ}‖σ_{Λ}), by D(A), respectively, D_{A}(Λ), since we are considering the same states ρ_{Λ} and σ_{Λ} in every (conditional) relative entropy.
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 A_{i} and B_{i}. For that, we need to consider that Assumption 1 holds true.
Finally, in the last step of the proof, we show that the conditional MLSI constants on every A_{i} and B_{i} 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.
Finally, putting together Steps 1, 2, 3, and 4, we conclude the proof of Theorem 7.
V. MIXING CONDITION AND STRONG QUASI-FACTORIZATION
A. Mixing condition
In this subsection, we will elaborate on the mixing condition introduced in Assumption 1 and provide sufficient conditions for it to hold. Consider $\Lambda \u2282\u2282Z$ as a finite chain and A, B ⊂ Λ as in the splitting of Λ in the proof of Theorem 7 (see Fig. 1). Denote C ≔ B^{c} and D ≔ A^{c} so that they can be expressed as the union of disjoint segments, $C=\u22c3i=1nCi$ and $D=\u22c3j=1nDj$, respectively. For every i = 1, …, n, respectively, i = 1, …, n − 1, denote by E_{i}, respectively, F_{i}, the connected set that separate C_{i} from D_{i}, respectively, D_{i} from C_{i+1} (see Fig. 3). Note that because of the construction of A and B described in Sec. IV, every E_{i} and F_{i} are composed of l sites and every C_{i} and D_{i} of, at least, 2k − 1 sites.
Let σ_{Λ} 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 K_{1}, K_{2} independent of Λ for which the following holds:
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 case^{13} 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.,
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:
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).
$\gamma i\u2254\gamma CE(i)\gamma ED(i)\gamma DF(i)\gamma FC(i)$, for i = 1, …, n − 1,
$\delta i\u2254\delta CE(i)\delta ED(i)\delta DF(i)\delta FC(i)$, for i = 1, …, n − 1,
$\gamma n\u2254\gamma CE(n)\gamma ED(n)$,
$\delta n\u2254\delta CE(n)\delta ED(n)$,
and for which each $\gamma GH(i)$, respectively, $\delta GH(i)$, is the minimum, respectively, maximum, eigenvalue of $\sigma (\u2202Gi)\u2229Hi$, then (A1b) holds.
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 σ_{CD} and σ_{C} ⊗ σ_{D}. Introducing those bounds in the inequalities appearing in (33), it is easy to see that this expression reduces to (34).□
B. Strong quasi-factorization
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 σ_{Λ} for it to hold.
Given Λ a finite chain and A a subset of Λ, if we denote by σ_{Λ} the Gibbs state of a k-local commuting Hamiltonian, Assumption 2 reads as
where 1 ≤ f_{A}(σ_{Λ}) < ∞ depends only on σ_{Λ} 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
implies inequality (35) by induction, taking
Therefore, we can pose the following natural question:
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.
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 f_{AB}(σ_{Λ}) 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 f_{AB} could possibly depend on σ on the whole lattice Λ (without depending on its size). The intuition behind this is that σ_{Λ} 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 σ_{Λ} 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
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 σ_{Λ} is a tensor product, this inequality holds with f = 1,^{9} as a consequence of strong subadditivity. Moreover, for a more general σ_{Λ}, 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 σ_{Λ}. 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 σ_{Λ} 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
for X = A, B, Λ, and thus,
and analogously for D_{A}(ρ_{Λ}‖σ_{Λ}) and D_{B}(ρ_{Λ}‖σ_{Λ}). Therefore, this allows us to reduce the expression above for f_{AB}(σ_{Λ}) to
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:
using the fact that $fAB(1\Lambda /d\Lambda )=1$. 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.
VI. CONCLUSIONS AND OPEN PROBLEMS
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 systems^{13,11} 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 of σ_{Λ} for which the strong quasi-factorization of Assumption 2 holds with f different from Assumption 1?
So far, the only example we have for this condition to hold is for σ_{Λ} 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 σ_{Λ} 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 of σ_{Λ} for 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 papers^{13,11} 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.
ACKNOWLEDGMENTS
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 AVAILABILITY
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
REFERENCES
This and other terms in italics used in the Introduction will be defined precisely later in the paper, in Sec. II.
The precise choice of the distance measure will be introduced later.