The modified logarithmic Sobolev inequality for quantum spin systems: classical and commuting nearest neighbour interactions

TL;DR

<3-5 sentence high-level summary> This work establishes a system-size-independent MLSI for quantum lattice spin systems at sufficiently high temperature by linking spatial clustering of Gibbs states to rapid entropy decay under a local quantum Markov semigroup. The authors develop a peeling technique together with a geometric tiling and approximate tensorization to translate a quantum L1-L∞ clustering condition into MLSI, with special, explicit results in 1D and 2D through rhombus-like geometries. The MLSI implies exponential convergence in relative entropy to equilibrium, and yields a suite of applications in quantum information, including robust energy concentration, eigenstate thermalization, finite-block hypothesis testing, and efficient local Gibbs-state preparation via logarithmic-depth quantum circuits. This provides a robust bridge between static correlation decay and dynamical mixing in quantum many-body systems, with broad implications for quantum annealing, transport inequalities, and thermodynamic control in noisy quantum devices.

Abstract

Given a uniform, frustration-free family of local Lindbladians defined on a quantum lattice spin system in any spatial dimension, we prove a strong exponential convergence in relative entropy of the system to equilibrium under a condition of spatial mixing of the stationary Gibbs states and the rapid decay of the relative entropy on finite-size blocks. Our result leads to the first examples of the positivity of the modified logarithmic Sobolev inequality for quantum lattice spin systems independently of the system size. Moreover, we show that our notion of spatial mixing is a consequence of the recent quantum generalization of Dobrushin and Shlosman's complete analyticity of the free-energy at equilibrium. The latter typically holds above a critical temperature Tc. Our results have wide-ranging applications in quantum information. As an illustration, we discuss four of them: first, using techniques of quantum optimal transport, we show that a quantum annealer subject to a finite range classical noise will output an energy close to that of the fixed point after constant annealing time. Second, we prove Gaussian concentration inequalities for Lipschitz observables and show that the eigenstate thermalization hypothesis holds for certain high-temperture Gibbs states. Third, we prove a finite blocklength refinement of the quantum Stein lemma for the task of asymmetric discrimination of two Gibbs states of commuting Hamiltonians satisfying our conditions. Fourth, in the same setting, our results imply the existence of a local quantum circuit of logarithmic depth to prepare Gibbs states of a class of commuting Hamiltonians.

Figures (11)

• Figure 1: Relations between static and dynamical properties of quantum Gibbs states. The main result of this paper is depicted by the red arrow connecting the notion of clustering of correlations to the existence of a modified logarithmic Sobolev constant. (i) The approximate tensorization of the variance was proved in [BK16] to lead to the non-closure of the gap, and an entropic strengthening of that statement is the subject of Theorem \ref{['ATAC']}. The interpolation (ii) was essentially proven in temme2015fast. We derive the clustering of correlations at high enough temperature and show (iii) by adapting techniques recently pioneered in harrow2020classical, whereas Lieb-Robinson bounds were employed in Kastoryano2013a to derive it from the non-closure of the gap (iv). Finally, the detectability lemma was employed in [BK16] to derive a stronger notion of clustering of correlation from the gap condition (v).
• Figure 2: For the construction of $\mathcal{A}_{A, \text{out}}$, given a site $k$ in the boundary of $A$, the algebra acts trivially on the site $k$ itself, as well as on the edge $l-k$, whereas it acts non-trivially in the other three edges.
• Figure 3: Relations between different notions of clustering of correlations and their link to thermalization times. (0) is proved in Theorem \ref{['thm:analyticity-high-temperature']} in the high temperature regime. (i) is the main result of [BK16]. (i') and (iii) are proved in Theorem \ref{['thm:differentclusterings']} whereas (ii) is proved in [BK16]. (ii') is proved in Proposition \ref{['propDScorrdecay']}. Finally ($\star$) is the subject of Section \ref{['sec:main']}.
• Figure 4: Decomposition of the boundaries of $C$ and $C \cup D$ for Proposition \ref{['propDScorrdecay']}. Here $\partial 1:=\partial C\backslash D$, $\partial 2:=\partial C \cap D$ and $\partial 3:=\partial D\backslash C\partial$. Therefore $\partial 1\cup \partial 2=\partial C$ whereas $\partial 1\cup \partial 3=\partial(C\cup D)$. Here, we considered nearest neighbour interactions.
• Figure 5: Tiling of $\mathbb{Z}^2$. Here, we assumed two-local interactions. The pixels in pink represent the non-planar sheet constructed out of translations of the pixels $A_0$ and $A_1$. The ones in red correspond to a column.

Theorems & Definitions (84)

• Theorem 1: MLSI for quantum lattice spin systems (informal)
• Theorem 2: Conditioned $\mathbb{L}_1- \mathbb{L}_\infty$ exponential decay of correlations (informal)
• Theorem 3: Approximate tensorization of the quantum relative entropy bardet2020approximate (informal)
• Theorem 4: maas2011gradient Theorem 3.1
• Definition 1: Uniform family of Lindbladians
• Lemma 1
• proof
• Lemma 2
• proof
• Definition 2: [KT13]CM15BarEID17gao2018fisher