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

TL;DR

This paper establishes a modified logarithmic Sobolev inequality for quantum spin systems with classical and commuting interactions, leading to exponential convergence to equilibrium and broad applications in quantum information processing.

Contribution

It proves the positivity of the modified logarithmic Sobolev inequality for quantum lattice systems independently of size, linking spatial mixing to analyticity of free energy.

Findings

Exponential convergence in relative entropy to equilibrium.

Quantum annealer outputs near-optimal energy after finite time.

Gaussian concentration inequalities for Lipschitz observables.

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.