Entropy decay for Davies semigroups of a one dimensional quantum lattice

Abstract

Given a finite-range, translation-invariant commuting system Hamiltonians on a spin chain, we show that the Davies semigroup describing the reduced dynamics resulting from the joint Hamiltonian evolution of a spin chain weakly coupled to a large heat bath thermalizes rapidly at any temperature. More precisely, we prove that the relative entropy between any evolved state and the equilibrium Gibbs state contracts exponentially fast with an exponent that scales logarithmically with the length of the chain. Our theorem extends a seminal result of Holley and Stroock to the quantum setting, up to a logarithmic overhead, as well as provides an exponential improvement over the non-closure of the gap proved by Brandao and Kastoryano. This has wide-ranging applications to the study of many-body in and out-of-equilibrium quantum systems. Our proof relies upon a recently derived strong decay of correlations for Gibbs states of one dimensional, translation-invariant local Hamiltonians, and tools from the theory of operator spaces.

arXiv comment44 pages, 6 figures

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• General Continuity Bounds for Quantum Relative Entropies
• Maximal Speed of Propagation in Open Quantum Systems
• Deviation bounds and concentration inequalities for quantum noises
• Polynomial Time Quantum Gibbs Sampling for Fermi-Hubbard Model at any Temperature
• Ricci curvature of quantum channels on non-commutative transportation metric spaces
• Area law for steady states of detailed-balance local Lindbladians
• Strong decay of correlations for Gibbs states in any dimension
• Dissipative ground state preparation in ab initio electronic structure theory
• Quantized Markov Chain Couplings that Prepare Qsamples
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• Efficient and simple Gibbs state preparation of the 2D toric code via duality to classical Ising chains