Entropy Decay for Davies Semigroups of a One Dimensional Quantum Lattice

Given a finite-range, translation-invariant commuting system Hamiltonian 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 (Commun Math Phys 123(1):85-93, 1989) to the quantum setting as well as provides an exponential improvement over the non-closure of the gap proved by Brandao and Kastoryano (Commun Math Phys 344(3):915-957, 2016). 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.