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 [40] 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 [43]. 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.

Quantum many-body systems in thermal equilibrium

“…On the other hand, a bound on the so-called log-Sobolev contant [168] implies that instead it takes a time logarithmic in the system size to thermalize. This was recently proven for 1D chains [169,170] and in [171,172] for other models of dissipation. These results show that the dissipative processes at hand can also be seen as an efficient quantum algorithms preparing thermal states, since in principle they can be simulated efficiently with a quantum computer [152].…”

Deviation bounds and concentration inequalities for quantum noises

“…In [44], it is proved that Davies dynamics are gapped at any inverse temperature β > 0 in 1D and on regular lattices below a threshold inverse temperature β c > 0. This result was extended to the MLSI in the 1D case in a paper to appear [5]. However, it is still open whether these dynamics also satisfy a transportation cost-information inequality under reasonable assumptions.…”

Complete Modified Logarithmic Sobolev inequality for sub-Laplacian on $SU(2)$

“…In recent decades, log-Sobolev inequalities has been studied for quantum systems on noncommutative spaces, and have attracted a lot of attention in quantum information theory and quantum many-body system, e.g. [4,12,13,21,32,41]. Motivated by the quantum information theory, we study modified log-Sobolev inequalities for matrixvalued functions for semigroups generated by sub-Laplacians, which has direct application to quantum Markov semigroups on matrix algebras.…”