On the modified logarithmic Sobolev inequality for the heat-bath dynamics for 1D systems

TL;DR

This paper establishes conditions under which the modified logarithmic Sobolev inequality holds for heat-bath dynamics in 1D quantum systems, linking quantum functional inequalities to Gibbs state properties.

Contribution

It introduces a strategy to prove the positivity of the modified logarithmic Sobolev constant for certain quantum systems based on clustering conditions of the Gibbs state.

Findings

Positivity of the modified logarithmic Sobolev constant for 1D heat-bath dynamics.

Link between Gibbs state clustering and quantum functional inequalities.

Conditions for rapid mixing in quantum many-body systems.

Abstract

The mixing time of Markovian dissipative evolutions of open quantum many-body systems can be bounded using optimal constants of certain quantum functional inequalities, such as the modified logarithmic Sobolev constant. For classical spin systems, the positivity of such constants follows from a mixing condition for the Gibbs measure, via quasi-factorization results for the entropy. Inspired by the classical case, we present a strategy to derive the positivity of the modified logarithmic Sobolev constant associated to the dynamics of certain quantum systems from some clustering conditions on the Gibbs state of a local, commuting Hamiltonian. In particular we show that for the heat-bath dynamics for 1D systems, the modified logarithmic Sobolev constant is positive under the assumptions of a mixing condition on the Gibbs state and a strong quasi-factorization of the relative entropy.