Relative entropy decay and complete positivity mixing time

TL;DR

This paper establishes bounds linking the complete modified logarithmic Sobolev constant of quantum Markov semigroups to their mixing times, with implications for classical and quantum systems, including concentration inequalities.

Contribution

It introduces bounds connecting the Sobolev constant and mixing time for quantum and classical Markov semigroups, extending to von Neumann algebras.

Findings

Bound on quantum Sobolev constant by inverse mixing time

Uniform modified log-Sobolev inequality for classical sub-Laplacians

Spectral gap approximation for quantum semigroups

Abstract

We prove that the complete modified logarithmic Sobolev constant of a quantum Markov semigroup is bounded by the inverse of its complete positivity mixing time. For classical Markov semigroups, this implies that every sub-Laplacian given by a H\"ormander system on a compact manifold satisfies a uniform modified log-Sobolev inequality for matrix-valued functions. For quantum Markov semigroups, we obtain that the complete modified logarithmic Sobolev constant is comparable to spectral gap up to a constant as logarithm of dimension constant. This estimate is asymptotically tight for a quantum birth-death process. Our results and the consequence of concentration inequalities apply to GNS-symmetric semigroups on general von Neumann algebras.