Quantum conditional relative entropy and quasi-factorization of the relative entropy

TL;DR

This paper introduces quantum conditional relative entropy and establishes quasi-factorization results, leading to new bounds on the mixing times of quantum dissipative systems with product fixed points.

Contribution

It defines quantum conditional relative entropy and proves quasi-factorization results, extending classical entropy techniques to quantum systems.

Findings

Proved quasi-factorization results for quantum conditional relative entropy.

Established a positive log-Sobolev constant for quantum heat-bath dynamics.

Connected quasi-factorization to mixing time bounds in quantum systems.

Abstract

The existence of a positive log-Sobolev constant implies a bound on the mixing time of a quantum dissipative evolution under the Markov approximation. For classical spin systems, such constant was proven to exist, under the assumption of a mixing condition in the Gibbs measure associated to their dynamics, via a quasi-factorization of the entropy in terms of the conditional entropy in some sub-$\sigma$-algebras. In this work we analyze analogous quasi-factorization results in the quantum case. For that, we define the quantum conditional relative entropy and prove several quasi-factorization results for it. As an illustration of their potential, we use one of them to obtain a positive log-Sobolev constant for the heat-bath dynamics with product fixed point.