Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance

TL;DR

This paper demonstrates that ergodic quantum Markov semigroups with detailed balance act as gradient flows for relative entropy under a non-commutative Wasserstein-like metric, leading to new entropy decay inequalities.

Contribution

It introduces a non-commutative Wasserstein metric under which quantum Markov semigroups are gradient flows, establishing entropy convexity and decay inequalities.

Findings

Quantum Markov semigroups are gradient flows for relative entropy.

The introduced metric is a non-commutative analog of the 2-Wasserstein metric.

New inequalities for entropy decay in quantum systems.

Abstract

We study a class of ergodic quantum Markov semigroups on finite-dimensional unital $C^*$-algebras. These semigroups have a unique stationary state $\sigma$, and we are concerned with those that satisfy a quantum detailed balance condition with respect to $\sigma$. We show that the evolution on the set of states that is given by such a quantum Markov semigroup is gradient flow for the relative entropy with respect to $\sigma$ in a particular Riemannian metric on the set of states. This metric is a non-commutative analog of the $2$-Wasserstein metric, and in several interesting cases we are able to show, in analogy with work of Otto on gradient flows with respect to the classical $2$-Wasserstein metric, that the relative entropy is strictly and uniformly convex with respect to the Riemannian metric introduced here. As a consequence, we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum Markov semigroups with detailed balance.