Complete gradient estimates of quantum Markov semigroups

TL;DR

This paper introduces a comprehensive gradient estimate for symmetric quantum Markov semigroups, linking entropy convexity with noncommutative Wasserstein distances and establishing stability and applications in free group factors.

Contribution

It presents a novel complete gradient estimate for quantum Markov semigroups, demonstrating its stability and applicability to free group factors.

Findings

Gradient estimate implies entropy semi-convexity

Stability under tensor and free products

Proves optimal modified logarithmic Sobolev inequality

Abstract

In this article we introduce a complete gradient estimate for symmetric quantum Markov semigroups on von Neumann algebras equipped with a normal faithful tracial state, which implies semi-convexity of the entropy with respect to the recently introduced noncommutative 2-Wasserstein distance. We show that this complete gradient estimate is stable under tensor products and free products and establish its validity for a number of examples. As an application we prove a complete modified logarithmic Sobolev inequality with optimal constant for Poisson-type semigroups on free group factors.