Curvature-dimension conditions for symmetric quantum Markov semigroups

TL;DR

This paper extends classical curvature-dimension bounds to the quantum setting, establishing new inequalities and geometric properties for symmetric quantum Markov semigroups on matrix algebras.

Contribution

It introduces two noncommutative curvature-dimension conditions and proves related functional inequalities, a Bonnet-Myers type theorem, and entropy power concavity in quantum systems.

Findings

Established dimension-dependent functional inequalities.

Proved a quantum version of the Bonnet-Myers theorem.

Provided examples including Schur and Herz-Schur multipliers.

Abstract

Following up on the recent work on lower Ricci curvature bounds for quantum systems, we introduce two noncommutative versions of curvature-dimension bounds for symmetric quantum Markov semigroups over matrix algebras. Under suitable such curvature-dimension conditions, we prove a family of dimension-dependent functional inequalities, a version of the Bonnet-Myers theorem and concavity of entropy power in the noncommutative setting. We also provide examples satisfying certain curvature-dimension conditions, including Schur multipliers over matrix algebras, Herz-Schur multipliers over group algebras and depolarizing semigroups.