Bimodule KMS Symmetric Quantum Markov Semigroups and Gradient Flows

TL;DR

This paper explores bimodule KMS symmetric quantum Markov semigroups, introduces directional matrices, and establishes a gradient-flow structure leading to new inequalities in quantum information theory.

Contribution

It introduces directional matrices for bimodule KMS symmetric semigroups and develops a gradient-flow framework with associated inequalities.

Findings

Established a gradient-flow structure for bimodule KMS symmetric semigroups.

Derived a modified logarithmic Sobolev inequality.

Proved a Talagrand inequality for these semigroups.

Abstract

The bimodule KMS symmetry of a bimodule quantum Markov semigroup extends the classical KMS symmetry of a quantum Markov semigroup. Compared with (bimodule) GNS symmetry, the (bimodule) KMS symmetry retains significantly more of the underlying noncommutativity. In this paper, we study bimodule KMS symmetric quantum Markov semigroups and introduce directional matrices for such semigroups, which reduce to diagonal matrices in the GNS symmetric setting. Using these directional matrices, we establish a corresponding gradient-flow structure. As a consequence, we obtain both a modified logarithmic Sobolev inequality and a Talagrand inequality for bimodule KMS symmetric quantum Markov semigroups.