Exponential Relative Entropy Decay Along Quantum Markov Semigroups

TL;DR

This paper proves that exponential decay of relative entropy in quantum Markov semigroups is equivalent to a modified logarithmic Sobolev inequality, extending the theory to infinite-dimensional algebras and GNS-symmetric cases.

Contribution

It establishes the equivalence between entropy decay and Sobolev inequalities for quantum Markov semigroups on general von Neumann algebras, including infinite-dimensional cases.

Findings

Proves the equivalence between exponential entropy decay and modified logarithmic Sobolev inequality.

Extends intertwining criterion to GNS-symmetric quantum Markov semigroups.

Applies to general von Neumann algebras, including infinite-dimensional cases.

Abstract

We establish the equivalence between exponential decay of the relative entropy along a quantum Markov semigroup and the modified logarithmic Sobolev inequality for general von Neumann algebras. We also extend an intertwining criterion for the modified logarithmic Sobolev inequality to GNS-symmetric quantum Markov semigroups on infinite-dimensional von Neumann algebras.