The KMS and GNS Spectral Gap of Quantum Markov Semigroups

TL;DR

This paper proves a relation between decay rates of quantum Markov semigroups under different inner products, extending previous conjectures from Gaussian cases to general von Neumann algebra settings.

Contribution

It confirms a conjecture about the lower bound of decay rates with respect to the KMS inner product, applicable to a broad class of quantum Markov semigroups.

Findings

Decay rate with respect to KMS inner product is bounded below by GNS decay rate.

The result holds for quantum Markov semigroups on arbitrary von Neumann algebras.

The KMS inner product bound extends to a class of inner products from operator monotone functions.

Abstract

We establish a relation between the exponential decay rates of quantum Markov semigroups with respect to different inner products. More precisely, it was conjectured by Fagnola, Poletti, Sasso and Umanit\`a that for a Gaussian quantum Markov semigroup, the exponential decay rate with respect to the KMS inner product is bounded below by the exponential decay rate for the GNS inner product. We show that this is indeed the case and not limited to Gaussian quantum Markov semigroups, but holds for quantum Markov semigroups with a faithful normal invariant state on arbitrary von Neumann algebras. Additionally, the KMS inner product can be replaced by a whole class of inner products induced by operator monotone functions.