The Spectral Gap of a Gaussian Quantum Markovian Generator

TL;DR

This paper explicitly computes the spectral gap of Gaussian quantum Markov semigroups, revealing how it depends on the non-commutative $L^2$ space and providing insights into convergence rates in open quantum systems.

Contribution

It provides an explicit formula for the spectral gap of Gaussian quantum Markov generators without assuming symmetry or detailed balance.

Findings

Spectral gap depends on the choice of non-commutative $L^2$ space.

Explicit examples show the spectral gap can be positive only with KMS multiplication.

The spectral gap is positive iff there are maximum independent noises.

Abstract

Gaussian quantum Markov semigroups are the natural non-commutative extension of classical Ornstein-Uhlenbeck semigroups. They arise in open quantum systems of bosons where canonical non-commuting random variables of positions and momenta come into play. If there exits a faithful invariant density we explicitly compute the optimal exponential convergence rate, namely the spectral gap of the generator, in non-commutative $L^2$ spaces determined by the invariant density showing that the exact value is the lowest eigenvalue of a certain matrix determined by the diffusion and drift matrices. The spectral gap turns out to depend on the non-commutative $L^2$ space considered, whether the one determined by the so-called GNS or KMS multiplication by the square root of the invariant density. In the first case, it is strictly positive if and only if there is the maximum number of linearly independent noises. While, we exhibit explicit examples in which it is strictly positive only with KMS multiplication. We do not assume any symmetry or quantum detailed balance condition with respect to the invariant density.