Spectral convergence bounds for classical and quantum Markov processes

TL;DR

This paper develops a new spectral bound framework for classical and quantum Markov processes, improving convergence estimates without requiring common assumptions like detailed balance or irreducibility.

Contribution

It introduces a novel approach using Wiener algebra functional calculus to derive sharper spectral convergence bounds for Markov chains.

Findings

Spectral bounds are improved over existing estimates.

The method applies to both classical and quantum Markov chains.

No need for detailed balance or irreducibility assumptions.

Abstract

We introduce a new framework that yields spectral bounds on norms of functions of transition maps for finite, homogeneous Markov chains. The techniques employed work for bounded semigroups, in particular for classical as well as for quantum Markov chains and they do not require additional assumptions like detailed balance, irreducibility or aperiodicity. We use the method in order to derive convergence bounds that improve significantly upon known spectral bounds. The core technical observation is that power-boundedness of transition maps of Markov chains enables a Wiener algebra functional calculus in order to upper bound any norm of any holomorphic function of the transition map. Finally, we discuss how general detailed balance conditions for quantum Markov processes lead to spectral convergence bounds.