The $\chi^2$-divergence and Mixing times of quantum Markov processes

TL;DR

This paper introduces quantum $oldsymbol{ ext{chi}^2}$-divergence, analyzes its properties, and uses it to derive bounds on the mixing times of quantum Markov processes, extending classical techniques to the quantum domain.

Contribution

It develops quantum $oldsymbol{ ext{chi}^2}$-divergence, explores its properties, and applies it to bound mixing times of quantum Markov processes, including spectral and geometric bounds.

Findings

Bounded trace-distance from steady state using quantum $ ext{chi}^2$-divergence.

Established spectral bounds for convergence rates of quantum Markov processes.

Analyzed contractive behavior of quantum $ ext{chi}^2$-divergence under completely positive maps.

Abstract

We introduce quantum versions of the $\chi^2$-divergence, provide a detailed analysis of their properties, and apply them in the investigation of mixing times of quantum Markov processes. An approach similar to the one presented in [1-3] for classical Markov chains is taken to bound the trace-distance from the steady state of a quantum processes. A strict spectral bound to the convergence rate can be given for time-discrete as well as for time-continuous quantum Markov processes. Furthermore the contractive behavior of the $\chi^2$-divergence under the action of a completely positive map is investigated and contrasted to the contraction of the trace norm. In this context we analyse different versions of quantum detailed balance and, finally, give a geometric conductance bound to the convergence rate for unital quantum Markov processes.