Perturbation Bounds for Quantum Markov Processes and their Fixed Points

TL;DR

This paper studies how small changes in quantum Markov processes affect their long-term behavior and fixed points, providing bounds and relations between stability, eigenvalues, and mixing times.

Contribution

It introduces a condition number for quantum channels' fixed points and relates process stability to eigenvalues and mixing times.

Findings

Bounds on the condition number in terms of eigenvalues

Linear relation between mixing time and fixed point sensitivity

Stability analysis of quantum Markov processes at finite times

Abstract

We investigate the stability of quantum Markov processes with respect to perturbations of their transition maps. In the first part, we introduce a condition number that measures the sensitivity of fixed points of a quantum channel to perturbations. We establish upper and lower bounds on this condition number in terms of subdominant eigenvalues of the transition map. In the second part, we consider quantum Markov processes that converge to a unique stationary state and we analyze the stability of the evolution at finite times. In this way we obtain a linear relation between the mixing time of a quantum Markov process and the sensitivity of its fixed point with respect to perturbations of the transition map.