Ergodic and Mixing Quantum Channels in Finite Dimensions

TL;DR

This paper characterizes ergodic and mixing properties of finite-dimensional quantum channels, explores their stability under randomization, and extends findings to continuous-time quantum evolutions via Lindblad generators.

Contribution

It provides a comprehensive structural analysis of quantum ergodic and mixing channels, including stability results and a characterization of ergodic Lindblad generators.

Findings

Ergodicity is stable under random mixtures with generic channels.

Conditions are identified under which ergodicity implies mixing.

Ergodic Lindblad generators are dense among all generators.

Abstract

The paper provides a systematic characterization of quantum ergodic and mixing channels in finite dimensions and a discussion of their structural properties. In particular, we discuss ergodicity in the general case where the fixed point of the channel is not a full-rank (faithful) density matrix. Notably, we show that ergodicity is stable under randomizations, namely that every random mixture of an ergodic channel with a generic channel is still ergodic. In addition, we prove several conditions under which ergodicity can be promoted to the stronger property of mixing. Finally, exploiting a suitable correspondence between quantum channels and generators of quantum dynamical semigroups, we extend our results to the realm of continuous-time quantum evolutions, providing a characterization of ergodic Lindblad generators and showing that they are dense in the set of all possible generators.