An Exponential Mixing Condition for Quantum Channels

TL;DR

This paper establishes a sufficient condition for exponential mixing in quantum channels using a quantum Markov-Dobrushin inequality, with applications to various channel classes including the qubit depolarizing channel.

Contribution

It introduces a new criterion based on the Markov-Dobrushin constant for determining exponential mixing in quantum channels.

Findings

Quantum Markov-Dobrushin constant > 0 implies exponential mixing.

Unistochastic channels are shown not to be mixing.

Results applied to the qubit depolarizing channel.

Abstract

Quantum channels, pivotal in information processing, describe transformations within quantum systems and enable secure communication and error correction. Ergodic and mixing properties elucidate their behavior. In this paper, we establish a sufficient condition for mixing based on a quantum Markov-Dobrushin inequality. We prove that if the Markov-Dobrushin constant of a quantum channel exceeds zero, it exhibits exponential mixing behavior. We explore limitations of some quantum channels, demonstrating that unistochastic channels are not mixing. Additionally, we analyze ergodicity of a class of mixed-unitary channels associated with finite groups of unitary operators. Finally, we apply our results to the qubit depolarizing channel.