Unital Quantum Channels - Convex Structure and Revivals of Birkhoff's Theorem

TL;DR

This paper explores the structure of doubly-stochastic quantum channels, revealing conditions under which channels outside the convex hull of unitaries can become correctable through multiple copies or depolarizing channels, thus reviving Birkhoff's theorem.

Contribution

It provides a detailed structural analysis and computable criteria for distinguishing between doubly-stochastic channels and mixtures of unitaries, including a complete characterization for O(d)-covariant channels.

Findings

Channels outside the convex hull of unitaries can become correctable with multiple copies.

Complete characterization of O(d)-covariant channels.

Instances where channels return to the convex hull after certain operations.

Abstract

The set of doubly-stochastic quantum channels and its subset of mixtures of unitaries are investigated. We provide a detailed analysis of their structure together with computable criteria for the separation of the two sets. When applied to O(d)-covariant channels this leads to a complete characterization and reveals a remarkable feature: instances of channels which are not in the convex hull of unitaries can return to it when either taking finitely many copies of them or supplementing with a completely depolarizing channel. In these scenarios this implies that a channel whose noise initially resists any environment-assisted attempt of correction can become perfectly correctable.