On regions of mixed unitarity for semigroups of unital quantum channels

TL;DR

This paper proves that all semigroups of unital quantum channels become mixed unitary after finite time, introduces the mixed unitary index, and explores conditions under which certain Schur semigroups remain within specific subclasses.

Contribution

It establishes the eventual mixed unitarity of semigroups of unital quantum channels and introduces the mixed unitary index, a novel measure of mixing time.

Findings

Semigroups of unital quantum channels are eventually mixed unitary.

No universal upper bound exists for the mixed unitary index in fixed dimensions.

Schur semigroups of correlation matrices become mixtures of rank-one matrices.

Abstract

It is established that both discrete and continuous semigroups of unital quantum channels are eventually mixed unitary. This result is novel even for the subclass of Schur maps and stands in sharp contrast to the resolution of the asymptotic quantum Birkhoff conjecture by Haagerup and Musat, who demonstrated that tensor powers of some unital quantum channels maintain a persistent positive distance from the set of mixed unitary channels. Remarkably, our results show that this gap vanishes in finite time when considering ordinary powers within a semigroup. Building on this, we define the mixed unitary index of a unital quantum channel as the minimum time (or power) beyond which all subsequent maps become mixed unitary. We demonstrate that for any fixed dimension $d \geq 3$, there is no universal upper bound for this index. Furthermore, we observe that if a continuous semigroup is not mixed unitary at some $t > 0$, it remains non-mixed unitary for all $t$ sufficiently close to the origin. Finally, we investigate quantum dynamical semigroups where mixed unitarity is restricted to specific families, such as Weyl or diagonal unitaries. We show that Schur semigroups of correlation matrices eventually become mixtures of rank-one correlation matrices, and we characterize the generators of Schur semigroups that remain within this set for all $t \geq 0$.