On regions of mixed unitarity for semigroups of unital quantum channels
B V Rajarama Bhat, Repana Devendra
TL;DR
This work addresses when and how unital quantum channels become mixed unitary under both discrete powers and continuous time evolution. It introduces the mixed unitary index and proves there is no universal bound for it in fixed dimension $d\ge 3$, while also establishing an eventual MU behavior for both discrete and continuous semigroups via a decomposition into a peripheral automorphism plus a vanishing dissipative part. A local criterion near the origin is provided, linking MU-ability to a cone condition on the GKLS generator, and the paper delivers a complete Weyl-unitary structure theorem, including extensions to $G$-mixed unitary channels for closed subgroups $G$. Collectively, the results illuminate the asymptotic and near-origin dynamics of unital quantum channels and offer precise algebraic characterizations of when such channels arise as convex combinations of Weyl unitary maps.
Abstract
It is established that both discrete and continuous semigroups of unital quantum channels are eventually mixed unitary. However, after introducing the mixed unitary index of a unital quantum channel as the least time beyond which all subsequent powers of the channel are mixed unitary, we demonstrate that for any fixed finite dimension $d\geq 3$, there exists no universal upper bound for this index. Furthermore, for a continuous semigroup that is not mixed unitary for some $t>0$, we prove it remains non-mixed unitary for all times $t>0$, sufficiently close to the origin. Finally, a necessary and sufficient condition is derived for a quantum dynamical semigroup to be a convex combination of maps implemented solely by Weyl unitaries.