Noninvertibility as a requirement for creating a semigroup under convex combinations of channels

TL;DR

This paper investigates the conditions for forming semigroups through convex combinations of quantum channels, revealing that most input channels must be noninvertible to produce a semigroup, especially in the case of Pauli channels.

Contribution

It establishes that convex combinations of channels forming a semigroup require most input channels to be noninvertible, challenging previous assumptions about invertibility in channel composition.

Findings

Convex combinations of channels rarely form a semigroup unless most are noninvertible.

Mixing only semigroups cannot generate a new semigroup.

Noninvertibility is a necessary condition for semigroup formation via convex combinations.

Abstract

We study the conditions under which a semigroup is obtained upon convex combinations of channels. In particular, we study the set of Pauli and generalized Pauli channels. We find that mixing only semigroups can never produce a semigroup. Counter-intuitively, we find that for a convex combination to yield a semigroup, most of the input channels have to be noninvertible.