Accessible maps in a group of classical or quantum channels

TL;DR

This paper explores the structure and properties of accessible channels within classical and quantum groups, revealing geometric characteristics and the influence of group structure on accessibility.

Contribution

It introduces a geometric analysis of accessible channels in group-structured channels, showing non-convexity and star-shaped properties independent of dimension or representation.

Findings

Accessible channels are determined by probability vectors of convex combinations.

The set of accessible channels is non-convex but star-shaped.

Accessible channels occupy a positive volume in the convex hull of the group.

Abstract

We study the problem of accessibility in a set of classical and quantum channels admitting a group structure. Group properties of the set of channels, and the structure of the closure of the analyzed group $G$ plays a pivotal role in this regard. The set of all convex combinations of the group elements contains a subset of channels that are accessible by a dynamical semigroup. We demonstrate that accessible channels are determined by probability vectors of weights of a convex combination of the group elements, which depend neither on the dimension of the space on which the channels act, nor on the specific representation of the group. Investigating geometric properties of the set $\mathcal{A}$ of accessible maps we show that this set is non-convex, but it enjoys the star-shape property with respect to the uniform mixture of all elements of the group. We demonstrate that the set $\mathcal{A}$ covers a positive volume in the polytope of all convex combinations of the elements of the group.