Exploring boundaries of quantum convex structures: special role of unitary processes

TL;DR

This paper investigates the concept of boundariness in quantum convex sets, revealing its operational meaning, the special role of unitary channels in decompositions, and its behavior under system composition.

Contribution

It establishes the operational interpretation of boundariness, shows that optimal decompositions involve unitary channels, and analyzes its properties under system composition.

Findings

Boundariness coincides with the distinguishability of elements.

Optimal decompositions of channels include unitary channels.

Boundariness is sub-multiplicative under system composition.

Abstract

We address the question of finding the most effective convex decompositions into boundary elements (so-called boundariness) for sets of quantum states, observables and channels. First we show that in general convex sets the boundariness essentially coincides with the question of the most distinguishable element, thus, providing an operational meaning for this concept. Unexpectedly, we discovered that for any interior point of the set of channels the optimal decomposition necessarily contains a unitary channel. In other words, for any given channel the best distinguishable one is some unitary channel. Further, we prove that boundariness is sub-multiplicative under composition of systems and explicitly evaluate its maximal value that is attained only for the most mixed elements of the considered convex structures.