On the convex structure of process POVMs

TL;DR

This paper explores the structure of process POVMs in quantum measurements, introducing A-convexity to better understand extremal points and their relation to measurement implementation.

Contribution

It introduces the A-convex structure of POVMs, generalizing convexity concepts, and characterizes extremal points and faces of process POVMs through this new framework.

Findings

A-convex structure generalizes convex and C*-convex structures.

Extremal points of process POVMs relate to A-extremal POVMs.

Characterization of A-extremal and A-pure POVMs provided.

Abstract

Measurements on quantum channels are described by so-called process operator valued measures, or process POVMs. We study implementing schemes of extremal process POVMs. As it turns out, the corresponding measurement must satisfy certain extremality property, which is stronger that the usual extremality given by the convex structure. This property motivates the introduction and investigation of the A-convex structure of POVMs, which generalizes both the usual convex and C*-convex structure. We show that extremal points and faces of the set of process POVMs are closely related to A-extremal points and A-faces of POVMs, for a certain subalgebra A. We give a characterization of A-extremal and A-pure POVMs in the Appendix.