$C^*$-extreme points of positive operator valued measures and unital completely positive maps

TL;DR

This paper explores the structure of quantum convexity in POVMs, revealing that $C^*$-extreme points are spectral measures and establishing a Krein-Milman type theorem, with implications for unital completely positive maps.

Contribution

It characterizes $C^*$-extreme points of POVMs as spectral measures and proves a Krein-Milman type theorem for these measures, linking $C^*$-extremity to *-homomorphisms.

Findings

$C^*$-extreme points of POVMs are spectral measures

Atomic $C^*$-extreme points are spectral

Unital *-homomorphisms are $C^*$-extreme maps on commutative $C^*$-algebras

Abstract

We study the quantum ($C^*$) convexity structure of normalized positive operator valued measures (POVMs) on measurable spaces. In particular, it is seen that unlike extreme points under classical convexity, $C^*$-extreme points of normalized POVMs on countable spaces (in particular for finite sets) are always spectral measures (normalized projection valued measures). More generally it is shown that atomic $C^*$-extreme points are spectral. A Krein-Milman type theorem for POVMs has also been proved. As an application it is shown that a map on any commutative unital $C^*$-algebra with countable spectrum (in particular ${\mathbb C}^n$) is $C^*$-extreme in the set of unital completely positive maps if and only if it is a unital $*$-homomorphism.