Extreme commutative quantum observables are sharp

TL;DR

This paper proves that the only extreme commutative quantum observables are sharp PVMs, highlighting the necessity of non-commutativity for more general and optimal quantum measurements.

Contribution

It establishes a precise characterization of extreme points among commutative POVMs, linking extremality to sharp PVMs and emphasizing non-commutativity's role.

Findings

Extreme commutative POVMs are exactly sharp PVMs

Non-commutativity is necessary for more general quantum measurements

Highlights limitations of PVMs and the role of non-commutativity

Abstract

It is well known that, in the description of quantum observables, positive operator valued measures (POVMs) generalize projection valued measures (PVMs) and they also turn out be more optimal in many tasks. We show that a commutative POVM is an extreme point in the convex set of all POVMs if and only if it is a PVM. This results implies that non-commutativity is a necessary ingredient to overcome the limitations of PVMs.