Extreme Covariant Quantum Observables in the Case of an Abelian Symmetry Group and a Transitive Value Space

TL;DR

This paper characterizes the extreme covariant quantum observables for Abelian symmetry groups with transitive value spaces, providing a detailed mathematical framework and applying it to position and time measurements.

Contribution

It offers a complete characterization of extreme covariant POVMs and extends the analysis to all quantum observables under Abelian symmetries.

Findings

Characterization of extreme covariant observables

Identification of extreme points in the set of all quantum observables

Applications to position, difference, and time observables

Abstract

We represent quantum observables as POVMs (normalized positive operator valued measures) and consider convex sets of observables which are covariant with respect to a unitary representation of a locally compact Abelian symmetry group $G$. The value space of such observables is a transitive $G$-space. We characterize the extreme points of covariant observables and also determine the covariant extreme points of the larger set of all quantum observables. The results are applied to position, position difference and time observables.