How sharp are PV measures?

TL;DR

This paper investigates the properties of sharp positive-operator valued measures (PV measures) and their relationship to smearing via Markov kernels, establishing equivalences among several conditions for sharp observables.

Contribution

It demonstrates the equivalence of conditions related to sharp observables and their smearing, clarifying the structure of PV measures in quantum measurement theory.

Findings

Range of P contained in range of M

P is a function of M

P is a smearing of M

Abstract

Properties of sharp observables (normalized PV measures) in relation to smearing by a Markov kernel are studied. It is shown that for a sharp observable $P$ defined on a standard Borel space, and an arbitrary observable $M$, the following properties are equivalent: (a) the range of $P$ is contained in the range of $M$; (b) $P$ is a function of $M$; (c) $P$ is a smearing of $M$.