On coexistence and joint measurability of rank-1 quantum observables

TL;DR

This paper characterizes when a rank-1 quantum observable can be measured together with another, showing that joint measurability is equivalent to being a smearing of the same observable, especially for extreme, rank-1, discrete cases.

Contribution

It provides a precise condition for joint measurability of rank-1 observables, linking it to the concept of smearing and extremality in quantum measurements.

Findings

M and M' are jointly measurable iff M' is a smearing of M.

For extreme, rank-1, discrete observables, coexistence and joint measurability are equivalent.

Characterization of joint measurability in terms of smearing for rank-1 observables.

Abstract

We show that a rank-1 quantum observable (POVM) M is jointly measurable with a quantum observable M' exactly when M' is a smearing of M. If M is extreme, rank-1 and discrete then M and M' are coexistent if and only if they are jointly measurable.