Uniqueness of the joint measurement and the structure of the set of compatible quantum measurements

TL;DR

This paper characterizes when a set of quantum measurements has a unique joint measurement, linking this property to extremal and boundary points of the compatible measurement set, and provides a complete structural description.

Contribution

It introduces a criterion for joint measurement uniqueness, characterizes extremal and boundary points of measurement sets, and clarifies their relation to compatibility in quantum measurements.

Findings

Extremal compatible tuples admit a unique joint measurement.

Tuples with a unique joint measurement lie on the boundary of the compatible set.

Counter-examples show the properties are not both necessary and sufficient.

Abstract

We address the problem of characterising the compatible tuples of measurements that admit a unique joint measurement. We derive a uniqueness criterion based on the method of perturbations and apply it to show that extremal points of the set of compatible tuples admit a unique joint measurement, while all tuples that admit a unique joint measurement lie in the boundary of such a set. We also provide counter-examples showing that none of these properties are both necessary and sufficient, thus completely describing the relation between joint measurement uniqueness and the structure of the compatible set. As a by-product of our investigations, we completely characterise the extremal and boundary points of the set of general tuples of measurements and of the subset of compatible tuples.