Post-processing minimal joint observables

TL;DR

This paper investigates the concept of minimal joint observables in quantum mechanics, providing characterizations and algorithms to identify minimal joint observables among compatible sets, with applications to qubit observables.

Contribution

It introduces criteria and algorithms for determining minimal joint observables in quantum systems, extending understanding of their structure and properties.

Findings

Any joint observable is lower bounded by a minimal one in the post-processing order.

Provided finite-step algorithms to check minimality of joint observables.

Applied results to non-commuting dichotomic qubit observables.

Abstract

A finite set of quantum observables (positive operator valued measures) is called compatible if these observables are marginals of a some observable, called a joint observable of them. For a given set of compatible observables, their joint observable is in general not unique and it is desirable to take a minimal joint observable in the post-processing order since a less informative observable disturbs less the system. We address the question of the minimality of finite-outcome joint observables and prove that any joint observable is lower bounded by a minimal joint observable in the post-processing order. We also give characterizations of the minimality of a joint observable that can be checked by finite-step algorithms and apply them to the case of non-commuting dichotomic qubit observables.