Operational measure of incompatibility of noncommuting observables

TL;DR

This paper introduces an operational measure of incompatibility for noncommuting observables that does not rely on uncertainty relations, capturing their non-orthogonality and providing bounds and exact values for specific cases.

Contribution

It proposes a novel, uncertainty-independent measure of incompatibility based on eigenstate non-orthogonality, with proven properties and exact calculations for mutually unbiased observables.

Findings

Measure is zero for commuting observables

Measure is maximal for mutually unbiased observables

Tight upper bounds derived for any set of noncommuting observables

Abstract

Uncertainty relations are often considered to be a measure of incompatibility of noncommuting observables. However, such a consideration is not valid in general, motivating the need for an alternate measure that applies to any set of noncommuting observables. We present an operational approach to quantifying incompatibility without invoking uncertainty relations. Our measure aims to capture the incompatibility of noncommuting observables as manifest in the non-orthogonality of their eigenstates. We prove that this measure has all the desired properties: it is zero when the observables commute, strictly greater than zero when they do not, and is maximum when they are mutually unbiased. We also obtain tight upper bounds on this measure for any N noncommuting observables and compute it exactly when the observables are mutually unbiased.