On the incompatibility of noncommuting observables

Somshubhro Bandyopadhyay, Prabha Mandayam

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 nonorthogonality 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.

Giuseppe Nisticò

Stephan De Bièvre

We provide an in-depth study of the recently introduced notion of completely incompatible observables and its links to the support uncertainty and to the Kirkwood–Dirac nonclassicality of pure quantum states. The latter notion has recently been proved central to a number of issues in quantum information theory and quantum metrology. In this last context, it was shown that a quantum advantage requires the use of Kirkwood–Dirac nonclassical states. We establish sharp bounds of very general validity that imply that the support uncertainty is an efficient Kirkwood–Dirac nonclassicality witness. When adapted to completely incompatible observables that are close to mutually unbiased ones, this bound allows us to fully characterize the Kirkwood–Dirac classical pure states as the eigenvectors of the two observables. We show furthermore that complete incompatibility implies several weaker notions of incompatibility, among which features a strong form of noncommutativity.

J C Matthews, F. W. Sheard

In any course of quantum mechanics it is necessary to explain the behaviour of incompatible dynamical variables when measurements are made of them and also the noncommutation of their corresponding operators. At the outset this presents the student with considerable conceptual difficulties since these ideas are quite outside his everyday experience. The authors present a simple model, an electronic device designed to simulate some of the features of incompatible observables, which can play a useful role in aiding the student to digest the mathematics of noncommuting operators and their eigenfunctions as they appear in an introductory course.

Arno Böhm