Approximate joint measurements of qubit observables

TL;DR

This paper investigates the conditions under which arbitrary pairs of qubit observables can be jointly measured, introducing optimal approximate measurements that respect Heisenberg-type uncertainty relations.

Contribution

It extends the understanding of joint measurability for qubit observables beyond covariant cases, proposing optimal approximate schemes and analyzing their fundamental limits.

Findings

Optimal approximate joint measurements are covariant.

Marginal observables are not coarse-grainings of original observables.

Approximation quality is limited by Heisenberg-type uncertainty relations.

Abstract

Joint measurements of qubit observables have recently been studied in conjunction with quantum information processing tasks such as cloning. Considerations of such joint measurements have until now been restricted to a certain class of observables that can be characterized by a form of covariance. Here we investigate conditions for the joint measurability of arbitrary pairs of qubit observables. For pairs of noncommuting sharp qubit observables, a notion of approximate joint measurement is introduced. Optimal approximate joint measurements are shown to lie in the class of covariant joint measurements. The marginal observables found to be optimal approximators are generally not among the coarse-grainings of the observables to be approximated. This yields scope for the improvement of existing joint measurement schemes. Both the quality of the approximations and the intrinsic unsharpness of the approximators are shown to be subject to Heisenberg-type uncertainty relations.