Optimal covariant quantum measurements

TL;DR

This paper investigates the properties of symmetric covariant quantum measurements, focusing on their optimality features such as extremality, informational completeness, and rank-1 characteristics, in both finite and infinite-dimensional settings.

Contribution

It provides a comprehensive characterization of covariant quantum measurements, including necessary and sufficient conditions for extremality, and explores their structure in finite and infinite dimensions.

Findings

Characterized extremality conditions for finite-outcome covariant measurements.

Showed that covariant observables include representatives from both optimality classes.

Demonstrated that slight deviations from rank-1 PVMs can produce extremal informationally complete measurements.

Abstract

We discuss symmetric quantum measurements and the associated covariant observables modelled, respectively, as instruments and positive-operator-valued measures. The emphasis of this work are the optimality properties of the measurements, namely, extremality, informational completeness, and the rank-1 property which contrast the complementary class of (rank-1) projection-valued measures. The first half of this work concentrates solely on finite-outcome measurements symmetric w.r.t. finite groups where we derive exhaustive characterizations for the pointwise Kraus-operators of covariant instruments and necessary and sufficient extremality conditions using these Kraus-operators. We motivate the use of covariance methods by showing that observables covariant with respect to symmetric groups contain a family of representatives from both of the complementary optimality classes of observables and show that even a slight deviation from a rank-1 projection-valued measure can yield an extreme informationally complete rank-1 observable. The latter half of this work derives similar results for continuous measurements in (possibly) infinite dimensions. As an example we study covariant phase space instruments, their structure, and extremality properties.