Simplifying measurement uncertainty with quantum symmetries

TL;DR

This paper introduces a framework leveraging quantum symmetries to simplify the calculation of measurement uncertainty regions, demonstrating its effectiveness on Pauli and phase space observables.

Contribution

It presents a systematic covariance-based approach showing that optimal compatible approximations are covariant, simplifying uncertainty region calculations.

Findings

Optimal approximations of covariant observables are themselves covariant.

The covariantisation map exists and can be characterized.

Measurement uncertainty regions are derived for Pauli and phase space observables.

Abstract

Determining the measurement uncertainty region is a difficult problem for generic sets of observables. For this reason the literature on exact measurement uncertainty regions is focused on symmetric sets of observables, where the symmetries are used to simplify the calculation. We provide a framework to systematically exploit available symmetries, formulated in terms of covariance, to simplify problems of measurement uncertainty. Our key result is that for a wide range figures of merit the optimal compatible approximations of covariant target observables are themselves covariant. This substantially simplifies the problem of determining measurement uncertainty regions for cases where it applies, since the space of covariant observables is typically much smaller than that of all observables. An intermediate result, which may be applicable more broadly, is the existence and characterisation of a covariantisation map, mapping observables to covariant observables. Our formulation is applicable to finite outcome observables on separable Hilbert spaces. We conjecture that the restriction of finite outcomes may be lifted, and explore some of the features a generalisation must have. We demonstrate the theorem by deriving measurement uncertainty regions for three mutually orthogonal Pauli observables, and for phase space observables in arbitrary finite dimensions.