Measurement Uncertainty for Finite Quantum Observables

TL;DR

This paper develops semidefinite programming methods to compute optimal measurement uncertainty bounds for finite quantum observables, using various error metrics based on optimal transport theory and physical testing scenarios.

Contribution

It introduces a general framework for calculating measurement uncertainty bounds in finite-dimensional quantum systems using semidefinite programming and optimal transport-based error measures.

Findings

The methods can compute tight bounds for joint measurement errors.

Different error metrics are applicable depending on the testing scenario.

The approach is illustrated with characteristic quantum examples.

Abstract

Measurement uncertainty relations are lower bounds on the errors of any approximate joint measurement of two or more quantum observables. The aim of this paper is to provide methods to compute optimal bounds of this type. The basic method is semidefinite programming, which we apply to arbitrary finite collections of projective observables on a finite dimensional Hilbert space. The quantification of errors is based on an arbitrary cost function, which assigns a penalty to getting result $x$ rather than y, for any pair (x,y). This induces a notion of optimal transport cost for a pair of probability distributions, and we include an appendix with a short summary of optimal transport theory as needed in our context. There are then different ways to form an overall figure of merit from the comparison of distributions. We consider three, which are related to different physical testing scenarios. The most thorough test compares the transport distances between the marginals of a joint measurement and the reference observables for every input state. Less demanding is a test just on the states for which a "true value" is known in the sense that the reference observable yields a definite outcome. Finally, we can measure a deviation as a single expectation value by comparing the two observables on the two parts of a maximally entangled state. All three error quantities have the property that they vanish if and only if the tested observable is equal to the reference. The theory is illustrated with some characteristic examples.