Distance-based resource quantification for sets of quantum measurements

TL;DR

This paper introduces a novel distance-based framework for quantifying resources in sets of quantum measurements, providing analytical bounds and a hierarchy of measurement resources, with implications for Bell experiments.

Contribution

It develops a new distance-based approach to quantify measurement resources, establishing a hierarchy and tight bounds, which was not previously done in this context.

Findings

Defined distance functions between sets of quantum measurements.

Established a hierarchy of measurement resources based on the diamond norm.

Derived tight analytical bounds on measurement incompatibility.

Abstract

The advantage that quantum systems provide for certain quantum information processing tasks over their classical counterparts can be quantified within the general framework of resource theories. Certain distance functions between quantum states have successfully been used to quantify resources like entanglement and coherence. Perhaps surprisingly, such a distance-based approach has not been adopted to study resources of quantum measurements, where other geometric quantifiers are used instead. Here, we define distance functions between sets of quantum measurements and show that they naturally induce resource monotones for convex resource theories of measurements. By focusing on a distance based on the diamond norm, we establish a hierarchy of measurement resources and derive analytical bounds on the incompatibility of any set of measurements. We show that these bounds are tight for certain projective measurements based on mutually unbiased bases and identify scenarios where different measurement resources attain the same value when quantified by our resource monotone. Our results provide a general framework to compare distance-based resources for sets of measurements and allow us to obtain limitations on Bell-type experiments.