Quantifying incompatibility of quantum measurements through non-commutativity

TL;DR

This paper introduces a family of measures based on non-commutativity to quantify quantum measurement incompatibility, linking it to operational tasks and analyzing their properties and behavior.

Contribution

It defines and studies new incompatibility measures based on non-commutativity, connecting them to existing measures and operational scenarios.

Findings

Measures satisfy natural information-processing requirements

Characterization of pairs with highest incompatibility in fixed dimensions

Relation of measures to robustness and operational tasks

Abstract

The existence of incompatible measurements, i.e. measurements which cannot be performed simultaneously on a single copy of a quantum state, constitutes an important distinction between quantum mechanics and classical theories. While incompatibility might at first glance seem like an obstacle, it turns to be a necessary ingredient to achieve the so-called quantum advantage in various operational tasks like random access codes or key distribution. To improve our understanding of how to quantify incompatibility of quantum measurements, we define and explore a family of incompatibility measures based on non-commutativity. We investigate some basic properties of these measures, we show that they satisfy some natural information-processing requirements and we fully characterize the pairs which achieve the highest incompatibility (in a fixed dimension). We also consider the behavior of our measures under different types of compositions. Finally, to link our new measures to existing results, we relate them to a robustness-based incompatibility measure and two operational scenarios: random access codes and entropic uncertainty relations.