# Incompatibility robustness of quantum measurements: a unified framework

## TL;DR

This paper introduces a comprehensive framework for quantifying quantum measurement incompatibility, analyzes existing measures, and identifies the most incompatible measurement pairs, revealing discrepancies among different measures.

## Contribution

It unifies various incompatibility measures under a robustness framework and evaluates their properties and implications for identifying highly incompatible measurements.

## Key findings

- Some measures do not meet proposed basic criteria.
- Different measures yield different most incompatible pairs.
- Mutually unbiased bases are among the most incompatible in all dimensions.

## Abstract

In quantum mechanics performing a measurement is an invasive process which generally disturbs the system. Due to this phenomenon, there exist incompatible quantum measurements, i.e., measurements that cannot be simultaneously performed on a single copy of the system. It is then natural to ask what the most incompatible quantum measurements are. To answer this question, several measures have been proposed to quantify how incompatible a set of measurements is, however their properties are not well-understood. In this work, we develop a general framework that encompasses all the commonly used measures of incompatibility based on robustness to noise. Moreover, we propose several conditions that a measure of incompatibility should satisfy, and investigate whether the existing measures comply with them. We find that some of the widely used measures do not fulfil these basic requirements. We also show that when looking for the most incompatible pairs of measurements, we obtain different answers depending on the exact measure. For one of the measures, we analytically prove that projective measurements onto two mutually unbiased bases are among the most incompatible pairs in every dimension. However, for some of the remaining measures we find that some peculiar measurements turn out to be even more incompatible.