# Quantifying the incompatibility of quantum measurements relative to a basis

## TL;DR

This paper introduces a new way to quantify the incompatibility of quantum measurements relative to a basis, linking it to quantum resource theories, majorization, and uncertainty relations.

## Contribution

It defines a novel measure of measurement incompatibility based on probabilistic post-processing and extends it to general POVMs, establishing connections with coherence and uncertainty.

## Key findings

- Established a quantitative link between incompatibility, coherence, and uncertainty.
- Developed families of monotones for measuring incompatibility.
- Generalized the incompatibility measure to arbitrary POVMs.

## Abstract

Motivated by quantum resource theories, we introduce a notion of incompatibility for quantum measurements relative to a reference basis. The notion arises by considering states diagonal in that basis and investigating whether probability distributions associated with different quantum measurements can be converted into one another by probabilistic post-processing. The induced preorder over quantum measurements is directly related to multivariate majorization and gives rise to families of monotones, i.e., scalar quantifiers that preserve the ordering. For the case of orthogonal measurement we establish a quantitative connection between incompatibility, quantum coherence and entropic uncertainty relations. We generalize the construction to arbitrary POVM measurements and report complete families of monotones.