# Necessary and sufficient condition for joint measurability

## TL;DR

This paper introduces a new operator-valued measure called W-measure to characterize joint measurability of quantum measurements, providing necessary and sufficient conditions and practical tests for joint measurability.

## Contribution

It establishes a novel W-measure framework that precisely determines joint measurability and offers operational criteria and negativity-based indicators for practical assessment.

## Key findings

- W-measure is a POVM if and only if its marginals are jointly measurable.
- Negatives of W-measure serve as indicators for non-joint measurability.
- Derived criteria for dichotomic and trichotomic variables.

## Abstract

In order to analyze joint measurability of given measurements, we introduce a Hermitian operator-valued measure, called $W$-measure, such that it has marginals of positive operator-valued measures (POVMs). We prove that ${W}$-measure is a POVM {\em if and only if} its marginal POVMs are jointly measurable. The proof suggests to employ the negatives of ${W}$-measure as an indicator for non-joint measurability. By applying triangle inequalities to the negativity, we derive joint measurability criteria for dichotomic and trichotomic variables. Also, we propose an operational test for the joint measurability in sequential measurement scenario.