# Measurement incompatibility does not give rise to Bell violation in general

## TL;DR

This paper demonstrates that, unlike the case with two measurements, incompatibility of measurements does not necessarily lead to Bell inequality violations when more than two measurements are involved, revealing a nuanced relationship.

## Contribution

It shows that for three measurements, incompatibility does not imply Bell violation, and provides examples of compatible measurements that still produce Bell violations.

## Key findings

- Incompatibility does not always lead to Bell violation for three measurements.
- A set of three incompatible measurements cannot violate Bell inequalities.
- Some compatible measurements can still produce Bell violations.

## Abstract

In the case of a pair of two-outcome measurements incompatibility is equivalent to Bell nonlocality. Indeed, any pair of incompatible two-outcome measurements can violate the Clauser-Horne-Shimony-Holt Bell inequality, which has been proven by Wolf et al. [Phys. Rev. Lett. 103, 230402 (2009)]. In the case of more than two measurements the equivalence between incompatibility and Bell nonlocality is still an open problem, though partial results have recently been obtained. Here we show that the equivalence breaks for a special choice of three measurements. In particular, we present a set of three incompatible two-outcome measurements, such that if Alice measures this set, independent of the set of measurements chosen by Bob and the state shared by them, the resulting statistics cannot violate any Bell inequality. On the other hand, complementing the above result, we exhibit a set of $N$ measurements for any $N>2$ that is $(N-1)$-wise compatible, nevertheless it gives rise to Bell violation.