Measurement incompatibility does not give rise to Bell violation in general

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.