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-HorneShimony-Holt Bell inequality, which has been proven by Wolf et al (2009 Phys. Rev. Lett. 103 230402). 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.

Measurement incompatibility and steering are necessary and sufficient for operational contextuality

“…Famously, by sharing entangled states and performing incompatible measurements that together steer the remote partner system, quantum theory can violate these inequalities and therefore provide an unequivocal demonstration of nonclassicality [3,4]. Surprisingly however, not all incompatible measurements, nor all steerable ensembles enable Bell inequality violations [5][6][7]. This motivates the question: is nonclassicality at the level of theoretical entities both necessary and sufficient for some form of operational nonclassicality?…”

Complete Resource Theory of Quantum Incompatibility as Quantum Programmability

“…For the demonstration of quantum nonlocality, it is not difficult to see that a Bell Inequality can be violated only if incompatible measurements are employed by each of the parties involved in the experiment [Fin82]. While for certain families of measurements the converse is true [WPGF09], only recently has it been found not to hold in general [QBHB16,BV18]. However, this asymmetry between measurement incompatibility and nonlocality vanishes when considering the more general task of quantum steering.…”

Incompatibility probability of random quantum measurements

“…The incompatibility probability for a pair of general qubit measurements is conjectured to be 1 4 by numerical simulation. As for the case of a pair of twooutcome measurements, the incompatibility is equivalent to Bell non-locality [12], this fact suggests that 25% of pairs of qubit measurements can lead to Bell non-locality. If we are restricted to the use of pairs of unbiased qubit measurements, the fraction increases to 60%.…”