Measurement incompatibility and quantum non-locality are two key features of
quantum theory. Violations of Bell inequalities require quantum entanglement
and incompatibility of the measurements used by the two parties involved in the
protocol. We analyze the converse question: for which Bell inequalities is the
incompatibility of measurements enough to ensure a quantum violation? We relate
the two questions by comparing two tensor norms on the space of dichotomic
quantum measurements: one characterizing measurement compatibility and the
second one characterizing violations of a given Bell inequality. We provide
sufficient conditions for the equivalence of the two notions in terms of the
matrix describing the correlation Bell inequality. We show that the CHSH
inequality and its variants are the only ones satisfying it.
