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Bell non-locality represents one of the most striking departures of quantummechanics from classical physics. It shows that correlations between space-likeseparated systems allowed by quantum mechanics are stronger than those presentin any classical theory. In a recent work [Sci. Adv. 7, eabc3847 (2021)], afamily of Bell functionals tailored to mutually unbiased bases (MUBs) isproposed. For these functionals, the maximal quantum violation is achieved ifthe two measurements performed by one of the parties are constructed out ofMUBs of a fixed dimension. Here, we generalize this construction to anarbitrary incompatible pair of rank-one projective measurements. Byconstructing a new family of Bell functionals, we show that for any such pairthere exists a Bell inequality that is maximally violated by this pair.Moreover, when investigating the robustness of these violations to noise, wedemonstrate that the realization which is most robust to noise is not generatedby MUBs.

Optimality of any pair of incompatible rank-one projective measurements for some non-trivial Bell inequality