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

Bell non-locality represents one of the most striking departures of quantum mechanics from classical physics. It shows that correlations between space-like separated systems allowed by quantum mechanics are stronger than those present in any classical theory. In a recent work [Sci. Adv. 7, eabc3847 (2021)], a family of Bell functionals tailored to mutually unbiased bases (MUBs) is proposed. For these functionals, the maximal quantum violation is achieved if the two measurements performed by one of the parties are constructed out of MUBs of a fixed dimension. Here, we generalize this construction to an arbitrary incompatible pair of rank-one projective measurements. By constructing a new family of Bell functionals, we show that for any such pair there exists a Bell inequality that is maximally violated by this pair. Moreover, when investigating the robustness of these violations to noise, we demonstrate that the realization which is most robust to noise is not generated by MUBs.