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

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