Device-independent certification of Majorana fermions and a new self-testing equivalence

Bell inequalities provide a fundamental tool for probing nonlocal correlations, yet their quantum bound, that is, the maximal value attainable by quantum strategies, is rarely accessible analytically. In this work, we introduce a general construction of Bell inequalities for which this bound can be computed exactly. The construction is based on a class of rectangular matrices with pairwise orthogonal rows and normalized columns, which we call row-orthogonal, column-normalized (ROCN) matrices. Our framework generalizes both the Clauser–Horne–Shimony–Holt inequality and Gisin’s elegant Bell inequality, yielding Bell expressions maximally violated by arbitrary families of pairwise anticommuting Clifford observables together with the corresponding maximally entangled state. Under suitable assumptions, these inequalities also enable the device-independent certification of Majorana fermions, understood as multiqubit realizations of Clifford algebra generators. Importantly, we identify an additional equivalence that must be incorporated into self-testing, beyond the standard equivalences generated by local isometries, the addition of auxiliary degrees of freedom, and complex conjugation. This equivalence arises from applying partial transposition jointly to the shared state and the measurements, which, whenever it is well-defined, preserves all observed correlations.