Tsirelson's bound and supersymmetric entangled states

A superqubit, belonging to a (2|1)-dimensional super-Hilbert space, constitutes the minimal supersymmetric extension of the conventional qubit. In order to see whether superqubits are more non-local than ordinary qubits, we construct a class of two-superqubit entangled states as a non-local resource in the CHSH game. Since super Hilbert space amplitudes are Grassmann numbers, the result depends on how we extract real probabilities and we examine three choices of map: (1) DeWitt (2) Trigonometric and (3) Modified Rogers. In cases (1) and (2), the winning probability reaches the Tsirelson bound pwin =cos²π/8≃0.8536 of standard quantum mechanics. Case (3) crosses Tsirelson's bound with pwin ≃0.9265. Although all states used in the game involve probabilities lying between 0 and 1, case (3) permits other changes of basis inducing negative transition probabilities.