New Bell inequalities for the singlet state: Going beyond the Grothendieck bound

Contemporary versions of Bell’s argument [Physics (Long Island City, N.Y.) 1, 195 (1964)] against local hidden variable (LHV) theories are based on the Clauser-Horne-Shimony-Holt (CHSH) [Phys. Rev. Lett. 23, 880 (1969)] inequality and various attempts to generalize it. The amount of violation of these inequalities cannot exceed the bound set by the Grothendieck constants. However, if we go back to the original derivation by Bell and use the perfect anticorrelation embodied in the singlet spin state, we can go beyond these bounds. In this paper, we derive two-particle Bell inequalities for traceless two-outcome observables, whose violation in the singlet spin state go beyond the Grothendieck constants both for the two and three dimensional cases. Moreover, creating a higher dimensional analog of perfect correlations and applying a recent result of Alon et al. [Invent. Math. 163, 499 (2006)], we prove that there are two-particle Bell inequalities for traceless two-outcome observables whose violation increas...

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