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

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

2007-02-21

arxiv.org/abs/quant-ph/0702199v3

Journal of Mathematical Physics 49, 012101 (2008)

Physics

Quantum Physics

18 pages, two figures, some corrections and additional references, published version

Scientific paper

10.1063/1.2826227

Contemporary versions of Bell's argument against local hidden variable (LHV) theories are based on the Clauser Horne Shimony and Holt (CHSH) 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 anti-correlation 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 and his associates (Invent. Math. 163 499 (2006)) we prove that there are two-particle Bell inequalities for traceless two-outcome observables whose violation increases to infinity as the dimension and number of measurements grow. Technically these result are possible because perfect correlations (or anti-correlations) allow us to transport the indices of the inequality from the edges of a bipartite graph to those of the complete graph. Finally, it is shown how to apply these results to mixed Werner states, provided that the noise does not exceed 20%.

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