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New Bell inequalities for the singlet state: going beyond the Grothendieck bound. (English) Zbl 1153.81419

Summary: 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 N. Alon and his associates [Invent. Math. 163, No. 3, 499–522 (2006; Zbl 1082.05051)] 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%.

MSC:

81P15 Quantum measurement theory, state operations, state preparations
81P68 Quantum computation

Citations:

Zbl 1082.05051
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References:

[1] Bell J. S., Physics (Long Island City, N.Y.) 1 pp 195– (1964)
[2] DOI: 10.1103/PhysRevLett.23.880 · Zbl 1371.81014 · doi:10.1103/PhysRevLett.23.880
[3] DOI: 10.1016/0375-9601(92)90949-M · doi:10.1016/0375-9601(92)90949-M
[4] Werner R. F., Quantum Inf. Comput. 1 pp 1– (2001)
[5] DOI: 10.1119/1.1976526 · doi:10.1119/1.1976526
[6] DOI: 10.1103/PhysRevA.73.062105 · doi:10.1103/PhysRevA.73.062105
[7] DOI: 10.1007/BF01663472 · Zbl 0617.46066 · doi:10.1007/BF01663472
[8] DOI: 10.1007/s00222-005-0465-9 · Zbl 1082.05051 · doi:10.1007/s00222-005-0465-9
[9] DOI: 10.1103/PhysRevLett.65.1838 · Zbl 0971.81507 · doi:10.1103/PhysRevLett.65.1838
[10] DOI: 10.1103/PhysRevA.40.4277 · Zbl 1371.81145 · doi:10.1103/PhysRevA.40.4277
[11] Grothendieck A., Bol. Soc. Mat. São Paulo 8 pp 1– (1953)
[12] DOI: 10.1137/S0895480191219350 · Zbl 0792.05030 · doi:10.1137/S0895480191219350
[13] DOI: 10.1007/BF01594946 · Zbl 0741.90054 · doi:10.1007/BF01594946
[14] DOI: 10.1103/PhysRevA.64.032112 · doi:10.1103/PhysRevA.64.032112
[15] DOI: 10.1016/S0195-6698(13)80049-2 · Zbl 0742.05077 · doi:10.1016/S0195-6698(13)80049-2
[16] DOI: 10.1007/978-3-642-04295-9 · Zbl 1210.52001 · doi:10.1007/978-3-642-04295-9
[17] Charikar M., Proceedings of the 45th IEEE Symp Foundations of Computer Science (2004)
[18] Pitowsky I., Lecture Notes in Physics 321, in: Quantum Probability-Quantum Logic (1989)
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