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Quantum analogues of the Bell inequalities. The case of two spatially separated domains

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Abstract

One Investigates inequalities for the probabilities and mathematical expectations which follow from the postulates of the local quantum theory. It turns out that the relation between the quantum and the classical correlation matrices is expressed] in terms of Grothendieck's known constant. It is also shown that the extremal quantum correlations characterize the Clifford algebra (i.e., canonical anticommutative relations).

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Literature cited

  1. B. I. Spasskii and A. V. Moskovskii, “On nonlocality in quantum physics,” Usp. Fiz. Nauk,142, No. 4, 599–617 (1984).

    Google Scholar 

  2. A. A. Grib, “Bell's inequalities and the experimental verification of quantum correlations at macroscopic distances,” Usp. Fiz. Nauk,142, No. 4, 619–634 (1984).

    Google Scholar 

  3. Zh.-P. Vizh'e (J.-P. Vigier), Lecture on the Einstein-Podolsky-Rosen paradox,” in: Problems of Physics: Classical and Modern (ed.: G.-Yu. Treder (H.-J. Treder)) [Russian translation], Mir, Moscow (1982), pp. 227–254.

    Google Scholar 

  4. J. P. Clauser and A. Shimony, “Bell's theorem: experimental tests and implications,” Rep. Progr. Phys.,41, No. 12, 1881–1927 (1978).

    Google Scholar 

  5. A. S. Wightman, “Hilbert's sixth problem: mathematical treatment of the axioms of physics,” Proc. Symp. Pure Mathematics,28, 147–240 (1976).

    Google Scholar 

  6. E. Vigner (E. Wigner), “On hidden parameters and quantum mechanics probabilities,” in: E. Vigner (E. Wigner) Studies in Symmetries [Russian translation], Mir, Moscow (1971), pp. 294–302.

    Google Scholar 

  7. M. Froissart, “Constructive generalization of Bell's inequalities,” Nuovo Cimento B,64, No. 2, 241–251 (1981).

    Google Scholar 

  8. A. Fine, “Hidden variables, joint probability, and the Bell inequalities,” Phys. Rev. Lett.,48, No. 5, 291–295 (1982).

    Google Scholar 

  9. B. S. Cirel'son, “Quantum generalizations of Bell's inequality,” Lett. Math. Phys.,4, No. 2, 93–100 (1980).

    Google Scholar 

  10. A. A. Kirillov, Elements of the Theory of Representations [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  11. J. L. Krivine, “Constantes de Grothendieck et fonctions de type positif sur les spheres,” in: Seminaire sur la Geometrie des Espaces de Banach (1977–1978), Exp. No. 1–2, Centre Math., Ecole Polytech., Palaiseau (1978).

    Google Scholar 

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Authors
  1. B. S. Tsirel'son
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Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 142, pp. 174–194, 1985.

The author is grateful to A. M. Vershik for formulating the problem of the quantum analogues of Bell's inequalities and to L. A. Khalfin for formulating the problem of the representations of extrernal quantum correlations.

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Tsirel'son, B.S. Quantum analogues of the Bell inequalities. The case of two spatially separated domains. J Math Sci 36, 557–570 (1987). https://doi.org/10.1007/BF01663472

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