Abstract
We consider the Bell and Bell-Clauser-Horne-Shimony-Holt inequalities for two-particle spin states. It is known that these inequalities are violated in experimental verification. We show that this can be explained because these inequalities are proved for correlation functions of random variables that are totally unrelated to one another, while the verification is done using correlation functions in which random variables refer to a pair of particles forming a two-particle state. In the case of entangled states, these random functions are dependent, and their correlation coefficient is nonzero. We give inequalities that explicitly involve this correlation coefficient. For factorable and separable states, these inequalities coincide with the standard Bell and Bell-Clauser-Horne-Shimony-Holt inequalities.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev., 47, 777–780 (1935).
N. Bohr, Phys. Rev., 2nd ser., 48, 696–702 (1935).
J. S. Bell, Physics, 1, 195–200 (1964).
J. S. Bell, Rev. Modern Phys., 38, 447–452 (1966).
J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett., 23, 880–884 (1969).
I. V. Volovich, “Towards quantum information theory in space and time,” arXiv:quant-ph/0203030v1 (2002); A. Khrennikov and I. Volovich, “Local realism, contextualism, and loopholes in Bell’s experiments,” arXiv:quant-ph/0212127v1 (2002); “Quantum nonlocality, EPR model, and Bell’s theorem,” in: Proc. 3rd Intl. Sakharov Conf. on Physics (A. Semikhatov, M. Vasiliev, and V. Zaikin, eds.), Vol. 2, World Scientific, Singapore (2003), pp. 260–267.
A. Khrennikov, Non-Kolmogorovian Theory of Probability and Quantum Physics [in Russian], Fizmatlit, Moscow (2003).
A. Bohm, Quantum Mechanics: Foundations and Applications, Springer, New York (1986).
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press, Cambridge (2000).
M. Genovese, Phys. Rep., 413, 319–396 (2005).
A. Khrennikov, “Bell’s inequality: Physics meets probability,” arXiv:0709.3909v2 [quant-ph] (2007).
F. J. Belinfante, A Survey of Hidden-Variables Theories, Pergamon, Oxford (1973).
S. J. Freedman and J. F. Clauser, Phys. Rev. Lett., 28, 938–941 (1972).
A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett., 47, 460–463 (1981).
A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett., 49, 91–94 (1982).
A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett., 49, 1804–1807 (1982).
Z. Y. Ou and L. Mandel, Phys. Rev. Lett., 61, 50–53 (1988).
T. E. Kiess, Y. H. Shih, A. V. Sergienko, and C. O. Alley, Phys. Rev. Lett., 71, 3893–3897 (1993).
M. Kupczynski, Phys. Lett. A, 121, 51–53 (1987); J. Russ. Laser Res., 26, 514–523 (2005).
D. N. Klyshko and A. V. Belinsky, Sov. Phys. Uspekhi, 36, 653–693.
V. A. Andreev and V. I. Man’ko, JETP Lett., 72, 93–96 (2000); V. A. Andreev and V. I. Man’ko, Phys. Lett. A, 281, 278–288 (2001).
V. A. Andreev, Theor. Math. Phys., 152, 1286–1298 (2007).
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 2, pp. 234–249, February, 2009.
Rights and permissions
About this article
Cite this article
Andreev, V.A. The correlation bell inequalities. Theor Math Phys 158, 196–209 (2009). https://doi.org/10.1007/s11232-009-0016-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11232-009-0016-4