Abstract
We use the reduced density matrix of the two-particle spin state to construct a generalized Bell-Clauser-Horne-Shimony-Holt inequality. For each specific state and under a special choice of the vectors \(\vec a, \vec b\), this inequality becomes an exact equality. We show how such vectors can be found using the reduced density matrix. Both sides of this equality have a specific numerical value. We indicate the connection of this number with the measure of entanglement of the two-particle spin state.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. S. Bell, Physics, 1, 195 (1964).
J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett., 23, 880 (1969).
S. J. Freedman and J. F. Clauser, Phys. Rev. Lett., 28, 938 (1972).
A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett., 47, 460 (1981).
A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett., 49, 91 (1982).
A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett., 49, 1804 (1982).
Z. Y. Ou and L. Mandel, Phys. Rev. Lett., 61, 50 (1988).
T. E. Kiess, Y. H. Shih, A. V. Sergienko, and C. O. Alley, Phys. Rev. Lett., 71, 3893 (1993).
V. A. Andreev and V. I. Man’ko, Theor. Math. Phys., 140, 1135 (2004).
I. V. Volovich, “Bell’s theorem and locality in space,” arXiv:quant-ph/0012010v1 (2001).
L. Accardi and M. Regoli, “Locality and Bell’s inequality,” arXiv:quant-ph/0007005v2 (2000).
M. B. Mensky, Quantum Measurements and Decoherence: Models and Phenomenology (Fund. Theories Phys., Vol. 110), Kluwer, Dordrecht (2000).
A. S. Kholevo, Introduction to the Quantum Theory of Information [in Russian], MCCME Publ., Moscow (2002).
A. Khrennikov, Found. Phys., 32, 1159 (2002).
A. Khrennikov and I. Volovich, “Local realism, contextualism, and loopholes in Bell’s experiments,” arXiv:quant-ph/0212127v1 (2002).
A. Yu. Khrennikov, Non-Kolmogorovian Probability Theories and Quantum Physics [in Russian], Fizmatlit, Moscow (2003).
A. Khrennikov and I. V. Volovich, “Quantum nonlocality, EPR model, and Bell’s theorem,” in: Proc. 3rd Intl. Sakharov Conf. on Physics (Moscow, Russia, 2002, A. Semikhatov, M. Vasiliev, and V. Zaikin, eds.), Vol. 2, World Scientific, Singapore (2003), p. 60.
V. A. Andreev and V. I. Man’ko, “The quantum tomography representation of Bell-CHSH inequalities,” in: Quantum Theory: Reconsideration of Foundations-2 (Math. Model. Phys. Eng. Cogn. Sci., Vol. 10, A. Khrennikov, ed.), Vaxjo Univ. Press, Växjö (2004), p. 47.
V. A. Andreev, V. I. Man’ko, O. V. Man’ko, and E. V. Shchukin, Theor. Math. Phys., 146, 140 (2006).
V. A. Andreev and V. I. Man’ko, JETP Lett., 72, 93 (2000).
V. A. Andreev and V. I. Man’ko, Phys. Lett. A, 281, 278 (2001).
R. F. Werner, Phys. Rev. A, 40, 4277 (1989).
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 3, pp. 488–501, September, 2007.
Rights and permissions
About this article
Cite this article
Andreev, V.A. Generalized Bell inequality and a method for its verification. Theor Math Phys 152, 1286–1298 (2007). https://doi.org/10.1007/s11232-007-0113-1
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11232-007-0113-1