Journal of Russian Laser Research

, Volume 36, Issue 1, pp 2–16 | Cite as

Modeling Tests Based on the Eberhard Inequality



Last year, the first experimental tests closing the detection loophole (also referred to as the fair sampling loophole) were performed by two experimental groups, one in Vienna and the other one in Urbana-Champaign. To violate the Bell-type inequalities (the Eberhard inequality in the first test and the Clauser–Horne inequality in the second test), one has to optimize a number of parameters involved in the experiment (angles of polarization beam splitters and quantum state parameters). We study this problem for the Eberhard inequality in detail, using the advanced method of numerical optimization, namely, the Nelder–Mead method.


Bell’s test fair sampling loophole Eberhard inequality Clauser–Horne inequality optimization of parameters fluctuations of angles coefficient of variation 


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  1. 1.
    J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press (1987).Google Scholar
  2. 2.
    M. Giustina, Al. Mech, S. Ramelow, et al., Nature, 497, 227 (2013).CrossRefADSGoogle Scholar
  3. 3.
    B. G. Christensen, K. T. McCusker, J. Altepeter, et al., Phys. Rev. Lett., 111, 1304 (2013).Google Scholar
  4. 4.
    J. Kofler, S. Ramelow, M. Giustina, and A. Zeilinger, “On Bell violation using entangled photons without the fair-sampling assumption,” arXiv: 1307.6475 [quant-ph].Google Scholar
  5. 5.
    J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett., 23, 880 (1969).CrossRefADSGoogle Scholar
  6. 6.
    J. F. Clauser and M. A. Horne, Phys. Rev. D, 10, 526 (1974).CrossRefADSGoogle Scholar
  7. 7.
    J. F. Clauser and A. Shimony, Rep. Prog. Phys., 41, 1881 (1978).CrossRefADSGoogle Scholar
  8. 8.
    A. Shimony, “Bell’s theorem,” in: E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy, Stanford (2012);
  9. 9.
    Ph. H. Eberhard, Phys. Rev. A, 477, 750 (1993).Google Scholar
  10. 10.
    N. D. Mermin, “The EPR experiment – thoughts about the ‘loophole,’ ” in: D. M. Greenberger (Ed.), Techniques and Ideas in Quantum Measurement Theory, New York Academy of Science, New York (2006), p. 422.Google Scholar
  11. 11.
    A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett., 49, 1804 (1982).CrossRefADSMathSciNetGoogle Scholar
  12. 12.
    A. Aspect, Three experimental tests of Bell inequalities by the measurement of polarization correlations between photons, Ph. D Thesis, Orsay, France (1983).Google Scholar
  13. 13.
    A. Aspect, “Bell’s theorem: The naive view of an experimentalist,” arXiv:quant-ph/0402001.Google Scholar
  14. 14.
    G. Jaeger, Entanglement, Information, and the Interpretation of Quantum Mechanics (The Frontiers Collection), Springer, Heidelberg-Berlin-New York (2009).CrossRefGoogle Scholar
  15. 15.
    G. Jaeger, A. Khrennikov, M. Schlosshauer, and G. Weihs (Eds.), Advances in Quantum Theory, AIP Conf. Proc., 1327 (2011).Google Scholar
  16. 16.
    A. Khrennikov (Ed.), Quantum Theory: Reconsideration of Foundations-5, AIP Conf. Proc., 1232 (2010).Google Scholar
  17. 17.
    L. Accardi, G. Adenier, C. A. Fuchs, et al. (Eds.), Foundations of Probability and Physics-5, AIP Conf. Proc., 1101 (2009).Google Scholar
  18. 18.
    G. Adenier, A. Yu. Khrennikov, P. Lahti, et al. (Eds.), Quantum Theory: Reconsideration of Foundations-4, AIP Conf. Proc., 962 (2008).Google Scholar
  19. 19.
    G. Adenier, C. Fuchs, and A. Khrennikov (Eds.), Foundations of Probability and Physics-4, AIP Conf. Proc., 889 (2007).Google Scholar
  20. 20.
    A. Khrennikov, Contextual Approach to Quantum Formalism, Springer, Berlin-Heidelberg-New York (2009).CrossRefMATHGoogle Scholar
  21. 21.
    S. Ramelow, “Quantum theory: Advances and problems,” Talk at A. Marcus Wallenberg Symposium “Quantum Theory: Advances and Problems” (Växjö, Sweden, 10–13 June, 2013).Google Scholar
  22. 22.
    J.-A. Larsson, M. Giustina, J. Kofler, et al., “Bell violation with entangled photons, free of the coincidence-time loophole,” Scholar
  23. 23.
    D. Schmid, T.-Y. Huang, R. Dirks, et al., “Polarization-dependent focusing,” Talk at the Workshop “Quantum Information and Measurement” (Rochester, New York, 17–20 June 2013).Google Scholar
  24. 24.
    S. V. Polyakov, E. B. Flagg, T. Thomay, and G. S. Solomon, AIP Conf. Proc., 1508, 67 (2012).CrossRefADSGoogle Scholar
  25. 25.
    G. Weihs, T. Jennewein, C. Simon, et al., Phys. Rev. Lett., 81, 5039 (1998).CrossRefADSMATHMathSciNetGoogle Scholar
  26. 26.
    R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, et al., Nature Phys., 3, 481 (2007).CrossRefADSGoogle Scholar
  27. 27.
    S. Lloyd, M. S. Shahriar, J. H. Shapiro, and P. R. Hemmer, Phys. Rev. Lett., 87, 167903 (2001).CrossRefADSGoogle Scholar
  28. 28.
    O. Landry, J. A. W. van Houwelingen, A. Beveratos, et al., J. Opt. Soc. Am. B, 24,398 (2007).CrossRefADSGoogle Scholar
  29. 29.
    H. Hübel, M. R. Vanner, T. Lederer, et al., Opt. Exp., 15, 7853 (2007).CrossRefADSGoogle Scholar
  30. 30.
    Q. Zhang, H. Takesue, S. W. Nam, et al., Opt. Exp., 16, 5776 (2008).CrossRefADSGoogle Scholar
  31. 31.
    I. Marcikic, H. de Riedmatten, W. Tittel, et al., Phys. Rev. Lett., 93, 180502 (2004).CrossRefADSGoogle Scholar
  32. 32.
    H. Takesue, Opt. Exp., 14, 3453 (2006).CrossRefADSGoogle Scholar
  33. 33.
    T. Honjo, H. Takesue, H. Kamada, et al., Opt. Exp., 15, 13957 (2007).CrossRefADSGoogle Scholar
  34. 34.
    A. Yu. Khrennikov and I. V. Volovich, “Local realism, contextualism and loopholes in Bell’s experiments,” in: A. Yu. Khrennikov (Ed.), Foundations of Probability and Physics-2, Ser. Math. Model., Växjö University Press, Växjö, Sweden (2002), Vol. 5, p. 325.Google Scholar
  35. 35.
    T. Ishiwatari, A. Khrennikov, B. Nilsson, and I. Volovich, “Quantum field theory and distance effects for polarization correlations in waveguides,” in: The third Conference on Mathematical Modeling of Wave Phenomena/20th Nordic Conference on Radio Science and Communications, AIP Conf. Proc., 1106, 276 (2009).Google Scholar
  36. 36.
    A. Khrennikov, B. Nilsson, S. Nordebo, and I. Volovich, “Distance dependence of entangled photons in waveguides,” in: Conference FPP6 – Foundations of Probability and Physics-6, AIP Conf. Proc., 1424, 262 (2012).Google Scholar
  37. 37.
    A. Khrennikov, B. Nilsson, S. Nordebo, and I. Volovich. Phys. Scr., 85, 06505 (2012).CrossRefGoogle Scholar
  38. 38.
    A. Khrennikov, B. Nilsson, S. Nordebo, and I. Volovich, “On the quantization of the electromagnetic field of a layered dielectric waveguide,” in: Conference QTRF6 – Quantim Theory: Reconsideration of Foundations–6, AIP Conf. Proc., 1508, 285 (2012).Google Scholar
  39. 39.
    J. A. Nelder and R. Mead, Comput. J., 7, 313 (1965).CrossRefGoogle Scholar
  40. 40.
    D. J. Schroeder, Astronomical Optics, 2nd ed., Academic Press (1999).Google Scholar
  41. 41.
    J. T. Bushberg, J. A. Seibert, E. M. Leidholdt, and J. M. Boone, The Essential Physics of Medical Imaging, 2nd ed., Lippincott Williams and Wilkins, Philadelphia (2006).Google Scholar

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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Moscow Institute of Electronic Technology ZelenogradMoscowRussia
  2. 2.International Center for Mathematical Modeling in Physics Engineering, Economics, and Cognitive Science Linnaeus UniversityVäxjöSweden

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