Bell’s Theorem, Bell Inequalities, and the “Probability Normalization Loophole”

  • John F. ClauserEmail author
Part of the The Frontiers Collection book series (FRONTCOLL)


Fifty years ago in 1964, John Bell [6], showed that deterministic local hidden-variables theories are incompatible with quantum mechanics for idealized systems. Inspired by his paper, Clauser, Horne, Shimony and Holt (CHSH) [12] in 1969 provided the first experimentally testable Bell Inequality and proposed an experiment to test it. That experiment was first performed in 1972 by Freedman and Clauser [20]. In 1974 Clauser and Horne (CH) [13] first showed that all physical theories consistent with “Local Realism” are constrained by an experimentally testable loophole-free Bell Inequality—the CH inequality. These theories were further clarified in 1976–1977 in “An Exchange on Local Beables”, a series of papers by Bell, Shimony, Horne, and Clauser [8] and by Clauser and Shimony (CS) [15] in their 1978 review article. In 2013, nearly fifty years after Bell’s original 1964 paper [6], two groups, Giustina et al. [24] and Christensen et al. [11] have finally tested the loophole-free CH inequality. Clauser and Shimony (CS) [15] also showed that the CHSH inequality is testable in a loophole-free manner by using a “heralded” source. It was first tested this way by Rowe et al. [35] in 2001, and more convincingly in 2008 by Matsukevich et al. [33]. To violate a Bell Inequality and thereby to disprove Local Realism, one must experimentally examine a two component entangled-state system, in a configuration that is analogous to a Gedankenexperiment first proposed by Bohm [9] in 1951. To be used, the configuration must generate a normalized coincidence rate with a large amplitude sinusoidal dependence upon adjustable apparatus settings. Proper normalization of this amplitude is critical for the avoidance of counterexamples and loopholes that can possibly invalidate the test. The earliest tests used the CHSH inequality without source heralding. The first method for normalizing coincidence rates without heralding was proposed by CHSH [12] in 1969. It consists of an experimental protocol in which coincidence rates measured with polarizers removed are used to normalize coincidence rates measured with polarizers inserted. Very high transmission polarizers are required when using this method. Highly reasonable and very weak supplementary assumptions by CHSH and by CH allow this protocol to work in a nearly loophole free manner. A second method for normalizing coincidence rates was offered by Garuccio and Rapisarda [22] in 1981. As will be discussed below, it allows experiments to be done more easily, but at a significant cost to the generality of their results. It was first used in the experiment by Aspect, Grangier, and Roge [3] in 1982. It uses “ternary-result” apparatuses and allows the use of highly absorbing polarizers, which would not work with other normalization methods. It normalizes using a sum of coincidence rates. Gerhardt et al. [23] in 2011 theoretically and experimentally demonstrated counterexamples for tests that use this normalization method. Their experiments thus obviate the validity of their counterexamples, and further indicate that very high transmission polarizers are necessary for convincing tests to be performed.


Quantum Mechanic Detection Probability Bell Inequality Coincidence Rate Local Hide Variable Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    M. Ansmann et al., Science 461, 504–506 (2009)Google Scholar
  2. 2.
    A. Aspect, P. Grangier, G. Roger, Phys. Rev. Lett. 47, 460 (1981)ADSCrossRefGoogle Scholar
  3. 3.
    A. Aspect, P. Grangier, G. Roger, Phys. Rev. Lett. 49, 91 (1982)ADSCrossRefGoogle Scholar
  4. 4.
    A. Aspect, J. Dalibard, G. Roger, Phys. Rev. Lett. 49, 1804–1807 (1982)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    F. Belinfante, A Survey of Hidden Variables Theories (Pergamon, Oxford, 1973)Google Scholar
  6. 6.
    J. Bell, Physics 1, 195 (1964)Google Scholar
  7. 7.
    J. Bell in Foundations of Quantum Mechanics, Proceedings of the International School of Physics “Enrico Fermi”, ed. by B. d’Espagnat (Academic Press, New York, 1971)Google Scholar
  8. 8.
    J. Bell, A. Shimony, M. Horne, J. Clauser (1976–1977) Epistemological Letters (Association Ferdinand Gonseth, Institut de la Methode, Case Postale 1081, CH-2501, Bienne.) republished as An Exchange on Local Beables in Dialectica, 39, 85−110 (1985)Google Scholar
  9. 9.
    D. Bohm, Quantum Theory (Prentis Hall, Englewood Cliffs, NJ, 1951)Google Scholar
  10. 10.
    D. Bohm, Y. Aharonov, Phys. Rev. 108, 1070 (1957)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    B. Christensen et al., Phys. Rev. Lett. 111, 130406–130409 (2013)Google Scholar
  12. 12.
    J. Clauser, M. Horne, A. Shimony, R. Holt, Phys. Rev. Lett. 23, 880–884 (1969)ADSCrossRefGoogle Scholar
  13. 13.
    J. Clauser, M. Horne, Phys. Rev. D 10, 526–535 (1974)ADSCrossRefGoogle Scholar
  14. 14.
    J. Clauser, Phys. Rev. Lett. 36, 1223 (1976)ADSCrossRefGoogle Scholar
  15. 15.
    J. Clauser, A. Shimony, Rep. Prog. Phys. 41, 1881–1927 (1978)ADSCrossRefGoogle Scholar
  16. 16.
    J. Clauser in Quantum [Un]speakables, From Bell to Quantum Information, Proceedings of the 1st Quantum [Un]speakables Conference, ed. by R. Bertlmann, A. Zeilinger (Springer, Berlin, 2002), pp 61−96Google Scholar
  17. 17.
    P. Eberhard, Phys. Rev. A 47, R747–R750 (1993)ADSCrossRefGoogle Scholar
  18. 18.
    U. Eichmann et al., Phys. Rev. Lett. 70, 2359–2362 (1993)ADSCrossRefGoogle Scholar
  19. 19.
    A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47, 777–780 (1935)ADSCrossRefGoogle Scholar
  20. 20.
    S. Freedman, J. Clauser, Phys. Rev. Lett. 28, 938–941 (1972)ADSCrossRefGoogle Scholar
  21. 21.
    E. Fry, R. Thompson, Phys. Rev. Lett. 37, 465 (1976)ADSCrossRefGoogle Scholar
  22. 22.
    A. Garuccio, V. Rapisarda, 1981. Nuovo Cimento A65, 269 (1981)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    I. Gerhardt et al., Phys. Rev. Lett. 107, 170404 (2011)ADSCrossRefGoogle Scholar
  24. 24.
    M. Giustina et al., Nature 497, 227–230 (2013)ADSCrossRefGoogle Scholar
  25. 25.
    J. Hofmann et al., Science 337, 72–75 (2012)ADSCrossRefGoogle Scholar
  26. 26.
    R. Holt, Ph. D. thesis, Harvard Univ. (1973)Google Scholar
  27. 27.
    R. Holt, F. Pipkin, unpublished preprint, Harvard University. (1973)Google Scholar
  28. 28.
    C. Kocher, E. Commins, Phys. Rev. Lett. 18, 575 (1967)ADSCrossRefGoogle Scholar
  29. 29.
    P. Kwiat et al., Phys. Rev. Lett. 75, 4337–4341 (1995)ADSCrossRefGoogle Scholar
  30. 30.
    W. Lamb, M. Scully, in Polarization: Matiere at Rayonnement, Société Française de Physique, Presses Universitaires de France, Paris (1968)Google Scholar
  31. 31.
    T. Marshall, E. Santos, F. Selleri, Phys. Lett. A 98, 5–9 (1983)ADSCrossRefGoogle Scholar
  32. 32.
    D. Matsukevich et al., Phys. Rev. Lett. 96, 030405 (2006)ADSCrossRefGoogle Scholar
  33. 33.
    D. Matsukevich et al., Phys. Rev. Lett. 100, 150404 (2008)ADSCrossRefGoogle Scholar
  34. 34.
    Z. Ou, L. Mandel, Phys. Rev. Lett. 61, 50–53 (1988)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    M. Rowe et al., Nature 409, 791–794 (2001)CrossRefGoogle Scholar
  36. 36.
    Y. Shih, C. Alley, Phys. Rev. Lett. 61, 2921–2924 (1988)ADSCrossRefGoogle Scholar
  37. 37.
    A. Shimony in Foundations of Quantum Mechanics, Proceedings of the International School of Physics “Enrico Fermi”, ed. by B. d’Espagnat (Academic Press, New York, 1971)Google Scholar
  38. 38.
    W. Tittel et al., Phys. Rev. Lett. 81, 3563–3566 (1998)ADSCrossRefGoogle Scholar
  39. 39.
    R. Ursin et al., Nat. Phys. 3, 481–486 (2007)Google Scholar
  40. 40.
    G. Weihs et al., Phys. Rev. Lett. 81, 5039–5043 (1998)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    E. Wigner, Am. J. Phys. 38, 1005–1009 (1970)ADSCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Walnut CreekUSA

Personalised recommendations