Quantum [Un]Speakables II pp 451-484 | Cite as

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

## Abstract

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.

## Keywords

Quantum Mechanic Detection Probability Bell Inequality Coincidence Rate Local Hide Variable Theory## References

- 1.M. Ansmann et al., Science
**461**, 504–506 (2009)Google Scholar - 2.A. Aspect, P. Grangier, G. Roger, Phys. Rev. Lett.
**47**, 460 (1981)ADSCrossRefGoogle Scholar - 3.A. Aspect, P. Grangier, G. Roger, Phys. Rev. Lett.
**49**, 91 (1982)ADSCrossRefGoogle Scholar - 4.A. Aspect, J. Dalibard, G. Roger, Phys. Rev. Lett.
**49**, 1804–1807 (1982)ADSMathSciNetCrossRefGoogle Scholar - 5.F. Belinfante,
*A Survey of Hidden Variables Theories*(Pergamon, Oxford, 1973)Google Scholar - 6.J. Bell, Physics
**1**, 195 (1964)Google Scholar - 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.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.D. Bohm,
*Quantum Theory*(Prentis Hall, Englewood Cliffs, NJ, 1951)Google Scholar - 10.D. Bohm, Y. Aharonov, Phys. Rev.
**108**, 1070 (1957)ADSMathSciNetCrossRefGoogle Scholar - 11.B. Christensen et al., Phys. Rev. Lett.
**111**, 130406–130409 (2013)Google Scholar - 12.J. Clauser, M. Horne, A. Shimony, R. Holt, Phys. Rev. Lett.
**23**, 880–884 (1969)ADSCrossRefGoogle Scholar - 13.J. Clauser, M. Horne, Phys. Rev. D
**10**, 526–535 (1974)ADSCrossRefGoogle Scholar - 14.J. Clauser, Phys. Rev. Lett.
**36**, 1223 (1976)ADSCrossRefGoogle Scholar - 15.J. Clauser, A. Shimony, Rep. Prog. Phys.
**41**, 1881–1927 (1978)ADSCrossRefGoogle Scholar - 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.P. Eberhard, Phys. Rev. A
**47**, R747–R750 (1993)ADSCrossRefGoogle Scholar - 18.U. Eichmann et al., Phys. Rev. Lett.
**70**, 2359–2362 (1993)ADSCrossRefGoogle Scholar - 19.A. Einstein, B. Podolsky, N. Rosen, Phys. Rev.
**47**, 777–780 (1935)ADSCrossRefGoogle Scholar - 20.S. Freedman, J. Clauser, Phys. Rev. Lett.
**28**, 938–941 (1972)ADSCrossRefGoogle Scholar - 21.E. Fry, R. Thompson, Phys. Rev. Lett.
**37**, 465 (1976)ADSCrossRefGoogle Scholar - 22.A. Garuccio, V. Rapisarda, 1981. Nuovo Cimento
**A65**, 269 (1981)ADSMathSciNetCrossRefGoogle Scholar - 23.I. Gerhardt et al., Phys. Rev. Lett.
**107**, 170404 (2011)ADSCrossRefGoogle Scholar - 24.M. Giustina et al., Nature
**497**, 227–230 (2013)ADSCrossRefGoogle Scholar - 25.J. Hofmann et al., Science
**337**, 72–75 (2012)ADSCrossRefGoogle Scholar - 26.R. Holt, Ph. D. thesis, Harvard Univ. (1973)Google Scholar
- 27.R. Holt, F. Pipkin, unpublished preprint, Harvard University. (1973)Google Scholar
- 28.C. Kocher, E. Commins, Phys. Rev. Lett.
**18**, 575 (1967)ADSCrossRefGoogle Scholar - 29.P. Kwiat et al., Phys. Rev. Lett.
**75**, 4337–4341 (1995)ADSCrossRefGoogle Scholar - 30.W. Lamb, M. Scully, in
*Polarization: Matiere at Rayonnement*, Société Française*de Physique*, Presses Universitaires de France, Paris (1968)Google Scholar - 31.T. Marshall, E. Santos, F. Selleri, Phys. Lett. A
**98**, 5–9 (1983)ADSCrossRefGoogle Scholar - 32.D. Matsukevich et al., Phys. Rev. Lett.
**96**, 030405 (2006)ADSCrossRefGoogle Scholar - 33.D. Matsukevich et al., Phys. Rev. Lett.
**100**, 150404 (2008)ADSCrossRefGoogle Scholar - 34.Z. Ou, L. Mandel, Phys. Rev. Lett.
**61**, 50–53 (1988)ADSMathSciNetCrossRefGoogle Scholar - 35.M. Rowe et al., Nature
**409**, 791–794 (2001)CrossRefGoogle Scholar - 36.Y. Shih, C. Alley, Phys. Rev. Lett.
**61**, 2921–2924 (1988)ADSCrossRefGoogle Scholar - 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.W. Tittel et al., Phys. Rev. Lett.
**81**, 3563–3566 (1998)ADSCrossRefGoogle Scholar - 39.R. Ursin et al., Nat. Phys.
**3**, 481–486 (2007)Google Scholar - 40.G. Weihs et al., Phys. Rev. Lett.
**81**, 5039–5043 (1998)ADSMathSciNetCrossRefGoogle Scholar - 41.E. Wigner, Am. J. Phys.
**38**, 1005–1009 (1970)ADSCrossRefGoogle Scholar