Skip to main content
SpringerLink
Log in

CHSH Inequality: Quantum Probabilities as Classical Conditional Probabilities

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

In this note we demonstrate that the results of observations in the EPR–Bohm–Bell experiment can be described within the classical probabilistic framework. However, the “quantum probabilities” have to be interpreted as conditional probabilities, where conditioning is with respect to fixed experimental settings. Our approach is based on the complete account of randomness involved in the experiment. The crucial point is that randomness of selections of experimental settings has to be taken into account within one consistent framework covering all events related to the experiment. This approach can be applied to any complex experiment in which statistical data are collected for various (in general incompatible) experimental settings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Experiments to test violation of the Bell-type inequalities can be treated as one special class of statistical tests of nonclassicality of quantum probabilistic data. As was emphasized by Feynman et al. [16] and Accardi [17], another important test is given by the two slit experiment, see also [18]. However, recently the viewpoint that entanglement is the main root of quantumness became very popular in the quantum foundational community, especially in its part closely linked to quantum information and technology. (In particular, this viewpoint was presented in the private discussions of the author and Anton Zeilinger.) However, this question needs further clarifications and debates.

  2. The quantum formalism does not account this sort of randomness. Random generators used in the experimental tests based on the Bell-type inequalities are not described by operators in the complex Hilbert space. They are considered as “technicalities”; often this sort of randomness is related to the freewill of an experimentalist.

  3. As a possible realization, we can consider the following experimental framework. The random generator \(a\) is coupled to a block which splits the channel going from the source of photons in the \(A\)-direction into two channels, which are also labeled by \(i=1,2.\) Each of the channels is coupled to its own polarization beam splitter (PBS) which has the fixed orientation given by the angle \(\theta _i\) and the PBS in the \(i\)th channel is coupled to its pair of the detectors \(D^A_i(-),\) polarization down, and \(D^A_i(+),\) polarization up. Thus at each side there are two PBSs (corresponding to the fixed orientations) and totally 4 detectors. The complete two-side experimental scheme is based on 4 PBSs and 8 detectors.

  4. The terminology might be misleading. To escape this problem, we recall that in our model detectors have 100 % efficiency; nondetection means just that this experimental setting was not selected.

References

  1. Bell, J.S.: On the Einstein Podolsky Rosen Paradox. Physics 1(3), 195200 (1964)

    Google Scholar 

  2. Bell, J.S.: Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press, Cambridge (1987)

  3. Khrennikov, A. (ed.): Foundations of Probability and Physics. Quantum Probability and White Noise Analysis, vol. 13. WSP, Singapore (2001)

  4. Khrennikov, A. (ed.): Quantum Theory: Reconsideration of Foundations. Ser. Math. Modelling, vol. 2. Växjö University Press, Växjö (2002)

  5. Khrennikov, A. (ed.): Quantum Theory: Reconsideration of Foundations-2. Ser. Math. Modelling, vol. 10. Växjö University Press, Växjö (2003)

  6. Khrennikov, A. (ed.): Foundations of Probability and Physics 3, vol. 750 (2005)

  7. Adenier, G., Fuchs, C., Khrennikov, A. (eds.): Foundations of Probability and Physics-4, American Institute of Physics, Ser. Conference Proceedings, 889, Melville, NY (2007)

  8. Adenier, G., Khrennikov, A. Yu., Lahti, P., Manko, V. I., Nieuwenhuizen, T.M. (eds.): Quantum Theory: Reconsideration of Foundations-4, American Institute of Physics, Ser. Conference Proceedings 962, Melville, NY (2008)

  9. L. Accardi, G. Adenier, C.A. Fuchs, G. Jaeger, A. Khrennikov, J.-A. Larsson, S. Stenholm (eds.): Foundations of Probability and Physics-5, American Institute of Physics, Ser. Conference Proceedings, 1101, Melville, NY (2009)

  10. Khrennikov, A. (ed.): Quantum Theory: Reconsideration of Foundations-5, vol. 1232. AIP, Melville, NY (2010)

    Google Scholar 

  11. Jaeger, G., Khrennikov, A., Schlosshauer, M., Weihs, G. (eds.): Advances in Quantum Theory, vol. 1327. AIP, Melville, NY (2011)

  12. Giustina, M., Mech, A., Ramelow, S., Wittmann, B., Kofler, B., Kofler, J., Beyer, J., Lita, A., Calkins, B., Gerrits, T., Woo Nam, S., Ursin, R., Zeilinger, A.: Bell violation using entangled photons without the fair-sampling assumption. Nature 497, 227 (2013)

  13. Christensen, B.G., et al.: Detection-loophole-free test of quantum nonlocality, and applications. Phys. Rev. Lett. 111, 1304 (2013)

  14. Khrennikov, A., Ramelow, S., Ursin, R., Wittmann, B., Kofler, J., Basieva, I.: On the equivalence of the Clauser–Horne and Eberhard inequality based tests, Phys. Scripta, to be published, 2014

  15. Kolmogoroff, A.N.: Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer, Berlin. [English translation: Kolmogorov A N 1956 Foundations of Theory of Probability. Chelsea Publishing Company, New York] (1933)

  16. Feynman, R., Hibbs, A.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965)

    MATH  Google Scholar 

  17. Accardi, L.: Urne e Camaleoni: Dialogo sulla Realta. le Leggi del Caso e la Teoria Quantistica. Il Saggiatore, Rome (1997)

  18. Khrennikov, A.: Interpretations of Probability. De Gruyter, Berlin, 2009, second edition (completed); first edition 1998

  19. Accardi, L.: The probabilistic roots of the quantum mechanical paradoxes. In: Diner, S., Fargue, D., Lochak, G., Selleri, F. (eds.) The Wave–Particle Dualism. A Tribute to Louis de Broglie on his 90th Birthday, pp. 47–55, D. Reidel Publ. Company, Dordrecht (1970)

  20. Accardi, L.: Topics in quantum probability. Phys. Rep. 77, 169–192 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  21. Kupczynski, M.: Bertrand’s paradox and Bell’s inequalities. Phys. Lett. A 121, 205 (1987)

  22. Accardi, L., Regoli, M.: Locality and Bell’s inequality. arXiv:quant-ph/0007005

  23. Khrennikov, A.: Frequency analysis of the EPR–Bell argumentation. Found. Phys. 32, 1159–1174 (2002)

    Article  MathSciNet  Google Scholar 

  24. De Muynck, W.M.: Foundations of Quantum Mechanics, An Empiricists Approach. Kluwer, Dordrecht (2002)

    Book  Google Scholar 

  25. De Muynck, W. M.: Interpretations of quantum mechanics, and interpretations of violations of Bell’s inequality. In: Khrennikov, A.Y. (ed.) Foundations of Probability and Physics, pp. 95–104. Series PQ-QP: Quantum Probability and White Noise Analysis 13. WSP, Singapore (2001).

  26. Hess, K., Philipp, W.: Exclusion of time in Mermin’s proof of Bell-type inequalities. In: Khrennikov, A.Y. (ed.) Quantum Theory: Reconsideration of Foundations-2, pp. 243–254. Ser. Math. Model. 10, Växjö University Press, Växjö (2003)

  27. Hess, K., Philipp, W.: Bell’s theorem: critique of proofs with and without inequalities. In: Adenier, G., Khrennikov, A.Y. (eds): Foundations of Probability and Physics-3, pp. 150–155. American Institute of Physics, Ser. Conference Proceedings 750, Melville, NY (2005)

  28. Hess, K.: In memoriam Walter Philipp. In: Adenier, G., Fuchs, C. and Khrennikov, A. Yu. (eds) Foundations of Probability and Physics-3, pp. 3–6. American Institute of Physics, Ser. Conference Proceedings 889, Melville, NY (2007).

  29. Kupczynski, M.: EPR paradox, locality and completeness of quantum theory. AIP Conf. Proc. 962, 274 (2007)

  30. Khrennikov, A., Smolyanov, O.G., Truman, A.: Kolmogorov probability spaces describing Accardi models for quantum correlations. Open Syst. Inf. Dyn. 12(4), 371–384 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  31. Accardi, L.: Some loopholes to save quantum nonlocality. Foundations of Probability and Physics-3, vol. 750, pp. 1–20. AIP, Melville, NY (2005)

  32. Kupczynski, M.: Entanglement and quantum nonlocality demystified. AIP Conf. Proc. 1508, 253 (2012)

  33. Kupczynski, M.: Causality and local determinism versus quantum nonlocality. J. Phys.: Conf. Ser. 504, 012015 (2014)

  34. Khrennikov, A.: Contextual Approach to Quantum Formalism. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  35. Adenier, G.: Local Realist Approach and Numerical Simulation of Nonclassical Experiments in Quantum Mechanics. \({\rm V}\ddot{{\rm x}}{\rm j}\ddot{{\rm o}}\) University Press, \({\rm V}\ddot{{\rm x}}{\rm j}\ddot{{\rm o}}\) (2008)

  36. Khrennikov, A.: Bell–Boole inequality: nonlocality or probabilistic incompatibility of random variables? Entropy 10, 19–32 (2008)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  37. De Raedt, K., Keimpema, K., De Raedt, H., Michielsen, K., Miyashita, S.: A local realist model for correlations of the singlet state. Eur. Phys. J. B 53, 139–142 (2006)

    Article  ADS  Google Scholar 

  38. De Raedt, H., De Raedt, K., Michielsen, K., Keimpema, K., Miyashita, S.: Event-based computer simulation model of aspect-type experiments strictly satisfying Einstein’s locality conditions. Phys. Soc. Jpn. 76, 104005 (2007)

    Article  ADS  Google Scholar 

  39. Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Proposed experiment to test local hidden-Variable theories. Phys. Rev. Lett. 23, 880–884 (1969)

  40. Avis, D., Fischer, P., Hilbert, A., Khrennikov, A.: Single, Complete, Probability Spaces Consistent With EPR–Bohm–Bell Experimental Data, Foundations of Probability and Physics 5, vol. 750, pp 294–301. AIP, Melville, NY (2009)

  41. A. Khrennikov 2014 Classical probability model for Bell inequality. EmQM13: Emergent Quantum Mechanics 3–6 October 2013, Vienna, Austria. J. Phys.: Conf. Ser., 504

  42. Khrennikov, A.: Ubiquitous quantum structure: from psychology to finances. Springer, Berlin (2010)

    Book  Google Scholar 

  43. Dzhafarov, E.N., Kujala, J.V.: Embedding quantum into classical: contextualization vs conditionalization. PLoS One 9(3), e92818 (2014)

    Article  MathSciNet  ADS  Google Scholar 

  44. Dzhafarov, E.N., Kujala, J.V.: No-forcing and no-matching theorems for classical probability applied to quantum mechanics. Found. Phys. 44, 248–265 (2014)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  45. Kofler, J., Paterek, T., Brukner, C.: Experimenter’s freedom in Bell’s theorem and quantum cryptography. Phys. Rev. A 73, 022104 (2006)

    Article  ADS  Google Scholar 

  46. Khrennikov, A., Nilsson, B., Nordebo, S., Volovich, I.: Photon flux and distance from the source: consequences for quantum communication. Found. Phys. doi:10.1007/s10701-014-9786-0

  47. Ishiwatari, T., Khrennikov, A., Nilsson, B., Volovich, I.: Quantum field theory and distance effects for polarization correlations in waveguides. In: 3rd Conference Mathematical Modeling of Wave Phenomena/20th Nordic Conference on Radio Science and Communications, vol. 1106 of AIP Conference Proceeding, pp. 276–285. American Institute of Physics (2009)

  48. Khrennikov, A., Nilsson, B., Nordebo, S., Volovich, I.: Distance dependence of entangled photons in waveguides. In: Conference FPP6-Foundations of Probability and Physics-6, vol. 1424 of AIP Conference Proceeding, pp. 262–269. American Institute of Physics, Melville, New York (2012)

  49. Kofler, J., Brukner, C.: Condition for macroscopic realism beyond the Leggett–Garg inequalities. Phys. Rev. A 87, 052115 (2013)

    Article  ADS  Google Scholar 

  50. Dzhafarov, E.N.: Selective influence through conditional independence. Psychometrika 68, 7–26 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  51. Dzhafarov, E.N., Kujala, J.V.: On selective influences, marginal selectivity, and Bell/CHSH inequalities. Top. Cogn. Sci. 12118, 6 (2014)

    Google Scholar 

  52. Asano, M., Hashimoto, T., Khrennikov, A., Ohya, M., Tanaka, Y.: Violation of contextual generalization of the Leggett-Garg inequality for recognition of ambiguous figures. arXiv:1401.2897 [q-bio.NC]

  53. Von Neumann, J. Mathematische Grundlagen der Quantenmechanik (Berlin-Heidelberg-New York: Springer) (1932) English translation, : Mathematical Foundations of Quantum Mechanics. Princeton Univ. Press, Princeton (1955)

  54. Zeilinger, A.: A foundational principle for quantum mechanics. Found. Phys. 29(4), 631–643 (1999)

    Article  MathSciNet  Google Scholar 

  55. Zeilinger, A.: Dance of the Photons: From Einstein to Quantum Teleportation. Farrar, Straus and Giroux, New York (2010)

    Google Scholar 

  56. Brukner, C., Zeilinger, A.: Malus’ law and quantum information. Acta Phys. Slovava 49(4), 647–652 (1999)

    Google Scholar 

  57. Brukner, C., Zeilinger, A.: Operationally invariant information in quantum mechanics. Phys. Rev. Lett. 83(17), 3354–3357 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  58. Brukner, C., Zeilinger, A.: Information invariance and quantum probabilities. Found. Phys. 39, 677 (2009)

    Article  MATH  MathSciNet  ADS  Google Scholar 

Download references

Acknowledgments

This paper was written during author’s visiting professor fellowship to the Institute for Quantum Optics and Quantum Information of Austrian Academy of Science (April–June, 2014); the main result of this paper was presented in the course of lectures on the inter-relation between classical and quantum randomness given for the graduate students of this institute. I would like to thank Anton Zeilinger for hospitality and critical discussions about the objective representation of quantum observables.

Author information

Authors and Affiliations

  1. International Center for Mathematical Modeling in Physics and Cognitive Sciences, Linnaeus University, Växjö-Kalmar, Sweden

    Andrei Khrennikov

Authors
  1. Andrei Khrennikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrei Khrennikov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khrennikov, A. CHSH Inequality: Quantum Probabilities as Classical Conditional Probabilities. Found Phys 45, 711–725 (2015). https://doi.org/10.1007/s10701-014-9851-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-014-9851-8

Keywords

Search

Navigation