Skip to main content
SpringerLink
Log in

No-Forcing and No-Matching Theorems for Classical Probability Applied to Quantum Mechanics

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

Correlations of spins in a system of entangled particles are inconsistent with Kolmogorov’s probability theory (KPT), provided the system is assumed to be non-contextual. In the Alice–Bob EPR paradigm, non-contextuality means that the identity of Alice’s spin (i.e., the probability space on which it is defined as a random variable) is determined only by the axis \(\alpha _{i}\) chosen by Alice, irrespective of Bob’s axis \(\beta _{j}\) (and vice versa). Here, we study contextual KPT models, with two properties: (1) Alice’s and Bob’s spins are identified as \(A_{ij}\) and \(B_{ij}\), even though their distributions are determined by, respectively, \(\alpha _{i}\) alone and \(\beta _{j}\) alone, in accordance with the no-signaling requirement; and (2) the joint distributions of the spins \(A_{ij},B_{ij}\) across all values of \(\alpha _{i},\beta _{j}\) are constrained by fixing distributions of some subsets thereof. Of special interest among these subsets is the set of probabilistic connections, defined as the pairs \(\left( A_{ij},A_{ij'}\right) \) and \(\left( B_{ij},B_{i'j}\right) \) with \(\alpha _{i}\not =\alpha _{i'}\) and \(\beta _{j}\not =\beta _{j'}\) (the non-contextuality assumption is obtained as a special case of connections, with zero probabilities of \(A_{ij}\not =A_{ij'}\) and \(B_{ij}\not =B_{i'j}\)). Thus, one can achieve a complete KPT characterization of the Bell-type inequalities, or Tsirelson’s inequalities, by specifying the distributions of probabilistic connections compatible with those and only those spin pairs \(\left( A_{ij},B_{ij}\right) \) that are subject to these inequalities. We show, however, that quantum-mechanical (QM) constraints are special. No-forcing theorem says that if a set of probabilistic connections is not compatible with correlations violating QM, then it is compatible only with the classical–mechanical correlations. No-matching theorem says that there are no subsets of the spin variables \(A_{ij},B_{ij}\) whose distributions can be fixed to be compatible with and only with QM-compliant correlations.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. A rigorous formulation [13, 27–29] requires that \(H\) be defined as \(\left( A_{ij}',B_{ij}':i,\!j\in \left\{ 1,2\right\} \right) \) such that each pair \(\left( A_{ij}',B_{ij}'\right) \) has the same distribution as (rather than is identical to) \(\left( A_{ij},B_{ij}\right) \) for \(i,\!j\in \left\{ 1,2\right\} \). Our lax notation is unlikely to cause confusion in the present paper.

  2. The theorem states that the single-indexed \(A_{1},A_{2},B_{1},B_{2}\) are jointly distributed if and only if \(p\) satisfies the CHSH inequalities. We use the fact that the single-indexation means that the connection vector for the double-indexed \(A\)’s and \(B\)’s is \(\varepsilon _{0}\), and that the existence of the joint distribution of these \(A\)’s and \(B\)’s means, by definition, that \(\varepsilon _{0}\) and \(p\) are compatible.

References

  1. Bell, J.: On the Einstein–Podolsky–Rosen paradox. Physics 1, 195–200 (1964)

    Google Scholar 

  2. 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)

    Article  ADS  Google Scholar 

  3. Clauser, J.F., Horne, M.A.: Experimental consequences of objective local theories. Phys. Rev. D 10, 526–535 (1974)

    Article  ADS  Google Scholar 

  4. Fine, A.: Hidden variables, joint probability, and the Bell inequalities. Phys. Rev. Lett. 48, 291–295 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  5. Kolmogorov, A.: Foundations of the Theory of Probability. Chelsea, New York (1956)

    MATH  Google Scholar 

  6. Kochen, S., Specker, F.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87 (1967)

    MATH  MathSciNet  Google Scholar 

  7. Laudisa, F.: Contextualism and nonlocality in the algebra of EPR observables. Philos. Sci. 64, 478–496 (1997)

    Article  MathSciNet  Google Scholar 

  8. Spekkens, R.W., Buzacott, D.H., Keehn, A.J., Toner, B., Pryde, G.J.: Universality of state-independent violation of correlation inequalities for noncontextual theories. Phys. Rev. Lett. 102, 010401 (2009)

    Article  ADS  Google Scholar 

  9. Kirchmair, G., Zhringer, F., Gerritsma, R., Kleinmann, M., Ghne, O., Cabello, A., Blatt, R., Roos, C.F.: State-independent experimental test of quantum contextuality. Nature 460, 494–497 (2009)

    Article  ADS  Google Scholar 

  10. Badzia̧g, P., Bengtsson, I., Cabello, A., Pitowsky, I.: Universality of state-independent violation of correlation inequalities for noncontextual theories. Phys. Rev. Lett. 103, 050401 (2009)

    Google Scholar 

  11. Khrennikov, A.Y.: Contextual approach to quantum formalism. In: Fundamental Theories of Physics, vol. 160. Springer, Dordrecht (2009)

  12. Cabello, A.: Simple explanation of the quantum violation of a fundamental inequality. Phys. Rev. Lett. 110, 060402 (2013)

    Article  ADS  Google Scholar 

  13. Dzhafarov, E.N., Kujala, J.V.: All-possible-couplings approach to measuring probabilistic context. PLoS ONE 8(5), e61712 (2013). doi:10.1371/journal.pone.0061712

  14. Dzhafarov, E.N., Kujala, J.V.: A qualified Kolmogorovian account of probabilistic contextuality. In: Lecture Notes in Computer Science (in press). arXiv:1304.4546

  15. Dzhafarov, E.N., Kujala, J.V.: Random variables recorded under mutually exclusive conditions: contextuality-by-Default. Adv. Cogn. Neurodyn. IV (in press). arXiv:1309.0962

  16. Bohm, D., Aharonov, Y.: Discussion of experimental proof for the paradox of Einstein. Rosen Podolski. Phys. Rev. 108, 1070–1076 (1957)

    Article  ADS  Google Scholar 

  17. Dzhafarov, E.N., Kujala, J.V.: On selective influences, marginal selectivity, and Bell/CHSH inequalities. Topics Cognit. Sci. 6, 121–128 (2014)

    Google Scholar 

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

    Google Scholar 

  19. Khrennikov, A.Y.: EPR–Bohm experiment and Bell’s inequality: quantum physics meets probability theory. Theor. Math. Phys. 157, 1448–1460 (2008)

    Google Scholar 

  20. Gudder, S.: On hidden-variable theories. J. Math. Phys. 2, 431–436 (1970)

    Article  ADS  MathSciNet  Google Scholar 

  21. Thorisson, H.: Coupling, Stationarity, and Regeneration. Springer, New York (2000)

    Book  MATH  Google Scholar 

  22. Landau, L.J.: Empirical two-point correlation functions. Found. Phys. 18, 449–460 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  23. Cabello, A.: How much larger quantum correlations are than classical ones. Phys. Rev. A 72, 12113 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  24. Kujala, J.V., Dzhafarov, E.N.: Testing for selectivity in the dependence of random variables on external factors. J. Math. Psychol. 52, 128–144 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. Tsirelson, B.S.: Quantum generalizations of Bell’s inequality. Lett. Math. Phys. 4, 93–100 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  26. Landau, L.J.: On the violation of Bell’s inequality in quantum theory. Phys. Lett. A 120, 54–56 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  27. Dzhafarov, E.N., Kujala, J.V.: Selectivity in probabilistic causality: where psychology runs into quantum physics. J. Math. Psychol. 56, 54–63 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  28. Dzhafarov, E.N., Kujala, J.V.: Quantum entanglement and the issue of selective influences in psychology: an overview. In: Lecture Notes in Computer Science, vol. 7620, pp. 184–195 (2012)

  29. Dzhafarov, E.N., Kujala, J.V.: Order-distance and other metric-like functions on jointly distributed random variables. Proc. Am. Math. Soc. 141, 3291–3301 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  30. Klyachko, A.A., Can, M.A., Binicioglu, S., Shumovsky, A.S.: Simple test for hidden variables in spin-1 systems. Phys. Rev. Lett. 101, 020403 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  31. Avis, D., Fischer, P., Hilbert, A., Khrennikov, A.: Single, complete, probability spaces consistent with EPR–Bohm–Bell experimental data. In: Khrennikov, A. (ed.) Foundations of Probability and Physics-5, AIP Conference Proceedings, vol. 750, pp. 294–301. AIP, Melville, New York (2009)

  32. Dzhafarov, E.N., Kujala, J.V.: Embedding quantum into classical: contextualization vs conditionalization. arXiv:1312.0097 (2013)

Download references

Acknowledgments

This work was supported by NSF Grant SES-1155956.

Author information

Authors and Affiliations

  1. Purdue University, West Lafayette, IN, USA

    Ehtibar N. Dzhafarov

  2. Department of Mathematical Information Technology, University of Jyväskylä, Jyväskylä, Finland

    Janne V. Kujala

Authors
  1. Ehtibar N. Dzhafarov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Janne V. Kujala
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ehtibar N. Dzhafarov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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). https://doi.org/10.1007/s10701-014-9783-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-014-9783-3

Keywords

Search

Navigation