Foundations of Physics

, Volume 45, Issue 7, pp 762–782 | Cite as

Contextuality in Three Types of Quantum-Mechanical Systems

  • Ehtibar N. Dzhafarov
  • Janne V. Kujala
  • Jan-Åke Larsson


We present a formal theory of contextuality for a set of random variables grouped into different subsets (contexts) corresponding to different, mutually incompatible conditions. Within each context the random variables are jointly distributed, but across different contexts they are stochastically unrelated. The theory of contextuality is based on the analysis of the extent to which some of these random variables can be viewed as preserving their identity across different contexts when one considers all possible joint distributions imposed on the entire set of the random variables. We illustrate the theory on three systems of traditional interest in quantum physics (and also in non-physical, e.g., behavioral studies). These are systems of the Klyachko–Can–Binicioglu–Shumovsky-type, Einstein–Podolsky–Rosen–Bell-type, and Suppes–Zanotti–Leggett–Garg-type. Listed in this order, each of them is formally a special case of the previous one. For each of them we derive necessary and sufficient conditions for contextuality while allowing for experimental errors and contextual biases or signaling. Based on the same principles that underly these derivations we also propose a measure for the degree of contextuality and compute it for the three systems in question.


CHSH inequalities Contextuality Klyachko inequalities Leggett–Garg inequalities  Probabilistic couplings Signaling 



This work is supported by NSF Grant SES-1155956 and AFOSR Grant FA9550-14-1-0318. We have benefited from collaboration with J. Acacio de Barros, and Gary Oas, as well as from discussions with Samson Abramsky, Guido Bacciagaluppi, and Andrei Khrennikov. An abridged version of this paper was presented at the Purdue Winer Memorial Lectures in November 2014.


  1. 1.
    Bacciagaluppi, G.: Leggett–Garg inequalities, pilot waves and contextuality. Int. J. Quant. Found. 1, 1–17 (2015)Google Scholar
  2. 2.
    Bell, J.: On the Einstein–Podolsky–Rosen paradox. Physics 1, 195–200 (1964)Google Scholar
  3. 3.
    Bell, J.: On the problem of hidden variables in quantum mechanics. Rev. Modern Phys. 38, 447–453 (1966)zbMATHMathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Cereceda, J.: Quantum mechanical probabilities and general probabilistic constraints for Einstein–Podolsky–Rosen–Bohm experiments. Found. Phys. Lett. 13, 427–442 (2000)MathSciNetCrossRefGoogle Scholar
  5. 5.
    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)ADSCrossRefGoogle Scholar
  6. 6.
    Clauser, J.F., Horne, M.A.: Experimental consequences of objective local theories. Phys. Rev. D 10, 526–535 (1974)ADSCrossRefGoogle Scholar
  7. 7.
    de Barros, J.A., Dzhafarov, E.N., Kujala, J.V., Oas, G.: Measuring observable quantum contextuality (2014). arXiv:1406.3088
  8. 8.
    Dzhafarov, E.N., Kujala, J.V.: The joint distribution criterion and the distance tests for selective probabilistic causality. Frontiers Quant. Psychol. Meas. 1, 151 (2010). doi: 10.3389/fpsyg.2010.00151 Google Scholar
  9. 9.
    Dzhafarov, E.N., Kujala, J.V.: Probability, random variables, and selectivity (2013). arXiv:1312.2239
  10. 10.
    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 ADSCrossRefGoogle Scholar
  11. 11.
    Dzhafarov, E.N., Kujala, J.V.: Order-distance and other metric-like functions on jointly distributed random variables. Proc. Am. Math. Soc. 141(9), 3291–3301 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Dzhafarov, E.N., Kujala, J.V.: Generalizing Bell-type and Leggett–Garg-type inequalities to systems with signaling (2014). arXiv:1407.2886
  13. 13.
    Dzhafarov, E.N., Kujala, J.V.: Probabilistic contextuality in EPR/Bohm-type systems with signaling allowed (2014). arXiv:1406.0243
  14. 14.
    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)zbMATHMathSciNetADSCrossRefGoogle Scholar
  15. 15.
    Dzhafarov, E.N., Kujala, J.V.: Embedding quantum into classical: contextualization vs conditionalization. PLoS One 9(3), e92818 (2014). doi: 10.1371/journal.pone.0092818 MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    Dzhafarov, E.N., Kujala, J.V.: A Qualified Kolmogorovian account of probabilistic contextuality. Lect. Not. Comp. Sci. Springer, Berlin (2014)Google Scholar
  17. 17.
    Dzhafarov, E.N., Kujala, J.V.: Contextuality is about identity of random variables. Phys. Scripta. T163, 014009 (2015)ADSCrossRefGoogle Scholar
  18. 18.
    Dzhafarov, E.N., Kujala, J.V.: Random variables recorded under mutually exclusive conditions: contextuality-by-default. In: Liljenström, H. (ed.) Advances in Cognitive Neurodynamics IV, pp. 405–410 (2015)Google Scholar
  19. 19.
    Fine, A.: Hidden variables, joint probability, and the Bell inequalities. Phys. Rev. Lett. 48, 291–295 (1982)MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    Khrennikov, A.: The principle of supplementarity: a contextual probabilistic viewpoint to complementarity, the interference of probabilities, and the incompatibility of variables in quantum mechanics. Found. Phys. 35, 1655–1693 (2005)zbMATHMathSciNetADSCrossRefGoogle Scholar
  21. 21.
    Khrennikov, A.: Bell–Boole inequality: nonlocality or probabilistic incompatibility of random variables? Entropy 10, 19–32 (2008)zbMATHMathSciNetADSCrossRefGoogle Scholar
  22. 22.
    Khrennikov, A.: Contextual Approach to Quantum Formalism. Springer, Berlin (2009)zbMATHCrossRefGoogle Scholar
  23. 23.
    Klyachko, A.A., Can, M.A., Binicioglu, S., Shumovsky, A.S.: A simple test for hidden variables in spin-1 system. Phys. Rev. Lett. 101, 020403 (2008)MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    Kujala, J.V., Dzhafarov, E.N., Larsson, J.-Å.: Necessary and sufficient conditions for maximal noncontextuality in a broad class of quantum mechanical systems (2014). arXiv:1412.4724
  25. 25.
    Larsson, J.-Å.: A Kochen–Specker inequality. Europhys. Lett. 58, 799–805 (2002)ADSCrossRefGoogle Scholar
  26. 26.
    Leggett, A.J., Garg, A.: Quantum mechanics versus macroscopic realism: is the flux there when nobody looks? Phys. Rev. Lett. 54, 857–860 (1985)MathSciNetADSCrossRefGoogle Scholar
  27. 27.
    Masanes, L., Acin, A., Gisin, N.: General properties of nonsignaling theories. Phys. Rev. A 73, 012112 (2006)ADSCrossRefGoogle Scholar
  28. 28.
    Popescu, S., Rohrlich, D.: Quantum nonlocality as an axiom. Found. Phys. 24, 379–385 (1994)MathSciNetADSCrossRefGoogle Scholar
  29. 29.
    Specker, E.: Die Logik Nicht Gleichzeitig Entscheidbarer Aussagen. Dialectica 14, 239–246 (1960). (English translation by Seevinck, M.P., available as arXiv:1103.4537.)
  30. 30.
    Suppes, P., Zanotti, M.: When are probabilistic explanations possible? Synthese 48, 191–199 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Svozil, K.: How much contextuality? Nat. Comput. 11, 261–265 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Ehtibar N. Dzhafarov
    • 1
  • Janne V. Kujala
    • 2
  • Jan-Åke Larsson
    • 3
  1. 1.Purdue UniversityWest LafayetteUSA
  2. 2.University of JyväskyläJyväskyläFinland
  3. 3.Linköping UniversityLinköpingSweden

Personalised recommendations