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Contextuality and Dichotomizations of Random Variables

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Abstract

The Contextuality-by-Default approach to determining and measuring the (non)contextuality of a system of random variables requires that every random variable in the system be represented by an equivalent set of dichotomous random variables. In this paper we present general principles that justify the use of dichotomizations and determine their choice. The main idea in choosing dichotomizations is that if the set of possible values of a random variable is endowed with a pre-topology (V-space), then the allowable dichotomizations split the space of possible values into two linked subsets (“linkedness” being a weak form of pre-topological connectedness). We primarily focus on two types of random variables most often encountered in practice: categorical and real-valued ones (including continuous random variables, greatly underrepresented in the contextuality literature). A categorical variable (one with a finite number of unordered values) is represented by all of its possible dichotomizations. If the values of a random variable are real numbers, then they are dichotomized by intervals above and below a variable cut point.

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Notes

  1. We use script letters, \({\mathcal{R}},{\mathcal{R}}_{q},\) etc., for a set of random variables if they are not necessarily jointly distributed. If, as in \(R^{c},\) all elements of a set are jointly distributed, then \(R^{c}\) is a random variable in its own right, and we can use ordinary italics.

  2. This means that the choice of a domain space for \(\{Y_{i}:i\in I\}\) is irrelevant. There is a canonical way of constructing this space. Let the set of values of \(X_{i}\) be \(E_{i},\) with the induced sigma-algebra \(\Sigma_{q}.\) Then the domain space for \(\{Y_{i}:i\in I\}\) can be chosen as the set \(\prod_{i\in I}E_{i}\) endowed with \(\bigotimes_{i\in I}\Sigma_{i}.\) With this choice, every \(Y_{i}\) is a coordinate projection function.

  3. Denoting the measurable spaces in which \(X_{1},X_{2}\) are taking their values by \(\left( E_{1},\Sigma_{1}\right)\) and \(\left( E_{2},\Sigma_{2}\right),\) the definition of maximal coupling is predicated on the assumption that the diagonal set \(\left\{ \left( x,y\right) \in E_{1}\times E_{2}:x=y\right\}\) is measurable in \(\Sigma_{1}\otimes \Sigma_{2}.\) For dichotomous random variables this condition is satisfied trivially.

  4. The notion of a coupling in the traditional approach is not used explicitly (see [12, 13] for difficulties this creates). To our knowledge, Thorisson [35, Chap. 1, Sect. 10.4, p. 29] was first to use couplings in contextuality analysis of a system. In CbD, they play a central role.

  5. There is a slight abuse of notation here: we use the same symbol \(\prec\) to indicate the format relation of both \({\mathcal{R}}\) and \({\mathcal{D}}.\)

  6. This is essentially a weak form of pre-topological connectedness, but we avoid using the latter word to prevent confusing it with its use in CbD, in such terms as “multimaximally connected” or “consistently connected,” derived from the term “connection” for the set of random variables sharing a content.

References

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Bell, J.: On the problem of hidden variables in quantum mechanics. Rev. Modern Phys. 38, 447 (1966)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  5. Dzhafarov, E.N., Kujala, J.V., Larsson, J.-Å.: Contextuality in three types of quantum-mechanical systems. Found. Phys. 7, 762–782 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  6. Abramsky, S., Brandenburger, A.: The sheaf-theoretic structure of non-locality and contextuality. New J. Phys. 13, 113036–113075 (2011)

    Article  ADS  Google Scholar 

  7. Abramsky, S., Barbosa, R.S., Mansfield, S.: The contextual fraction as a measure of contextuality. Phys. Rev. Lett. 119, 050504 (2017)

    Article  ADS  Google Scholar 

  8. Kujala, J.V., Dzhafarov, E.N., Larsson, J.-Å.: Necessary and sufficient conditions for extended noncontextuality in a broad class of quantum mechanical systems. Phys. Rev. Lett. 115, 150401 (2015)

    Article  ADS  Google Scholar 

  9. Kujala, J.V., Dzhafarov, E.N.: Proof of a conjecture on contextuality in cyclic systems with binary variables. Found. Phys. 46, 282 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  10. Dzhafarov, E.N., Cervantes, V.H., Kujala, J.V.: Contextuality in canonical systems of random variables. Philos. Trans. R. Soc. A 375, 20160389 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  11. Dzhafarov, E.N.: Replacing nothing with something special: contextuality-by-Default and dummy measurements. In: Quantum Foundations, Probability and Information, pp. 39–44. Springer, Berlin (2017)

  12. Dzhafarov, E. N., Kujala, J. V.: Probabilistic foundations of contextuality. Fortsch. Phys. - Prog. Phys. 65, 1600040 (1-11) (2017)

  13. Dzhafarov, E.N.: On joint distributions, counterfactual values, and hidden variables in understanding contextuality. Phil. Trans. Roy. Soc. A 377, 20190144 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  14. Kujala, J.V., Dzhafarov, E.N.: Measures of contextuality and noncontextuality. Phil. Trans. Roy. Soc. A 377, 20190149 (2019)

    Article  ADS  Google Scholar 

  15. Dzhafarov, E. N., Kujala, J. V., Cervantes, V. H.: Contextuality and noncontextuality measures and generalized Bell inequalities for cyclic systems. Phys. Rev. A 101, 042119 (2020). [Erratum Notes in Phys. Rev. A 101, 069902 (2020) and Phys. Rev. A 103, 059901 (2021)]

  16. Dzhafarov, E.N., Kujala, J.V.: Contextuality analysis of the double slit experiment (with a glimpse into three slits). Entropy 20, 278 (2018). https://doi.org/10.3390/e20040278

    Article  ADS  Google Scholar 

  17. Dzhafarov, E.N., Zhang, R., Kujala, J.V.: Is there contextuality in behavioral and social systems? Philos. Trans. R. Soc. A 374, 20150099 (2016)

    Article  ADS  Google Scholar 

  18. Wang, Z., Solloway, T., Shiffrin, R.M., Busemeyer, J.R.: Context effects produced by question orders reveal quantum nature of human judgments. Proc. Natl Acad. Sci. U.S.A. 111, 9431 (2014)

    Article  ADS  Google Scholar 

  19. Lapkiewicz, R., Li, P., Schaeff, C., Langford, N.K., Ramelow, S., Wieśniak, M., Zeilinger, A.: Experimental non-classicality of an indivisible quantum system. Nature 474, 490 (2011)

    Article  Google Scholar 

  20. Klyachko, A.A., Can, M.A., Binicioğlu, S., Shumovsky, A.S.: A simple test for hidden variables in spin-1 system. Phys. Rev. Lett. 101, 020403 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  21. Arias, M., Canas, G., Gomez, E.S., Barra, J.B., Xavier, G.B., Lima, G., D’Ambrosio, V., Baccari, F., Sciarrino, F., Cabello, A.: Testing noncontextuality inequalities that are building blocks of quantum correlations. Phys. Rev. A 92, 032126 (2015)

    Article  ADS  Google Scholar 

  22. Crespi, A., Bentivegna, M., Pitsios, I., Rusc, D., Poderini, D., Carvacho, G., D’Ambrosio, V., Cabello, A., Sciarrino, F., Osellame, R.: Single-photon quantum contextuality on a chip. ACS Photonics 4, 2807–2812 (2017)

    Article  Google Scholar 

  23. Zhan, X., Kurzyński, P., Kaszlikowski, D., Wang, K., Bian, Zh., Zhang, Y., Xue, P.: Experimental detection of information deficit in a photonic contextuality scenario. Phys. Rev. Lett. 119, 220403 (2017)

    Article  ADS  Google Scholar 

  24. Flühmann, C., Negnevitsky, V., Marinelli, M., Home, J.P.: Sequential modular position and momentum measurements of a trapped ion mechanical oscillator. Phys. Rev. 8, 021001 (2018)

    Article  Google Scholar 

  25. Leupold, F.M., Malinowski, M., Zhang, C., Negnevitsky, V., Alonso, J., Cabello, A., Home, J.P.: Sustained state-independent quantum contextual correlations from a single ion. Phys. Rev. Lett. 120, 180401 (2018)

    Article  ADS  Google Scholar 

  26. Malinowski, M., Zhang, C., Leupold, F. M., Alonso, J., Home, J. P., Cabello, A.: Probing the limits of correlations in an indivisible quantum system (2018). arXiv:1712.06494v2

  27. Bacciagaluppi, G. (2015) Einsten, Bohm, and Leggett-Garg. In: Dzhafarov, E.N., Jordan, S., Zhang, R., Cervantes, V. (eds.) Contextuality from Quantum Physics to Psychology, pp. 63–76. World Scientific, New Jersey (2015)

  28. Bacciagaluppi, G.: Leggett–Garg inequalities, pilot waves and contextuality. Int. J. Quant. Found. 1, 1 (2015)

    Google Scholar 

  29. Leggett, A.J., Garg, A.: Quantum mechanics versus macroscopic realism: is the flux there when nobody looks? Phys. Rev. Lett. 54, 857 (1985)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  31. Budroni, C.: Temporal Quantum Correlations and Hidden Variable Models. Springer, Heidelberg (2016)

    Book  Google Scholar 

  32. Cervantes, V.H., Dzhafarov, E.N.: Snow Queen is evil and beautiful: experimental evidence for probabilistic contextuality in human choices. Decision 5, 193–204 (2018)

    Article  Google Scholar 

  33. Basieva, I., Cervantes, V.H., Dzhafarov, E.N., Khrennikov, A.: True contextuality beats direct influences in human decision making. J. Exp. Psychol. Gen. 148, 1925–1937 (2019)

    Article  Google Scholar 

  34. Dzhafarov, E.N., Cervantes, V.H.: True contextuality in a psychophysical experiment. J. Math. Psychcol. 91, 119–127 (2019)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  36. Williams, D.: Probability with Martingales. Cambridge University Press, Cambridge (1991)

    Book  Google Scholar 

  37. Suppes, P., Zanotti, M.: When are probabilistic explanations possible? Synthesis 48, 191 (1981)

    Article  MathSciNet  Google Scholar 

  38. Fine, A.: Joint distributions, quantum correlations, and commuting observables. J. Math. Phys. 23, 1306 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  39. Araújo, M., Quintino, M.T., Budroni, C., Cunha, M.T., Cabello, A.: All noncontextuality inequalities for the n-cycle scenario. Phys. Rev. A 88, 022118 (2013)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  41. Kurzynski, P., Cabello, A., Kaszlikowski, D.: Fundamental monogamy relation between contextuality and nonlocality. Phys. Rev. Lett. 112, 100401 (2014)

    Article  ADS  Google Scholar 

  42. Kurzynski, P., Ramanathan, R., Kaszlikowski, D.: Entropic test of quantum contextuality. Phys. Rev. Lett. 109, 020404 (2012)

    Article  ADS  Google Scholar 

  43. Dzhafarov, E.N., Kujala, J.V.: Contextuality-by-Default 2.0: systems with binary random variables. In: de Barros, J.A., Coecke, B., Pothos, E. (eds.) Lecture Notes in Computer Science, vol. 10106, pp. 16–32. Springer, Dordrecht (2017)

  44. Sierpinski, W.: General Topology. University of Toronto Press, Toronto (1956)

    MATH  Google Scholar 

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Authors and Affiliations

  1. University of Turku, Turku, Finland

    Janne V. Kujala

  2. Purdue University, West Lafayette, USA

    Ehtibar N. Dzhafarov

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  1. Janne V. Kujala
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Kujala, J.V., Dzhafarov, E.N. Contextuality and Dichotomizations of Random Variables. Found Phys 52, 13 (2022). https://doi.org/10.1007/s10701-021-00527-9

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