On Extension of Joint Distribution Functions on Quantum Logics


The problem of extension of joint distribution functions (s-maps) on quantum logics is studied. Necessary and sufficient conditions for the extension of bivariate s-maps to trivariate s-maps are given. However, it is shown that these conditions are not sufficient for extending trivariate s-maps to 4-variate s-maps.

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  1. 1.

    pn(a,...,a) = pn(π(a,..., 1L)) = pn(π(a,..., 1L, 1L)) = pn(π(a, 1L,..., 1L)), where π is arbitrary permutation


  1. 1.

    Al-Adilee, A. M.: A note to a common cause in quantum logic. Slovak Journal of Civil Engineering XVIII(4), 24–29 (2010).

  2. 2.

    Al-Adilee, A. M., Nánásiová, O.: Copula and s-map on a quantum logic, vol. 179. (2009)

  3. 3.

    Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Annals Math. 37, 823–843 (1936).

  4. 4.

    Bohdalová, M., Kalina, M., Nánásiová, O.: Granger causality from a different viewpoint. Informační, bulletin České statistické společnosti 27(2), 23–28 (2016)

  5. 5.

    Dvurečenskij, A., Pulmannová, S.: Connection between joint distribution and compatibility. Reports on Mathematical Physics 19, 349–359 (1984).

  6. 6.

    Dvurečenskij, A., Pulmannová, S.: New trends in quantum structures. Kluwer Academic Publishers, Dordrecht and Ister Science, Bratislava (2000)

  7. 7.

    Garg, A., Mermin, N. D., Farkas’s, Lemma: the nature of reality statistical implications of quantum correlations. Foundation of Physsics 14(1), 1–39 (1984).

  8. 8.

    Garola, C.: An epistemic interpretation of quantum probability via contextuality. Foundations of Science. (2018)

  9. 9.

    Greechie, R.J.: Orthomodular lattices admitting no states. Journal of Combinatorial Theory Series A 10, 119–132 (1971).

  10. 10.

    Gleason, A.: Measures on the closed subspaces of a Hilbert space. Indiana Univ. Math. J. 6(4), 885–893 (1957).

  11. 11.

    Gudder, S.: Joint distributions of observables. J. Math. Mech. 18, 325–335 (1968)

  12. 12.

    Kalmbach, G.: Orthomodular lattices. Academic Press, London (1983)

  13. 13.

    Khrennikov, A., Basieva, I.: Possibility to agree on disagree from quantum information and decision making. J. Math. Psychol. 62(3), 1–5 (2014).

  14. 14.

    Khrennikov, A.: CHSH inequality: quantum probabilities as classical conditional probabilities. Foundation of Physics 45, 711–725 (2015).

  15. 15.

    Khrennikov, A., Alodjants, A.: Classical (local and contextual) probability model for Bohm-Bell type experiments: no-signaling as independence of random variables. Entropy 21(2), 157 (2019).

  16. 16.

    Malvestuto, F.M.: Existence of extensions and product extensions for discrete probability distributions. Discret. Math. 69, 61–77 (1988).

  17. 17.

    Nánásiová, O.: Map for simultaneous measurements for a quantum logic. Int. J. Theor. Phys. 42, 1889–1903 (2003)

  18. 18.

    Nánásiová, O.: Principle conditioning. Int. J. Theor. Phys. 43, 1383–1395 (2004)

  19. 19.

    Nánásiová, O., Kalina, M.: Calculus for non-compatible observables, construction through conditional states. Int. J. Theor. Phys. 54, 506–518 (2014).

  20. 20.

    Nánásiová, O., Khrennikov, A.: Representation theorem of observables on a quantum system. Int. J. Theor. Phys. 45, 469–482 (2006)

  21. 21.

    Nánásiová, O., Khrennikov, A.: Compatibility and marginality. Int. J. Theor. Phys. 46, 1083–1095 (2007)

  22. 22.

    Nánásiová, O., Pulmannová, S.: S-map and tracial states. Inf. Sci. 179, 515–520 (2009).

  23. 23.

    Nánásiová, O., Valášková, L.: Maps on a quantum logic. Soft. Comput. 14, 1047–1052 (2010).

  24. 24.

    Nánásiová, O., Valášková, L.: Marginality and triangle inequality. Int. J. Theor. Phys. 49(12), 3199–3208 (2010)

  25. 25.

    Nánásiová, O., Valášková, L., Čerňanová, V.: Probability measures and projections on quantum logics. In: Kulczycki, P., et al. (eds.) ITSRCP 2018. Advances in intelligent systems and computing, 945. Springer, Cham (2020)

  26. 26.

    Nielsen, R.B.: An introduction to copulas. Springer, New York (1999)

  27. 27.

    Pavičić, M.: Exhaustive generation of orthomodular lattices with exactly one nonquantum state. Reports on Mathematical Physics 64, 417–428 (2009).

  28. 28.

    Pták, P., Pulmannová, S.: Orthomodular structures as quantum logics. Kluwer, Dortrecht and Veda, Bratislava (1991)

  29. 29.

    Pulmannová, S.: Relative compatibility and joint distributions of observables. Foundation of Physics 10, 641–653 (1980)

  30. 30.

    Pykacz, J., Fra̧ckiewicz, P: The problem of conjunction and disjunction in quantum logics. Int. J. Theor. Phys. 56(12), 3963–3970 (2017).

  31. 31.

    Pykacz, J., Valášková, L., Nánásiová, O.: Bell-type inequalities for bivariate maps on orthomodular lattices. Foundation of Physics 45, 900–913 (2015).

  32. 32.

    Sozzo, S.: Conjunction and negation of natural concepts: A quantum-theoretic modeling. J. Math. Psychol. 66, 83–102 (2015).

  33. 33.

    Svozil, K.: Faithful orthogonal representations of graphs from partition logics. arXiv:1810.10423

  34. 34.

    Vlach, M.: Conditions for the existence of solutions of the three-dimensional planar transportation problem. Discrete Appplied Mathematics 13, 61–78 (1986).

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O. N. and K. Č. thank for the support of the VEGA grant no. 1/0159/17, L’. V. thanks for the support of the VEGA grant no. 1/0420/15, and J. P. thanks Polish National Agency for Academic Exchange for the support which allowed him to visit Slovak Technical University in Bratislava, no. PPN/BIL/2018/1/87/SVK/UMOWA/1.

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Correspondence to Ol’ga Nánásiová.

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Nánásiová, O., Pykacz, J., Valášková, L. et al. On Extension of Joint Distribution Functions on Quantum Logics. Int J Theor Phys 59, 274–291 (2020) doi:10.1007/s10773-019-04322-1

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  • Quantum logic
  • Quantum probability
  • Joint distribution
  • S-map
  • State