Advertisement

Probability Measures and Projections on Quantum Logics

  • Oľga NánásiováEmail author
  • Viera Čerňanová
  • Ľubica Valášková
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 945)

Abstract

The present paper is devoted to modelling of a probability measure of logical connectives on a quantum logic (QL), via a G-map, which is a special map on it. We follow the work in which the probability of logical conjunction, disjunction and symmetric difference and their negations for non-compatible propositions are studied.

We study such a G-map on quantum logics, which is a probability measure of a projection and show, that unlike classical (Boolean) logic, probability measure of projections on a quantum logic are not necessarilly pure projections.

We compare properties of a G-map on QLs with properties of a probability measure related to logical connectives on a Boolean algebra.

Keywords

Logical connectives Orthomodular lattice Quantum logic Probability measure State 

Notes

Acknowledgement

The author (O. Nánásiová) would like to thank for the support of the VEGA grant agency by means of grant VEGA 1/0710/15 and VEGA 1/0159/17 and the author (L. Valášková) would like to thank for the support of VEGA 1/0420/15.

References

  1. 1.
    Nánásiová, O., Čerňanová, V., Valášková, Ľ.: Probability measures and projections on quantum logics. In: Kulczycki, P., Kowalski, P.A., Łukasik, S. (eds.) Contemporary Computational Science, p. 78. AGH-UST Press, Cracow (2018)Google Scholar
  2. 2.
    Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37(4), 823–843 (1936). second seriesMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bunce, L.J., Navara, M., Pták, P., Maitland Wright, D.: Quantum logics with Jauch-Piron states. Q. J. Math. 36(3), 261–271 (1985)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Springer, Dordrecht (2000). ISBN 978-94-017-2422-7CrossRefGoogle Scholar
  5. 5.
    Dvurečenskij, A., Pulmannová, S.: Connection between joint distribution and compatibility. Rep. Math. Phys. 19(3), 349–359 (1984)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Sozzo, S.: Conjunction and negation of natural concepts: a quantum-theoretic modeling S Sozzo. J. Math. Psychol. 66, 83–102 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Herman, L., Marsden, L., Piziak, R.: Implication connectives in orthomodula lattices. Notre Dame J. Formal Logic XVI(3), 305–326 (1975)CrossRefGoogle Scholar
  8. 8.
    Jauch, J.M., Piron, C.: On the structure of quantal proposition systems. Helv. Phys. Acta 42, 842–848 (1969)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Kalina, M., Nánásiová, O.: Calculus for non-compatible observables, construction through conditional states. Int. J. Theor. Phys. 54(2), 506–518 (2014)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Khrennikov, A.Y.: EPR-Bohm experiment and Bell’s inequality: quantum physics meets probability theory. TMF 157(1), 99–115 (2008). (Mi tmf6266)CrossRefGoogle Scholar
  11. 11.
    Khrennikov, A.: Violation of Bell’s inequality and non-Kolmogorovness. In: Accardi, L., et al. (eds.) Foundations of Probability and Physics-5. American Institute of Physics, Mellville (2009)Google Scholar
  12. 12.
    Nánásiová, O.: Principle conditionig. Int. J. Theor. Phys. 43(7–8), 1757–1768 (2004)CrossRefGoogle Scholar
  13. 13.
    Nánásiová, O.: Map for simultaneous measurements for a quantum logic. Int. J. Theor. Phys. 42(9), 1889–1903 (2003)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Nánásiová, O., Drobná, E., Valášková, Ľ.: Quantum logics and bivariable functions. Kybernetika 46(6), 982–995 (2010)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Nánásiová, O., Khrennikov, A.: Representation theorem of observables on a quantum system. Int. J. Theor. Phys. 45(3), 469–482 (2006)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Nánásiová, O., Pykacz, J.: Modelling of uncertainty and bi-variable maps. J. Electr. Eng. 67(3), 169–176 (2016)Google Scholar
  17. 17.
    Nánásiová, O., Valášková, Ľ.: Maps on a quantum logic. Soft Comput. 14(10), 1047–1052 (2010)CrossRefGoogle Scholar
  18. 18.
    Nánásiová, O., Valášková, Ľ.: Marginality and triangle inequality. Int. J. Theor. Phys. 49(12), 3199–3208 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Pavičić, M., Megill, N.D.: Is quantum logic a logic? In: Engesser, K., Gabbay, D., Lehmann, D. (eds.) Handbook of Quantum Logic and Quantum Structures, pp. 23–47. Elsevier, Amsterdam (2009)CrossRefGoogle Scholar
  20. 20.
    Pavičić, M.: Classical logic and quantum logic with multiple and common lattice models. Hindawi Publishing Corporation Advances in Mathematical Physics volume 2016, Article ID 6830685, 12 pages (2016)Google Scholar
  21. 21.
    Pavičić, M.: Exhaustive generation of orthomodular lattices with exactly one nonquantum state. Rep. Math. Phys. 64, 417–428 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics. Springer, Netherlands (1991)zbMATHGoogle Scholar
  23. 23.
    Pitovsky, I.: Quantum Probability-Quantum Logic. Springer, Berlin (1989)Google Scholar
  24. 24.
    Pykacz, J., Frackiewicz, P.: The problem of conjunction and disjunction in quantum logics. Int. J. Theor. Phys. 56(12), 3963–3970 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Pykacz, J., Valášková, L., Nánásiová, O.: Bell-type inequalities for bivariate maps on orthomodular lattices. Found. Phys. 45(8), 900–913 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Sergioli, G., Bosyk, G.M., Santucci, E., Giuntini, R.: A quantum-inspired version of the classification problem. Int. J. Theor. Phys. 56, 3880–3888 (2017).  https://doi.org/10.1007/s10773-017-3371-1MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Oľga Nánásiová
    • 1
    Email author
  • Viera Čerňanová
    • 2
  • Ľubica Valášková
    • 3
  1. 1.Institute of Computer Science and MathematicsSlovak University of TechnologyBratislavaSlovakia
  2. 2.Department of Mathematics and Computer Science, Faculty of EducationTrnava UniversityTrnavaSlovakia
  3. 3.Department of Mathematics and Descriptive GeometrySlovak University of TechnologyBratislavaSlovakia

Personalised recommendations