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Quantum Logics Defined by Divisibility Conditions

  • Michal Hroch
  • Mirko Navara
  • Pavel Pták
Article

Abstract

Let p be a prime number and let S be a countable set. Let us consider the collection \({Div}_{p}^{S}\) of all subsets of S whose cardinalities are multiples of p and the complements of such sets. Then the collection \({Div}_{p}^{S}\) constitutes a (set-representable) quantum logic (i.e., \({Div}_{p}^{S}\) is an orthomodular poset). We show in this note that each state on \({Div}_{p}^{S}\) can be extended over the Boolean algebra expS of all subsets of S. We also prove that all pure states on \({Div}_{p}^{S}\) are two-valued. (If we lend to a main result a possible interpretation in terms of quantum entities, the logics \({Div}_{p}^{S}\) have higher degree of noncompatibility but somewhat classical states.)

Keywords

Set-representable quantum logic State Extensions of states 

Notes

Acknowledgments

The first author was supported by the Grant Agency of the Czech Technical University in Prague, grant No. SGS18/131/OHK3/2T/13. The second and the third author were supported by the project OPVVV CAAS CZ.02.1.01/0.0/0.0/16_019/0000778.

References

  1. 1.
    Bikchentaev, A., Navara, M.: States on symmetric logics: extensions. Mathematica Slovaca 66(2), 359–366 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Gudder, S.: Stochastic Methods in Quantum Mechanics. North-Holland, New York (1979)zbMATHGoogle Scholar
  3. 3.
    Horn, A., Tarski, A.: Measures in Boolean algebras. Trans. Am. Math. Soc. 64, 467–497 (1948)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hroch, M., Pták, P.: Concrete quantum logics, Δ-logics, states and Δ-states. Int. J. Theor. Phys. 56(12), 3852–3659 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Navara, M., Pták, P.: Two-valued measures on σ-classes. Časopis pro Pěstování Matematiky 108, 225–229 (1983)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Ovchinnikov, P.G.: Measures on finite concrete logics. Proc. Am. Math. Soc. 127 (7), 1957–1966 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Prather, R.E.: Generating the k-subsets of an n-set. Am. Math. Mon. 87(9), 740–743 (1980)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics. Kluwer, Dordrecht (1991)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Electrical EngineeringCzech Technical University in PraguePragueCzech Republic
  2. 2.CMP, Department of Cybernetics, Faculty of Electrical EngineeringCzech Technical University in PraguePragueCzech Republic

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