Quantum Logics Defined by Divisibility Conditions

  • Michal HrochEmail author
  • Mirko Navara
  • Pavel Pták


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.)


Set-representable quantum logic State Extensions of states 



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.


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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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