Advertisement

Quantum Structures Without Group-Valued Measures

  • Mirko NavaraEmail author
  • Václav Voráček
Article

Abstract

Quantum structures with small dimensions of state spaces are not only mathematical curiosities. They enriched the mathematical theory by new tools. We have significantly optimized some of these constructions. Related questions have been studied also in graph theory.

Keywords

Orthomodular lattice Orthomodular poset Orthoalgebra Effect algebra State Probability measure Group-valued measure Girth 

Notes

Acknowledgements

The authors thank to Mladen Pavičić and Foat F. Sultanbekov for their valuable remarks. The work was supported from European Regional Development Fund-Project “Center for Advanced Applied Science” (No. CZ.02.1.01/0.0/0.0/16_/0000778).

References

  1. 1.
    Beran, L.: Orthomodular Lattices. Algebraic Approach. Academia, Praha (1984)zbMATHGoogle Scholar
  2. 2.
    Dichtl, M.: Astroids and pastings. Algebra Universalis 18, 380–385 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer/Dordrecht, Ister/Bratislava (2000)CrossRefzbMATHGoogle Scholar
  4. 4.
    Greechie, R.J.: Orthomodular lattices admitting no states. J. Combin. Theory Ser. A 10, 119–132 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gudder, S.P.: Quantum Probability. Academic Press (1988)Google Scholar
  6. 6.
    Hamhalter, J.: Quantum Measure Theory. Kluwer, Dordrecht (2003)CrossRefzbMATHGoogle Scholar
  7. 7.
    Hamhalter, J., Navara, M., Pták, P.: States on orthoalgebras. Internat. J. Theoret. Phys. 34(8), 1439–1465 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Harding, J., Navara, M.: Embeddings into orthomodular lattices with given centers, state spaces and automorphism groups. Order 17(3), 239–254 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jenča, G.: The block structure of complete lattice ordered effect algebras. J. Aust. Math. Soc. 83, 181–216 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kalmbach, G.: Orthomodular Lattices. Academic Press, London (1983)zbMATHGoogle Scholar
  11. 11.
    Lazebnik, F., Verstraëte, J.: On hypergraphs of girth five. Electr. J. Comb. 10, R25 (2003)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Mayet, R.: Personal communication (1993)Google Scholar
  13. 13.
    Navara, M.: State space properties of finite logics. Czechoslovak Math. J. 37(112), 188–196 (1987)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Navara, M.: Independence of automorphism group, center and state space of quantum logics. Internat. J. Theoret. Phys. 31, 925–935 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Navara, M.: Descriptions of state spaces of orthomodular lattices. Math. Bohemica 117, 305–313 (1992)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Navara, M.: An orthomodular lattice admitting no group-valued measure. Proc. Amer. Math. Soc. 122, 7–12 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Navara, M.: Two descriptions of state spaces of orthomodular structures. Internat. J. Theoret. Phys. 38(12), 3163–3178 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Navara, M.: State spaces of quantum structures. Rend. Istit. Mat. Univ. Trieste 31(Suppl. 1), 143–201 (2000)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Navara, M.: Small quantum structures with small state spaces. Internat. J. Theoret. Phys. 47(1), 36–43 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Navara, M.: Existence of states on quantum structures. Information Sci. 179, 508–514 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Navara, M., Pták, P., Rogalewicz, V.: Enlargements of quantum logics. Pacific J. Math. 135, 361–369 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ovchinnikov, P.G.: On homogeneous finite Greechie’s logics permitting two-valued state. In: Teor. Funktsii, Prilozh. i Smezhnye Voprosy, pp. 167–168. Kazansk. Gos. Univ., Kazan (1999)Google Scholar
  23. 23.
    Pavičić, M.: Exhaustive generation of orthomodular lattices with exactly one nonquantum state. Rep. Math. Phys. 64, 417–428 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Pavičić, M., McKay, B.D., Megill, N.D., Fresl, K.: Graph approach to quantum systems. J. Math. Phys. 51(102103), 1–31 (2010).  https://doi.org/10.1063/1.3491766 MathSciNetzbMATHGoogle Scholar
  25. 25.
    Pták, P.: Exotic logics. Colloquium Math. 54, 1–7 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics. Kluwer Academic Publishers, Dordrecht (1991)zbMATHGoogle Scholar
  27. 27.
    Ray-Chaudhuri, D., Wilson, R.: The existence of resolvable block designs. Survey of combinatorial theory. In: Proc. Internat. Sympos., pp. 361–375. North-Holland (1973)Google Scholar
  28. 28.
    Riečanová, Z.: Generalization of blocks for D-lattices and lattice ordered effect algebras. Internat. J. Theoret. Phys. 39, 231–237 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Riečanová, Z.: Proper effect algebras admitting no states. Internat. J. Theoret. Phys. 40, 1683–1691 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Riečanová, Z.: Smearings of states defined on sharp elements onto effect algebras. Internat. J. Theoret. Phys. 41, 1511–1524 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Riečanová, Z.: The existence of states on every Archimedean atomic lattice effect algebra with at most five blocks. Kybernetika 44, 430–440 (2008)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Shultz, F.W.: A characterization of state spaces of orthomodular lattices. J. Comb. Theory A 17, 317–328 (1974)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Sultanbekov, F.F.: On (3,3)-homogeneous Greechie orthomodular posets. Internat. J. Theoret. Phys. 44, 957–963 (2002).  https://doi.org/10.1007/s10773-005-7072-9. arXiv:math.LO/0211311 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Sultanbekov, F. F.: Boolean Algebras and Quantum Logics. Kazan State University, Kazan (2007)Google Scholar
  35. 35.
    Sultanbekov, F.F.: Automorphism groups of small (3,3)-homogeneous logics. Internat. J. Theoret. Phys. 49, 3271–3278 (2010).  https://doi.org/10.1007/s10773-010-0439-6 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Sultanbekov, F.F.: (3,3)-homogeneous quantum logics with 18 atoms. I. Russian Math. (Izv. vuzov. Matematika) 56, 62–66 (2012).  https://doi.org/10.3103/S1066369X12110072 MathSciNetzbMATHGoogle Scholar
  37. 37.
    Weber, H.: There are orthomodular lattices without non-trivial group valued states; a computer-based construction. J. Math. Anal. Appl. 183, 89–94 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Center for Machine Perception, Department of Cybernetics, Faculty of Electrical EngineeringCzech Technical University in PraguePrahaCzech Republic

Personalised recommendations