International Journal of Theoretical Physics

, Volume 47, Issue 1, pp 125–148 | Cite as

Sharp and Fuzzy Observables on Effect Algebras

  • A. Jenčová
  • S. Pulmannová
  • E. Vinceková


Observables on effect algebras and their fuzzy versions obtained by means of confidence measures (Markov kernels) are studied. It is shown that, on effect algebras with the (E)-property, given an observable and a confidence measure, there exists a fuzzy version of the observable. Ordering of observables according to their fuzzy properties is introduced, and some minimality conditions with respect to this ordering are found. Applications of some results of classical theory of experiments are considered.


Effect algebra Observable Hilbert space effects PV-measure POV-measure Sufficient Markov kernel Smearing 


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  1. 1.
    Barbieri, G., Weber, H.: Measures on clans and on MV-algebras. In: Pap, E. (ed.) Handbook of Measure Theory, vol. II, pp. 911–945. Elsevier, Amsterdam (2002) Google Scholar
  2. 2.
    Busch, P., Lahti, P., Mittelstaedt, P.: The Quantum Theory of Measurement. Lecture Notes in Physics. Springer, Berlin (1991) Google Scholar
  3. 3.
    Bugajski, S.: Statistical maps, I: basic properties. Math. Slovaca 51, 321–342 (2001) MATHMathSciNetGoogle Scholar
  4. 4.
    Bugajski, S., Hellwig, K.E., Stulpe, W.: On fuzzy random variables and statistical maps. Rep. Math. Phys. 41, 1–11 (1998) MATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Busemi, F., D’Ariano, G.M., Keyl, M., Perinotti, P., Werner, R.F.: Ordering of measurements according to quantum noise. Lecture on QUIT, Budmerice, 2 December 2004 Google Scholar
  6. 6.
    Butnariu, D., Klement, E.: Triangular-norm-based measures and their Markov kernel representation. J. Math. Anal. Appl. 162, 111–143 (1991) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cignoli, R., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Fundations of Many-Valued Reasoning. Kluwer, Dordrecht (2000) Google Scholar
  8. 8.
    Chang, C.C.: Algebraic analysis of many valued logic. Trans. Am. Math. Soc. 88, 467–490 (1958) MATHCrossRefGoogle Scholar
  9. 9.
    Chovanec, F., Kôpka, F.: Boolean D-posets. Tatra Mt. Math. Publ. 10, 183–197 (1997) MATHMathSciNetGoogle Scholar
  10. 10.
    Davies, E.B.: Quantum Theory of Open Systems. Academic, London (1976) MATHGoogle Scholar
  11. 11.
    Duchoň, M., Dvurečenskij, A., De Lucia, P.: Moment problem for effect algebras, Moment problem for effect algebras. Int. J. Theor. Phys. 36, 1941–1958 (1997) CrossRefMATHGoogle Scholar
  12. 12.
    Dvurečenskij, A.: Loomis-Sikorski theorem for σ-complete MV-algebras and -groups. J. Austral. Math. Soc. Ser. A 68, 261–277 (2000) MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer, Dordrecht (2000) MATHGoogle Scholar
  14. 14.
    Dvurečenskij, A., Pulmannová, S.: Difference posets, effects and quantum measurements. Int. J. Theor. Phys. 33, 819–850 (1994) CrossRefMATHGoogle Scholar
  15. 15.
    Foulis, D., Bennett, M.K.: Effect algebras and unsharp quantum logics. Found. Phys. 24, 1325–1346 (1994) CrossRefMathSciNetADSGoogle Scholar
  16. 16.
    Giuntini, R., Greuling, H.: Toward a formal language for unsharp properties. Found. Phys. 19, 931–945 (1989) CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Gudder, S.: Lattice properties of quantum effects. J. Math. Phys. 37, 2637–2642 (1996) MATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Halmos, P.R., Savage, L.J.: Applications of the Radon–Nikodym theorem to the theory of sufficient statistics. Ann. Math. Stat. 20, 225–241 (1949) CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Heinonen, T.: Optimal measurement in quantum mechanics. Phys. Lett. A 346, 77–86 (2005) CrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Heinonen, T., Lahti, P., Ylinen, K.: Covariant fuzzy observables and coarse-grainings. Rep. Math. Phys. 53, 425–441 (2004) MATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Heyer, H.: Theory of Statistical Experiments. Springer, New York (1982) MATHGoogle Scholar
  22. 22.
    Kôpka, F., Chovanec, F.: D-posets. Math. Slovaca 44, 21–34 (1994) MATHMathSciNetGoogle Scholar
  23. 23.
    Lahti, P.J., Ma̧cziński, M.J.: On the order structure of the set of effects in quantum mechanics. J. Math. Phys. 36, 1673–1680 (1995) MATHCrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Liese, L., Vajda, I.: Convex Statistical Distances. Teubner-Texte zur Mathematik. Leipzig (1987) Google Scholar
  25. 25.
    Mundici, D.: Tensor product and the Loomis-Sikorski theorem for MV-algebras. Adv. Appl. Math. 22, 227–248 (1999) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics. Kluwer, Dordrecht (1991) MATHGoogle Scholar
  27. 27.
    Stěpán, J.: Probability Theory (Teorie pravděpodobnosti, in Czech). Academia, Prague (1987) Google Scholar
  28. 28.
    Strasser, H.: Mathematical Theory of Statistics. de Gruyter, Berlin (1985) MATHGoogle Scholar
  29. 29.
    Varadarajan, V.S.: Geometry of Quantum theory. Springer, Berlin (1985) MATHGoogle Scholar
  30. 30.
    Holevo, A.S.: Statistical Structures of Quantum Theory, LNP m67, p. 43. Springer, New York (2001) Google Scholar
  31. 31.
    Ali, S.T., Antoine, J.-P., Gazeau, J.-P.: Coherent states, Wavelets and Their Generalizations. Springer, New York (2000) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Mathematical InstituteSlovak Academy of SciencesBratislavaSlovakia

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