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Compatibility of observables on effect algebras

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Abstract

We compare different notions of simultaneous measurability (compatibility) of observables on lattice \(\sigma \)-effect algebras and more generally, on \(\sigma \)-effect algebras that can be covered by \(\sigma \)-MV-algebras. We prove that every \(\sigma \)-MV-algebra is the range of a \(\sigma \)-additive observable, and we compare the following notions of compatibility of observables: joint measurability, coexistence, joint measurability of binarizations, coexistence of binarizations, smearings of the same observable. We prove that if there is a faithful state on the effect algebra, then any two standard observables that are smearings of the same (sharp) observable admit a generalized joint observable.

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References

  • Barbieri J, Weber H (2002) Measures on clans on MV-algebras. In: Pap E (ed) Handbook of Measure Theory, vol IIs. Elsevier, Amsterdam, pp 911–945

  • Beran L (1984) Orthomodular Lattices—Algebraic Approach. Academia, D. Reidel, Dordrecht, Holland

  • Boccuto A (1995) On Stone-type extensions for group-valued measures. Math Slovaca 45:309–315

    MathSciNet  MATH  Google Scholar 

  • Busch P, Lahti P, Mittelstaedt P (1991) The Quantum Theory of Measurement. Lecture Notes in Physics. Springer, Berlin

    MATH  Google Scholar 

  • Butnariu D, Klement EP (1991) Triangular norm-based measures and their Markov kernel representation. J Math Anal Appl 162:111–143

    Article  MathSciNet  MATH  Google Scholar 

  • Chang C (1958) Algebraic analysis of many-valued logic. Trans Am Math Soc 88:467–490

    Article  MATH  Google Scholar 

  • Cignoli R, D’Ottaviano IML, Mundici D (2000) Algebraic foundations of many-valued reasoning. Kluwer, Dordrecht

    Book  MATH  Google Scholar 

  • Dvurečenskij A (2000) Loomis–Sikorski theorem for \(\sigma \)-complete MV-algebras and \(\ell \)-groups. J Austral Math Soc Ser A 68:261–277

    Article  MathSciNet  MATH  Google Scholar 

  • Dvurečenskij A (2002) On effect algebras which can be covered by MV-algebras. Inter J Theor Phys 41:221–229

    Article  MathSciNet  MATH  Google Scholar 

  • Dvurečenskij A (2014) Representable effect algebras and observables. Inter J Theor Phys 53:2855–2866

    Article  MathSciNet  MATH  Google Scholar 

  • Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Ister Science, Bratislava and Kluwer, Dordrecht

    Book  MATH  Google Scholar 

  • Foulis DJ, Greechie R, Rüttimann GT (1992) Filters and supports in orthoalgebras. Inter J Theor Phys 31:789–807

    Article  MathSciNet  MATH  Google Scholar 

  • Foulis DJ (2004) Spectral resolutions in a Rickart comgroup. Rep Math Phys 54:229–250

    Article  MathSciNet  MATH  Google Scholar 

  • Foulis DJ, Bennett MK (1993) Tensor product of orthoalgebras. Order 10:271–282

    Article  MathSciNet  MATH  Google Scholar 

  • Foulis DJ, Bennett MK (1994) Effect algebras and unsharp quantum logics. Found Phys 24:1331–1352

    Article  MathSciNet  MATH  Google Scholar 

  • Giuntini R, Greuling H (1989) Toward a formal language for unsharp properties. Found Phys 19:931–945

    Article  MathSciNet  Google Scholar 

  • Goodearl KR (1986) Partially ordered abelian groups with interpolation. Math. Surveys and Monographs No 20, Amer. Math. soc., providence, Rhode Island

  • Grätzer G (1971) Lattice Theory. First Concepts and Distributive Lattices. W.H. Freeman and Co., San Francisco

    MATH  Google Scholar 

  • Haapsalo E, Pelonpää J P, Uola R (2014) Compatibility properties of extreme quantum observables. arXiv:1404.4172v1[quant-ph]

  • Jenča G (2001) Blocks in homogeneous effect algebras. Bull Austral Math Soc 64:81–98

    Article  MathSciNet  MATH  Google Scholar 

  • Jenča G (2004) Boolean algebras R-generated by MV-algebras. Fuzzy Sets Syst 154:279–285

    MathSciNet  MATH  Google Scholar 

  • Jenča G, Riečanová Z (1999) On sharp elements in lattice ordered effect algebras. Busefal 80:24–29

    Google Scholar 

  • Jenčová A, Pulmannová S, Vinceková E (2008) Sharp and fuzzy observables on effect algebras. Int J Theor Phys 47:125–148

    Article  MathSciNet  MATH  Google Scholar 

  • Jenčová A, Pulmannová S, Vinceková E (2011) Observables on \(\sigma \)-MV-algebras and \(\sigma \)-lattice effect algebras. Kybernetika 47:541–559

    MathSciNet  MATH  Google Scholar 

  • Ji W (2014) Characterization of homogeneity in orthocomplete atomic effect algebras. Fuzzy Sets Syst 236:113–121

    Article  MathSciNet  MATH  Google Scholar 

  • Kalmbach G (1983) Orthomodular Lattices. Academic Press, London

    MATH  Google Scholar 

  • Kôpka F (1995) Compatibility in D-posets. Int J Theor Phys 34:1525–1531

    Article  MathSciNet  MATH  Google Scholar 

  • Kôpka F, Chovanec F (1994) D-posets. Math Slovaca 44:21–34

    MathSciNet  MATH  Google Scholar 

  • Lahti P (2003) Coexistence and joint measurability in quantum mechanics. Int J Theor Phys 42:893–906

    Article  MathSciNet  MATH  Google Scholar 

  • Lahti P, Ylinen K (2004) Dilatations of positive operator valued measures and bimeasures related to quantum mechanics. Math Slovaca 54:169–189

  • Mundici D (1986) AF C*-algebras in Łukasziewicz sentential calculus. J Funct Anal Appl 65:15–63

  • Mundici D (1999) Tensor products and the Loomis–Sikorski theorem for MV-algebras. Adv Appl Math 22:227–248

    Article  MathSciNet  MATH  Google Scholar 

  • Mundici D, Riečan B (2002) Probability on MV-algebras. In: Pap E (ed) Handbook of Measure Theory, vol II. Elsevier, Amsterdam, pp 869–910

  • Niederle J, Paseka J (2013) Homogeneous orthocomplete effect algebras are covered by MV-algebras. Fuzzy Sets Syst 210:89–101

    Article  MathSciNet  MATH  Google Scholar 

  • Pták P, Pulmannová S (1991) Orthomodular structures as quantum logics. Kluwer, Dordrecht

    MATH  Google Scholar 

  • Pulmannová S (2000) A note on observables on MV-algebras. Soft Comput. 4:45–48

    Article  MATH  Google Scholar 

  • Pulmannová S (2002) Compatibility and decompositions of effects. J Math Phys 43:2817–2830

    Article  MathSciNet  MATH  Google Scholar 

  • Pulmannová S (2005) A spectral theorem for \(\sigma \)-MV-algebras. Kybernetika 41:361–374

    MathSciNet  MATH  Google Scholar 

  • Pulmannová S (2005) Spectral resolutions in Dedekind \(\sigma \)-complete \(\ell \)-groups. J Math Anal Appl 309:322–335

    Article  MathSciNet  MATH  Google Scholar 

  • Pulmannová S (2005) Spectral resolutions for \(\sigma \)-complete lattice effect algebras. Math Slovaca 56:557–575

    MathSciNet  Google Scholar 

  • Ravindran K (1996) On a structure theory of effect algebras. PhD thesis, Kansas State Univ., Manhattan, Kansas

  • Reeb O, Reitzner D, Wolf MM (2013) Coexistence does not imply joint measurability. J Phys A Math Theor 46:462002

    Article  MathSciNet  MATH  Google Scholar 

  • Riečanová Z (2000) Generalizations of blocks for D-lattices and lattice ordered effect algebras. Int J Theor Phys 39:231–237

    Article  MATH  Google Scholar 

  • Sikorski R (1960) Boolean Algebras. Springer, New York

    Book  MATH  Google Scholar 

  • Strasser H (1985) Math. Theory of Statistics. W De Gruyter, Berlin

    Book  MATH  Google Scholar 

  • Tkadlec J (2010) Common generalization of orthocomplete and lattice effect algebras. Int J Theor Phys 49:3279–3285

    Article  MathSciNet  MATH  Google Scholar 

  • Varadarajan VS (1985) Geometry of quantum theory. Springer, Berlin

    MATH  Google Scholar 

  • Ylinen K (1996) Positive operator bimeasures and a noncommutative generalization. Studia Math 118:157–168

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors are thankful to both referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73, Bratislava, Slovakia

    S. Pulmannová & E. Vinceková

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  1. S. Pulmannová
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  2. E. Vinceková
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Correspondence to S. Pulmannová.

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The authors declare that they have no conflict of interest.

Additional information

Communicated by A. Di Nola.

The authors were supported by Research and Development Support Agency under the contract No. APVV-0178-11 and grant VEGA 2/0059/12. Pulmannová, S., Vinceková, E.Compatibility of observables.

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Pulmannová, S., Vinceková, E. Compatibility of observables on effect algebras. Soft Comput 20, 3957–3967 (2016). https://doi.org/10.1007/s00500-015-1984-y

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