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Compatibility for probabilistic theories

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Published/Copyright: July 4, 2016
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 2

Abstract

We define an index of compatibility for a probabilistic theory (PT). Quantum mechanics with index 0 and classical probability theory with index 1 are at the two extremes. In this way, quantum mechanics is at least as incompatible as any PT. We consider a PT called a concrete quantum logic that may have compatibility index strictly between 0 and 1, but we have not been able to show this yet. Finally, we show that observables in a PT can be represented by positive, vector-valued measures.

MSC 2010: Primary 46G12; 81P15
Keywords: probabilistic theory; compatibility; compatibility index; observable; joint observable; measure of compatibility; concrete quantum logic

Dedicated to Professor Anatolij Dvurečdenskij on the occasion of his 65th birthday

(Communicated by Sylvia Pulmannová)

Acknowledgement

The author thanks two anonymous referees for their helpful comments.

References

[1] Busch, P.—Heinosaari, T.—Schultz, J.—Stevens, N.: Quantum theory contains maximally incompatible observables, Europhysics Letts. 103 (2013), 1002.10.1209/0295-5075/103/10002Search in Google Scholar

[2] Busch, P.—Schmidt, H. J.: Coexistence of qubit effects, Quantum Inf. Process. 9 (2010), 143–169.10.1007/s11128-009-0109-xSearch in Google Scholar

[3] Gudder, S.: Quantum Probability, Academic Press, San Diego, 1988.10.1007/BF00670748Search in Google Scholar

[4] Heinosaari, T.—Reitzner, D.—Stano, P.: Notes on joint measurability of quantum observables, Found. Phys. 38 (2008), 1133–1142.10.1007/s10701-008-9256-7Search in Google Scholar

Received: 2014-2-12
Accepted: 2014-3-17
Published Online: 2016-7-4
Published in Print: 2016-4-1

© 2016 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2015-0149
Keywords for this article
probabilistic theory; compatibility; compatibility index; observable; joint observable; measure of compatibility; concrete quantum logic

Articles in the same Issue

  1. Research Article
  2. Holistic logical arguments in quantum computation
  3. Research Article
  4. States with values in the Łukasiewicz groupoid
  5. Research Article
  6. On realization of effect algebras
  7. Research Article
  8. States on symmetric logics: extensions
  9. Research Article
  10. Extensions and measurability in quantum measure spaces
  11. Research Article
  12. Convex subalgebras of upper bounded GMV-algebras
  13. Research Article
  14. Combining boolean algebras and ℓ-groups in the variety generated by chang’s mv-algebra
  15. Research Article
  16. Large rigid sets of algebras with respect to embeddability
  17. Research Article
  18. Conformal algebras, vertex algebras, and the logic of locality
  19. Research Article
  20. Extension of measures on pseudo-D-lattices
  21. Research Article
  22. Note on a parameter switching method for nonlinear ODEs
  23. Research Article
  24. Compatibility for probabilistic theories
  25. Research Article
  26. Completeness of Gelfand-Neumark-Segal inner product space on Jordan algebras
  27. Research Article
  28. Commutativity in a synaptic algebra
  29. Research Article
  30. Discrete averaged mixing applied to the logarithmic distributions
  31. Research Article
  32. Automorphisms of decompositions
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