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Foundations of Physics

, Volume 45, Issue 10, pp 1379–1393 | Cite as

On the Possibility to Combine the Order Effect with Sequential Reproducibility for Quantum Measurements

  • Irina BasievaEmail author
  • Andrei Khrennikov
Article

Abstract

In this paper we study the problem of a possibility to use quantum observables to describe a possible combination of the order effect with sequential reproducibility for quantum measurements. By the order effect we mean a dependence of probability distributions (of measurement results) on the order of measurements. We consider two types of the sequential reproducibility: adjacent reproducibility (\(A-A\)) (the standard perfect repeatability) and separated reproducibility(\(A-B-A\)). The first one is reproducibility with probability 1 of a result of measurement of some observable A measured twice, one A measurement after the other. The second one, \(A-B-A\), is reproducibility with probability 1 of a result of A measurement when another quantum observable B is measured between two A’s. Heuristically, it is clear that the second type of reproducibility is complementary to the order effect. We show that, surprisingly, this may not be the case. The order effect can coexist with a separated reproducibility as well as adjacent reproducibility for both observables A and B. However, the additional constraint in the form of separated reproducibility of the \(B-A-B\) type makes this coexistence impossible. The problem under consideration was motivated by attempts to apply the quantum formalism outside of physics, especially, in cognitive psychology and psychophysics. However, it is also important for foundations of quantum physics as a part of the problem about the structure of sequential quantum measurements.

Keywords

Order effect Repeatability Quantum-like models POVMs in cognitive science 

Notes

Acknowledgments

The authors would like to thank J. Busemeyer and E. Dzhafarov for fruitful discussions and M. D’ Ariano, P. Lahti, W.M. de Muynck, and M. Ozawa for fruitful comments and advices. This project was supported by researcher-fellowship at Mathematical Institute of Linnaeus University (I. Basieva, 2014-15).

References

  1. 1.
    Davies, E., Lewis, J.: An operational approach to quantum probability. Commun. Math. Phys. 17, 239–260 (1970)zbMATHMathSciNetCrossRefADSGoogle Scholar
  2. 2.
    Busch, P., Grabowski, M., Lahti, P.: Operational Quantum Physics. Springer, Berlin (1995)zbMATHGoogle Scholar
  3. 3.
    Holevo, A.S.: Statistical Structure of Quantum Theory (Lecture Notes in Physics Monographs). Springer, Heidelberg (2001)CrossRefGoogle Scholar
  4. 4.
    Ozawa, M.: An operational approach to quantum state reduction. Ann. Phys. 259, 121–137 (1997)zbMATHCrossRefADSGoogle Scholar
  5. 5.
    Busch, P., Cassinelli, G., Lahti, P.J.: On the quantum theory of sequential measurements. Found. Phys. 20(7), 757–775 (1990)MathSciNetCrossRefADSGoogle Scholar
  6. 6.
    Buscemi, F., D’ Ariano, G.M., Perinotti, P.: There exist nonorthogonal quantum measurements that are perfectly repeatable. Phys. Rev. Lett. 92, 070403-1–070403-4 (2004)CrossRefADSGoogle Scholar
  7. 7.
    Jaeger, G.: Quantum Information: An Overview. Springer, Berlin (2007)Google Scholar
  8. 8.
    Khrennikov, A., Basieva, I., Dzhafarov, E.N., Busemeyer, J.R.: Quantum Models for psychological measurements: an unsolved problem. PLOS One 9, 0110909 (2014)CrossRefGoogle Scholar
  9. 9.
    Khrennikov, A.: Information Dynamics in Cognitive, Psychological, Social, and Anomalous Phenomena. Fundamental Theories of Physics. Kluwer, Dordreht (2004)Google Scholar
  10. 10.
    Khrennikov, A.: Ubiquitous Quantum Structure: From Psychology to Finances. Springer, Berlin (2010)CrossRefGoogle Scholar
  11. 11.
    Busemeyer, J.R., Bruza, P.D.: Quantum Models of Cognition and Decision. Cambridge Press, Cambridge (2012)CrossRefGoogle Scholar
  12. 12.
    Haven, E., Khrennikov, A.: Quantum Social Science. Cambridge Press, Cambridge (2013)CrossRefGoogle Scholar
  13. 13.
    Haven, E., Khrennikov, A.: Quantum mechanics and violation of the sure-thing principle: the use of probability interference and other concepts. J. Math. Psychol. 53, 378–388 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Asano, M., Ohya, M., Tanaka, Y., Khrennikov, A., Basieva, I.: On application of Gorini–Kossakowski–Sudarshan–Lindblad equation in cognitive psychology. Open Syst. Inf. Dyn. 18, 55–69 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Asano, M., Ohya, M., Tanaka, Y., Khrennikov, A., Basieva, I.: Dynamics of entropy in quantum-like model of decision making. J. Theor. Biol. 281, 56–64 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Dzhafarov, E.N., Kujala, J.V.: Quantum Entanglement and the Issue of Selective Influences in Psychology: An Overview. Lecture Notes in Computer Science, pp. 184–195. Springer, Berlin (2012)Google Scholar
  17. 17.
    Aerts, D., Sozzo, S., Tapia, J.: A Quantum Model for the Elsberg and Machina Paradoxes. Quantum Interaction. Lecture Notes in Computer Science, pp. 48–59. Springer, Berlin (2012)Google Scholar
  18. 18.
    Cheon, T., Takahashi, T.: Interference and inequality in quantum decision theory. Phys. Lett. A 375, 100–104 (2010)zbMATHMathSciNetCrossRefADSGoogle Scholar
  19. 19.
    Cheon, T., Tsutsui, I.: Classical and quantum contents of solvable game theory on Hilbert space. Phys. Lett. A 348, 147–152 (2006)zbMATHCrossRefADSGoogle Scholar
  20. 20.
    D’ Ariano, G.M.: Operational axioms for quantum mechanics. In: Foundations of Probability and Physics-4. G. Adenier, C. Fuchs, and A. Khrennikov (eds.). AIP Conf. Proc. vol. 889, pp. 79–105 (2007)Google Scholar
  21. 21.
    Chiribella, G., D’Ariano, G.M., Perinotti, P.: Informational Axioms for Quantum Theory. In Foundations of Probability and Physics—6, AIP Conf. Proc. vol. 1424, pp. 270–279 (2012)Google Scholar
  22. 22.
    D’Ariano, M.: Physics as Information Processing. In Advances in Quantum Theory, AIP Conf. Proc. vol. 1327, pp. 7–16 (2011)Google Scholar
  23. 23.
    D’Ariano, G.M., Jaeger, G.: Entanglement, Information, and the Interpretation of Quantum Mechanics (The Frontiers Collection). Springer, Berlin (2009)Google Scholar
  24. 24.
    Sinha, U., Couteau, Ch., Medendorp, Z., Sllner, I., Laflamme, R., Sorkin, R., Weihs, G.: Testing Born’s Rule in Quantum Mechanics with a Triple Slit Experiment. In: Foundations of Probability and Physics-5, L. Accardi, G. Adenier, C.A. Fuchs, G. Jaeger, A. Khrennikov, J.-A. Larsson, S. Stenholm (eds.), American Institute of Physics, vol. 1101, pp. 200–207 (2009)Google Scholar
  25. 25.
    Khrennikov, A.: Towards Violation of Born’s Rule: Description of a Simple Experiment. In: Advances in Quantum Theory, G. Jaeger, A. Khrennikov, M. Schlosshauer, G. Weihs (eds.), American Institute of Physics, vol. 1327, pp. 387–394 (2011)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Prokhorov General Physics InstituteMoscowRussian Federation
  2. 2.International Center for Mathematical Modeling in Physics, Engineering, Economics, and Cognitive ScienceLinnaeus UniversityVäxjö, KalmarSweden

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