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

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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.

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Notes

  1. Typically in quantum measurement theory one uses the terminology “perfect repeatability” [6]. Since we shall consider two types of reproducibility (repeatability), it is useful to characterize them on the basis of their structure, so “adjacent” matches better with the situation.

  2. For observables with discrete spectra, effects encode the probabilities of the concrete results of observations.

  3. We remark that such instruments need not be of the von Neumann–Lüders type. In general, the quantum operations of are not reduced to projectors.

References

  1. Davies, E., Lewis, J.: An operational approach to quantum probability. Commun. Math. Phys. 17, 239–260 (1970)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. Busch, P., Grabowski, M., Lahti, P.: Operational Quantum Physics. Springer, Berlin (1995)

    MATH  Google Scholar 

  3. Holevo, A.S.: Statistical Structure of Quantum Theory (Lecture Notes in Physics Monographs). Springer, Heidelberg (2001)

    Book  Google Scholar 

  4. Ozawa, M.: An operational approach to quantum state reduction. Ann. Phys. 259, 121–137 (1997)

    Article  MATH  ADS  Google Scholar 

  5. Busch, P., Cassinelli, G., Lahti, P.J.: On the quantum theory of sequential measurements. Found. Phys. 20(7), 757–775 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  7. Jaeger, G.: Quantum Information: An Overview. Springer, Berlin (2007)

    Google Scholar 

  8. Khrennikov, A., Basieva, I., Dzhafarov, E.N., Busemeyer, J.R.: Quantum Models for psychological measurements: an unsolved problem. PLOS One 9, 0110909 (2014)

    Article  Google Scholar 

  9. Khrennikov, A.: Information Dynamics in Cognitive, Psychological, Social, and Anomalous Phenomena. Fundamental Theories of Physics. Kluwer, Dordreht (2004)

  10. Khrennikov, A.: Ubiquitous Quantum Structure: From Psychology to Finances. Springer, Berlin (2010)

    Book  Google Scholar 

  11. Busemeyer, J.R., Bruza, P.D.: Quantum Models of Cognition and Decision. Cambridge Press, Cambridge (2012)

    Book  Google Scholar 

  12. Haven, E., Khrennikov, A.: Quantum Social Science. Cambridge Press, Cambridge (2013)

    Book  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. 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. Cheon, T., Takahashi, T.: Interference and inequality in quantum decision theory. Phys. Lett. A 375, 100–104 (2010)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  19. Cheon, T., Tsutsui, I.: Classical and quantum contents of solvable game theory on Hilbert space. Phys. Lett. A 348, 147–152 (2006)

    Article  MATH  ADS  Google Scholar 

  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)

  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)

  22. D’Ariano, M.: Physics as Information Processing. In Advances in Quantum Theory, AIP Conf. Proc. vol. 1327, pp. 7–16 (2011)

  23. D’Ariano, G.M., Jaeger, G.: Entanglement, Information, and the Interpretation of Quantum Mechanics (The Frontiers Collection). Springer, Berlin (2009)

  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)

  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)

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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).

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Authors and Affiliations

  1. Prokhorov General Physics Institute, Vavilov str. 38D, Moscow, Russian Federation

    Irina Basieva

  2. International Center for Mathematical Modeling in Physics, Engineering, Economics, and Cognitive Science, Linnaeus University, Växjö, Kalmar, Sweden

    Andrei Khrennikov

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  1. Irina Basieva
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Correspondence to Irina Basieva.

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Basieva, I., Khrennikov, A. On the Possibility to Combine the Order Effect with Sequential Reproducibility for Quantum Measurements. Found Phys 45, 1379–1393 (2015). https://doi.org/10.1007/s10701-015-9932-3

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  • DOI: https://doi.org/10.1007/s10701-015-9932-3

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