Skip to main content
SpringerLink
Log in

Non-Kolmogorovian Approach to the Context-Dependent Systems Breaking the Classical Probability Law

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

There exist several phenomena breaking the classical probability laws. The systems related to such phenomena are context-dependent, so that they are adaptive to other systems. In this paper, we present a new mathematical formalism to compute the joint probability distribution for two event-systems by using concepts of the adaptive dynamics and quantum information theory, e.g., quantum channels and liftings. In physics the basic example of the context-dependent phenomena is the famous double-slit experiment. Recently similar examples have been found in biological and psychological sciences. Our approach is an extension of traditional quantum probability theory, and it is general enough to describe aforementioned contextual phenomena outside of quantum physics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In this paper we proceed pragmatically. We do not discuss arguments for and against hidden variables. We want just proceed mathematically. In the literature on quantum foundations it is generally claimed that Kolmogorov description of the double-slit experiment is impossible, cf., however, [19].

  2. Elaboration of such generalized quantum(-like) dynamics is not based on just a rather common wish to consider so general situation as possible. Already the simplest examples from biology, see Sect. 4, demonstrate that for such biological systems the dynamical state change cannot be described in the conventional quantum framework. Elaboration of such a new mathematical apparatus and its application to biology differs this paper from our previous publications [7, 8] in which the standard theory of open dynamical systems was in use. By the same reason we presented AD-theory in the framework of C ⋆-algebras (and not simply complex Hilbert space): in general there are no reasons to expect that the probabilistic structure of all possible biological phenomena can be embedded into complex Hilbert space model of probability.

  3. We point out that by using liftings we operate with entangled quantum states. By the conventional interpretation of quantum mechanics the corresponding probabilistic structure cannot be represented classically, in a Kolmogorov space. However, we again remark that inter-relation between classical and quantum probabilistic descriptions is still actively debated.

References

  1. Accardi, L.: Urne e Camaleonti: Dialogo Sulla Realta, le Leggi del Caso e la Teoria Quantistica. Il Saggiatore, Perugia (1997). English edition: World Scientific (2002); Japanese edition: Makino (2002), Russian edition, Regular and Chaotic dynamics (2002)

    Google Scholar 

  2. Accardi, L., Ohya, M.: Compound channels, transition expectations, and liftings. Appl. Math. Optim. 39, 33–59 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Accardi, L., Ohya, M.: A stochastic limit approach to the SAT problem. Open Syst. Inf. Dyn. 11, 1–16 (2004)

    Article  MathSciNet  Google Scholar 

  4. Accardi, L., Khrennikov, A., Ohya, M.: The problem of quantum-like representation in economy, cognitive science, and genetics. In: Accardi, L., Freudenberg, W., Ohya, M. (eds.) Quantum Bio-Informatics II: from Quantum Information to Bio-Informatics, pp. 1–8. WSP, Singapore (2008)

    Chapter  Google Scholar 

  5. Accardi, L., Khrennikov, A., Ohya, M.: Quantum Markov model for data from Shafir–Tversky experiments in cognitive psychology. Open Syst. Inf. Dyn. 16, 371–385 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Allahverdyan, A.E., Balian, R., Nieuwenhuizen, Th.M.: The quantum measurement process in an exactly solvable model. In: Foundations of Probability and Physics-3. American Institute of Physics, Ser. Conference Proceedings, vol. 750, pp. 16–24. Melville, New York (2005)

    Google Scholar 

  7. Asano, M., Ohya, M., Khrennikov, A.: Quantum-like model for decision making process in two players game. Found. Phys. 41(3), 538–548 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  8. 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(1), 55–69 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Asano, M., Ohya, M., Tanaka, Y., Khrennikov, A., Basieva, I.: Quantum-like representation of Bayesian updating. In: Proceedings of the International Conference on Advances in Quantum Theory. American Institute of Physics, vol. 1327, pp. 57–62 (2011)

    Google Scholar 

  10. Basieva, I., Khrennikov, A., Ohya, M., Yamato, I.: Quantum-like interference effect in gene expression: glucose-lactose destructive interference. Syst. Synth. Biol. (2011). doi:10.1007/s11693-011-9081-8

    Google Scholar 

  11. Busemeyer, J.R., Matthews, M., Wang, Z.: A quantum information processing explanation of disjunction effects. In: Sun, R., Myake, N. (eds.) The 29th Annual Conference of the Cognitive Science Society and the 5th International Conference of Cognitive Science, pp. 131–135. Erlbaum, Mahwah (2006)

    Google Scholar 

  12. Busemeyer, J.B., Wang, Z., Townsend, J.T.: Quantum dynamics of human decision making. J. Math. Psychol. 50, 220–241 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Busemeyer, J.R., Santuy, E., Lambert-Mogiliansky, A.: Comparison of Markov and quantum models of decision making. In: Bruza, P., Lawless, W., van Rijsbergen, K., Sofge, D.A., Coeke, B., Clark, S. (eds.) Quantum Interaction: Proceedings of the Second Quantum Interaction Symposium, pp. 68–74. College Publications, London (2008)

    Google Scholar 

  14. Caves, C.M., Fuchs, Ch.A., Schack, R.: Quantum probabilities as Bayesian probabilities. Phys. Rev. A 65, 022305 (2002)

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  ADS  MATH  Google Scholar 

  17. Conte, E., Khrennikov, A., Todarello, O., Federici, A., Zbilut, J.P.: Mental states follow quantum mechanics during perception and cognition of ambiguous figures. Open Syst. Inf. Dyn. 16, 1–17 (2009)

    Article  MathSciNet  Google Scholar 

  18. D’ Ariano, G.M.: In: Operational Axioms for Quantum Mechanics. Foundations of Probability and Physics-3. American Institute of Physics, Ser. Conference Proceedings, vol. 889, pp. 79–105. Melville, New York (2007)

    Google Scholar 

  19. De Muynck, W.M.: Foundations of Quantum Mechanics, an Empiricists Approach. Kluwer Academic, Dordrecht (2002)

    Google Scholar 

  20. Fichtner, K.-H., Fichtner, L., Freudenberg, W., Ohya, M.: On a quantum model of the recognition process. In: QP-PQ: Quantum Prob. White Noise Analysis, vol. 21, pp. 64–84 (2008)

    Google Scholar 

  21. Fuchs, Ch.A., Schack, R.: A quantum-Bayesian route to quantum-state space. Found. Phys. 41, 345–356 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Garola, C., Sozzo, S.: The ESR model: a proposal for a noncontextual and local Hilbert space extensions of QM. Europhys. Lett. 86, 20009–20015 (2009)

    Article  ADS  Google Scholar 

  23. Garola, C., Sozzo, S.: Generalized observables, Bell’s inequalities and mixtures in the ESR model. Found. Phys. 41, 424–449 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Inada, T., Kimata, K., Aiba, H.: Mechanism responsible for glucose-lactose diauxie in Escherichia coli challenge to the cAMP model. Genes Cells 1, 293–301 (1996)

    Article  Google Scholar 

  25. Inoue, K., Ohya, M., Sato, K.: Application of chaos degree to some dynamical systems. Chaos Solitons Fractals 11, 1377–1385 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. Inoue, K., Ohya, M., Volovich, I.V.: Semiclassical properties and chaos degree for the quantum baker’s map. J. Math. Phys. 43(1), 734 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. Khrennikov, A.: Open Syst. Inf. Dyn. 11(3), 267–275 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Khrennikov, A.: Biosystems 84, 225–241 (2006)

    Article  Google Scholar 

  29. Khrennikov, A.: Contextual Approach to Quantum Formalism (Fundamental Theories of Physics). Springer, Berlin (2009)

    Book  Google Scholar 

  30. Khrennikov, A.: Ubiquitous Quantum Structure: from Psychology to Finance. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  31. Khrennikov, A., Haven, E.: Quantum mechanics and violations of the sure-thing principle: the use of probability interference and other concepts. J. Math. Psychol. 53, 378–388 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kossakowski, A., Ohya, M., Togawa, Y.: How can we observe and describe chaos? Open Syst. Inf. Dyn. 10(3), 221–233 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ohya, M.: Note on quantum proability. Lett. Nuovo Cimento 38(11), 203–206 (1983)

    Article  MathSciNet  Google Scholar 

  34. Ohya, M.: On compound state and mutual information in quantum information theory. IEEE Trans. Inf. Theory 29, 770–777 (1983)

    Article  MATH  Google Scholar 

  35. Ohya, M.: Some aspects of quantum information theory and their applications to irreversible processes. Rep. Math. Phys. 27, 19–47 (1989)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. Ohya, M.: Complexities and their applications to characterization of chaos. Int. J. Theor. Phys. 37(1), 495–505 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ohya, M.: Adaptive dynamics and its applications to chaos and NPC problem. In: QP-PQ: Quantum Probability and White Noise Analysis, Quantum Bio-Informatics 2007, vol. 21, pp. 181–216 (2007)

    Google Scholar 

  38. Ohya, M., Volovich, I.V.: New quantum algorithm for studying NP-complete problems. Rep. Math. Phys. 52(1), 25–33 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  39. Ohya, M., Volovich, I.V.: Mathematical Foundations of Quantum Information and Computation and Its Applications to Nano- and Bio-Systems. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  40. Plotnitsky, A.: Reading Bohr: Physics and Philosophy. Springer, Berlin (2006)

    MATH  Google Scholar 

  41. Plotnitsky, A.: Epistemology and Probability: Bohr, Heisenberg, Schrödinger, and the Nature of Quantum-Theoretical Thinking. Springer, Berlin (2009)

    Google Scholar 

  42. Plotnitsky, A.: On the reasonable and unreasonable effectiveness of mathematics in classical and quantum physics. Found. Phys. 41, 466–491 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references

Acknowledgements

Two authors (I. B. and A. Kh.) were supported by the grant Quantum Bio-Informatics, Tokyo University of Science (visiting fellowships, 2010, 11, 13); they would like to thank Noboru Watanabe and their coauthors for hospitality.

Author information

Authors and Affiliations

  1. Tokuyama College of Technology, Gakuendai, Shunan, Yamaguchi, 745-8585, Japan

    Masanari Asano

  2. International Center for Mathematical Modeling in Physics and Cognitive Sciences, Linnaeus University, 35195, Växjö, Sweden

    Irina Basieva & Andrei Khrennikov

  3. Department of Biological Science and Technology, Tokyo University of Science, Tokyo, Japan

    Masanori Ohya & Ichiro Yamato

Authors
  1. Masanari Asano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Irina Basieva
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andrei Khrennikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Masanori Ohya
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Ichiro Yamato
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrei Khrennikov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Asano, M., Basieva, I., Khrennikov, A. et al. Non-Kolmogorovian Approach to the Context-Dependent Systems Breaking the Classical Probability Law. Found Phys 43, 895–911 (2013). https://doi.org/10.1007/s10701-013-9725-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-013-9725-5

Keywords

Search

Navigation