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Quantum Information Biology: From Information Interpretation of Quantum Mechanics to Applications in Molecular Biology and Cognitive Psychology


We discuss foundational issues of quantum information biology (QIB)—one of the most successful applications of the quantum formalism outside of physics. QIB provides a multi-scale model of information processing in bio-systems: from proteins and cells to cognitive and social systems. This theory has to be sharply distinguished from “traditional quantum biophysics”. The latter is about quantum bio-physical processes, e.g., in cells or brains. QIB models the dynamics of information states of bio-systems. We argue that the information interpretation of quantum mechanics (its various forms were elaborated by Zeilinger and Brukner, Fuchs and Mermin, and D’ Ariano) is the most natural interpretation of QIB. Biologically QIB is based on two principles: (a) adaptivity; (b) openness (bio-systems are fundamentally open). These principles are mathematically represented in the framework of a novel formalism— quantum adaptive dynamics which, in particular, contains the standard theory of open quantum systems.

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  1. QIB is not about quantum physics of bio-systems (see [11] for extended review), in particular, not about quantum physical modeling of cognition, see Sect. 2 for details. Another terminological possibility would be “quantum bio-information”. However, the latter term, bio-information, has been reserved for a special part of biological information theory studying computer modeling of sequencing of DNA. We do not want to be associated with this activity having totally different aims and mathematical tools.

  2. This algorithm transfers the probabilistic data, collection of unconditional and conditional probabilities for complementary variables, into a complex (or hyperbolic) probability amplitude. Complex amplitudes can be physically processed as classical (e.g., electromagnetic) signals. Thus the quantum-like representation algorithm if used by a cognitive system can lead to classical physical processing exhibiting quantum probabilistic features.

  3. N. Bohr always pointed out that quantum theory predicts the results of measurements and pointed to the role of an observer; he stressed that the whole experimental arrangement has to be taken into account; W. Heisenberg and W. Pauli had similar views, we remark that the role of (human) observer was emphasized by Pauli, see A. Plotnitsky [37, 38] for detailed analysis of their views.

  4. In a series of private discussions with A. Khrennikov he expressed satisfaction that by exploring ideas of QM for modeling of cognition one finally breaks up the realistic attitude dominating in cognitive science. At the same time he did not find reasons to use precisely the quantum formalism to model cognition: novel operational formalisms representing information processing might be more appropriate, see also Sect. .

  5. We remark that quantum master equation (as well as the Schrödinger equation) is a linear first order (with respect to time) differential equation. Linearity is one of the fundamental features of quantum theory. It is very attractive even from the purely operational viewpoint, since it simplifies essentially calculations. In fact, mathematically we proceed with matrix-calculus. The question whether QM can be treated as linearization of more complex nonlinear theory was actively discussed in quantum foundations. For a moment, it is commonly accepted that QM is fundamentally linear, although there were presented strong reasons in favor of the linearization hypothesis. This problem is very important for cognition as well. As opposed to physics, even preliminary analysis of this problem has not yet been performed.

  6. At the same time we do not advertise QBism as the basic probabilistic interpretation of the state in quantum physics. It seems that, for experiments with physical quantum systems, the statistical interpretation of probability matches better with the real physical and statistical situation.

  7. As was pointed in the first paragraph of this section, contextuality is present even in the ideal formulation, i.e., for detectors having 100% efficiency. Thus the low efficiency of detection just makes the contextual structure of Bell’s experiment more visible.


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

    MathSciNet  Article  ADS  Google Scholar 

  2. 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. 17, 1–15 (2010)

    MathSciNet  Article  Google Scholar 

  3. Basieva, I., Khrennikov, A., Ohya, M., Yamato, I.: Quantum-like interference effect in gene expression glucose-lactose destructive interference. Syst. Synth. Biol. 5, 1–10 (2010)

    Google Scholar 

  4. 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)

    MathSciNet  Article  Google Scholar 

  5. Asano, M., Basieva, I., Khrennikov, A., Ohya, M., Yamato, I.: Non-Kolmogorovian approach to the context-dependent systems breaking the classical probability law. Found. Phys. 43, 895–911 (2013). 2083–2099 (2012)

    MATH  MathSciNet  Article  ADS  Google Scholar 

  6. Asano, M., Basieva, I., Khrennikov, A., Ohya, M., Tanaka, Y.: Quantum-like generalization of the Bayesian updating scheme for objective and subjective mental uncertainties. J. Math. Psychol. 166–175, 56 (2012)

    MathSciNet  Google Scholar 

  7. Asano, M., Basieva, I., Khrennikov, A., Ohya, M., Tanaka, Y., Yamato, I.: Quantum-like model for the adaptive dynamics of the genetic regulation of E. coli’s metabolism of glucose/lactose. Syst. Synth. Biol. 6, 1–7 (2012)

    Article  Google Scholar 

  8. Asano, M., Basieva, I.I., Khrennikov, A., Ohya, M., Tanaka, Y., Yamato, I.: A model of epigenetic evolution based on theory of open quantum systems. Syst. Synth. Biol. 7, 161–173 (2013)

    Article  Google Scholar 

  9. Asano, M., Khrennikov, A., Ohya, M., Tanaka, Y., Yamato, I.: Violation of contextual generalization of the LeggettGarg inequality for recognition of ambiguous figures. Phys. Scr. T 163, 014006 (2014)

    Article  ADS  Google Scholar 

  10. Asano, M., Khrennikov, A., Ohya, M., Tanaka, Y., Yamato, I.: Quantum Adaptivity in Biology: from Genetics to Cognition. Springer, Heidelberg (2015)

    Google Scholar 

  11. Arndt, M., Juffmann, T.H., Vedral, V.: Quantum physics meets biology. HFSP J 3(6), 386400 (2009)

    Article  Google Scholar 

  12. Khrennikov, A.: On quantum-like probabilistic structure of mental information. Open Syst. Inf. Dyn. 11(3), 267–275 (2004)

    MATH  MathSciNet  Article  Google Scholar 

  13. Khrennikov, A.: Quantum-like brain: interference of minds. BioSystems 84, 225–241 (2006)

    Article  Google Scholar 

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

    Book  Google Scholar 

  15. Conte, E., Todarello, O., Federici, A., Vitiello, F., Lopane, M., Khrennikov, A., Zbilut, J.P.: Some remarks on an experiment suggesting quantum-like behavior of cognitive entities and formulation of an abstract quantum mechanical formalism to describe cognitive entity and its dynamics. Chaos Solitons Fractals 31, 1076–1088 (2006)

    Article  ADS  Google Scholar 

  16. Conte, E., Khrennikov, A., Todarello, O., Federici, A., Mendolicchio, L., Zbilut, J.P.: A preliminary experimental verification on the possibility of Bell inequality violation in mental states. Neuroquantology 6, 214–221 (2008)

    Google Scholar 

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

    Book  Google Scholar 

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

    MATH  MathSciNet  Article  Google Scholar 

  19. Dzhafarov, E.N., Kujala, J.V.: Selectivity in probabilistic causality: where psychology runs into quantum physics. J. Math. Psychol. 56, 54–63 (2012)

    MATH  MathSciNet  Article  Google Scholar 

  20. Acacio de Barros, J., Suppes, P.: Quantum mechanics, interference, and the brain. J. Math. Psychcol. 53, 306–313 (2009)

    MATH  MathSciNet  Article  Google Scholar 

  21. Acacio de Barros, J.: Quantum-like model of behavioral response computation using neural oscillators. Biosystems 110, 171–182 (2012)

    Article  Google Scholar 

  22. Acacio de Barros, J.: 2012 Joint probabilities and quantum cognition. In: A. Khrennikov, H. Atmanspacher, A. Migdall and S. Polyakov. (eds.) Quantum Theory: Reconsiderations of Foundations 6. Special Section: Quantum-Like Decision Making: from Biology to Behavioral Economics, AIP Conference Proceedings 1508, pp. 98–104

  23. Atmanspacher, H., Filk, T., Römer, H.: Complementarity in Bistable Perception, Recasting Reality, pp. 135–150. Springer, Berlin (2009)

    Book  Google Scholar 

  24. Atmanspacher, H., Filk, Th.: 2012 Temporal nonlocality in bistable perception. In: A. Khrennikov, H. Atmanspacher, A. Migdall and S. Polyakov. (eds.) Quantum Theory: Reconsiderations of Foundations—6, Special Section: Quantum-like decision making: from biology to behavioral economics, AIP Conference Proeeding. 1508, pp. 79–88

  25. Busemeyer, J.R., Pothos, E.M., Franco, R., Trueblood, J.: A quantum theoretical explanation for probability judgment errors. Psychol. Rev. 118, 193–218 (2011)

    Article  Google Scholar 

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

    MATH  MathSciNet  Article  ADS  Google Scholar 

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

    MATH  Article  ADS  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  29. Haven, E., Khrennikov, A.: Quantum Social Science. Cambridge Press, Cambridge (2012)

    Google Scholar 

  30. Khrennikova, P., Haven, E., Khrennikov, A.: An application of the theory of open quantum systems to model the dynamics of party governance in the US political system. Int. J. Theor. Phys. 53, 1346–1360 (2014)

    MathSciNet  Article  Google Scholar 

  31. Khrennikova, P.: Evolution of quantum-like modeling in decision making processes. AIP Conf. Proc. 1508, 108–112 (2012)

    Article  ADS  Google Scholar 

  32. Ohya, M., Volovich, I.: Mathematical foundations of quantum information and computation and its applications to nano- and bio-systems. Springer, Heidelberg (2011)

    MATH  Book  Google Scholar 

  33. Tegmark, M.: Importance of quantum decoherence in brain processes. Phys. Rev. E 61(4), 41944206 (2000)

    Article  Google Scholar 

  34. Penrose, R.: The Emperor’s New Mind. Oxford University Press, New York (1989)

    Google Scholar 

  35. Hameroff, S.: Quantum coherence in microtubules. A neural basis for emergent consciousness? J. Cons. Stud. 1, 91–118 (1994)

    Google Scholar 

  36. Khrennikov, A.: Quantum-like model of processing of information in the brain based on classical electromagnetic field. Biosystems 105(3), 250–262 (2011)

    Article  Google Scholar 

  37. Plotnitsky, A.: Reading Bohr: Physics and Philosophy. Springer, Heidelberg (2006)

    Google Scholar 

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

    Google Scholar 

  39. Chiribella, G., D’Ariano, G.M., Perinotti, P.: Probabilistic theories with purification phys. Rev. A 81, 062348 (2010)

    Article  Google Scholar 

  40. D’ Ariano, G.M.: Operational axioms for quantum mechanics, in Adenier et al., Foundations of Probability and Physics-3.AIP Conference Proceeding vol. 889, pp. 79–105 (2007)

  41. Chiribella, G., D’Ariano, G.M., Perinotti, P.: Informational axioms for quantum theory in Foundations of Probability and Physics—6. AIP Conference Proceeding vol. 1424, p. 270 (2012)

  42. D’Ariano, G.M.: Physics as Information Processing, in Advances in Quantum Theory, AIP Conference Proceeding 1327 7 (2011); arXiv:1012.0535

  43. Zeilinger, A.: A foundational principle for quantum mechanics. Found. Phys. 29(4), 631–643 (1999)

    MathSciNet  Article  Google Scholar 

  44. Zeilinger, A.: Dance of the Photons: from Einstein to Quantum Teleportation. Farrar, Straus and Giroux, New York (2010)

    Google Scholar 

  45. Brukner, C., Zeilinger, A.: Malus’ law and quantum information. Acta Phys. Slovava 49(4), 647–652 (1999)

    Google Scholar 

  46. Brukner, C., Zeilinger, A.: Operationally invariant information in quantum mechanics. Phys. Rev. Lett. 83(17), 3354–3357 (1999)

    MATH  MathSciNet  Article  ADS  Google Scholar 

  47. Brukner, C., Zeilinger, A.: Information invariance and quantum probabilities. Found. Phys. 39, 677 (2009)

    MATH  MathSciNet  Article  ADS  Google Scholar 

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

    MathSciNet  Article  ADS  Google Scholar 

  49. Fuchs, C.A.: Quantum mechanics as quantum information (and only a little more). In: Khrennikov, A. (ed.), Quantum Theory: Reconsideration of Foundations, Ser. Math. Modeling 2, Växjö University Press, Växjö, pp. 463–543 (2002)

  50. Fuchs, C. A.: The anti-Växjö interpretation of quantum mechanics. Quantum Theory: Reconsideration of Foundations, pp. 99–116. Ser. Math. Model. 2, Växjö University Press, Växjö (2002)

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

    MATH  MathSciNet  Article  ADS  Google Scholar 

  52. E. Schrdinger, Die gegenwrtige Situation in der Quantenmechanik. Naturwissenschaften 23,807–812; 823–828; 844–849 (1935)

  53. Fuchs, Ch.A.: QBism, the perimeter of quantum Bayesianism. arXiv:1003.5209

  54. Von Neuman, J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton (1955)

    Google Scholar 

  55. ’t Hooft, G.: The free-will postulate in quantum mechanics. Herald of Russian Academy of Science vol. 81, pp. 907–911 (2011); ArXiv: arXiv:quant-ph/0701097v1 (2007)

  56. ’t Hooft, G.: Quantum gravity as a dissipative deterministic system. ArXiv: arXiv:gr-qc/9903084 (1999)

  57. ’t Hooft, G.: The mathematical basis for deterministic quantum mechanics. ArXiv: arXiv:quant-ph/0604008 (2006)

  58. Kolmogoroff, A.N.: Grundbegriffe der Wahrscheinlichkeitsrechnung (Berlin: Springer Verlag); English translation: Kolmogorov A N 1956 Foundations of Theory of Probability. Chelsea Publishing Company, New York (1933)

    Google Scholar 

  59. Khrennikov, A.: Interpretations of Probability, 2nd edn. De Gruyter, Berlin (2010)

    Google Scholar 

  60. Khrennikov A, Introduction to foundations of probability and randomness (for students in physics). Lectures given at the Institute of Quantum Optics and Quantum Information, Austrian Academy of Science, Lecture-1: Kolmogorov and von Mises. arXiv:1410.5773 [quant-ph]

  61. Khrennikov, A.: Fundamental Theories of Physics. Information dynamics in cognitive, psychological, social, and anomalous phenomena. Kluwer, Dordreht (2004)

    Google Scholar 

  62. Khrennikov, A.: Modelling of psychological behavior on the basis of ultrametric mental space: encoding of categories by balls. P-Adic Numbers Ultrametric Anal. Appl. 2(1), 1–20 (2010)

    MATH  MathSciNet  Article  Google Scholar 

  63. Khrennikov, A., Basieva, I., Dzhafarov, E.N., Busemeyer, J.R.: Quantum models for psychological measurements : an unsolved problem. PLoS One. 9. Article ID: e110909 (2014)

  64. Acacio de Barros, J.: Beyond the quantum formalism: consequences of a neural-oscillator model to quantum cognition. Advances in Cognitive Neurodynamics (IV). pp. 401–404. Springer, Netherlands (2015)

  65. Acacio de Barros, J.: Decision making for inconsistent expert judgments using negative probabilities. In Quantum Interaction, pp. 257–269. Springer, Berlin-Heidelberg, (2014)

  66. Khrennikov, A.: Bell-boole inequality: nonlocality or probabilistic incompatibility of random variables? Entropy 10, 19–32 (2008)

    MATH  MathSciNet  Article  ADS  Google Scholar 

  67. Hasegawa, Y., Loidl, R., Badurek, G., Filipp, S., Klepp, J., Rauch, H.: Quantum contextuality induced by spin-path entanglement in single-neutrons. Acata Phys. Hung. A Heavy Ion Phys. 26(1–2), 157–164 (2006)

    Google Scholar 

  68. Weihs, G., Jennewein, T., Simon, C., Weinfurter, R., Zeilinger, A.: Phys. Rev. Lett. 81, pp. 5039–5043 (1998)

  69. Giustina, M., Mech, A.l., Ramelow, S., Wittmann, B., Kofler, J., Beyer, J., Lita, A., Calkins, B., Gerrits, Th., Woo Nam, S., Ursin, R., Zeilinger, A.: Nature 497, pp. 227–230 (2013)

  70. Christensen, B.G., McCusker, K.T., Altepeter, J., Calkins, B., Gerrits, T., Lita, A., Miller, A., Shalm, L.K., Zhang, Y., Nam, S.W., Brunner, N., Lim, C.C.W., Gisin, N., Kwiat, P.G.: Phys. Rev. Lett. 111, pp. 1304–1306 (2013)

  71. Khrennikov, Y.A., Volovich, I.V.: Local Realism, Contextualism and Loopholes in Bell‘s Experiments. Proc. Conf. Foundations of Probability and Physics-2, Ser. Math. Modelling in Phys., Eng., and Cogn. Sc., 5, pp. 325–344, Växjö University Press, 2002

  72. Dragovich, B., Khrennikov, A., Kozyrev, S.V., Volovich, I.V.: On p-adic mathematical physics P-Adic Numbers, Ultrametric Analysis, and Applications, 1, N 1, pp. 1–17 (2009); arXiv:org/pdf/0904.4205.pdf

  73. Acacio de Barros, J., Oas, G.: Negative probabilities and counter-factual reasoning in quantum cognition. Phys. Scr. T163, 014008 (2014)

    Article  ADS  Google Scholar 

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Appendix 1: Quantum Nonlocality in Physics and Biology

Nonlocal interpretation of violation of Bell’s inequality led to revolutionary rethinking of foundations of QM. The impossibility to create theories with local hidden variables makes quantum randomness even more mystical than it was seen by fathers of QM, for example, by von Neumann [54]. In biology we have not yet seen violations of Bell’s inequality in so to say nonlocal setting. Nevertheless, it may be useful to comment possible consequences of such possible violations. Consider, for example, quantum modeling of cognition and the issue of mental nonlocality. The quantum-mental analogy has to be used with some reservations. The brain is a small physical system (comparing with distances covered by propagating light). Therefore “mental-nonlocality” (restricted to information states produced by a single brain) is not as mystical as physical nonlocality of QM. One may expect that in future cognition can be “explained”, e.g., in terms of “nonlocal” hidden variables. This position was presented by J. Acacio de Barros in [21] and it is reasonable.

“Quantum nonlocality” is still the subject of intense debates about interpretational and experimental issues. From the interpretation side, the main counter-argument against the common nonlocal interpretation is that, in fact, the main issue is not nonlocality, but contextuality of Bell’s experiments [66]: s tatistical data collected in a few (typically four) experimental contexts \(C_i\) are embedded in one inequality. However, from the viewpoint of CP each context \(C_i\) has to be represented by its own probability space. It is not surprising or mystical that such contextual data can violate the inequality which was derived for a single probability space [14, 59]. This viewpoint is confirmed by experiments in neutron interferometry [67] showing a violation of Bell’s inequality for a single neutron source, but in multi-context setting. In cognitive psychology a similar experiment, on recognition of ambiguous figures, was performed by Conte et al. [16], see [12] for the theoretical basis of this experiment.

The experimental situation in quantum physics is not so simple as it is presented by the majority of writers about violations of Bell’s inequality. There are known various loopholes in Bell’s experimental setups. A loophole appears in the process of physical realization of the ideal theoretically described experimental setup. One suddenly finds that what is possible to do in reality differs essentially from the textbook description. To close each loophole needs tremendous efforts and it is costly. The main problem is to close a few (in future all possible) loopholes in one experiment. Unfortunately, this “big problem” of combination of loophole closing is practically ignored. The quantum physical community is factually fine with the situation that different loopholes are closed in different types of experiments. Of course, for any logically thinking person this situation is totally unacceptable. The two main loopholes which disturb the project of the experimental justification of violation of Bell’s inequality are the detection efficiency loophole also known as fair sampling loophole and the locality loophole.

Locality loophole: It is very difficult, if possible at all, to remove safely two massive particles (i.e., by escaping decoherence of entanglement) to a distance which is sufficiently large to exclude exchange by signals having the velocity of light and modifying the initially prepared correlations. Therefore the locality loophole was closed by Weihs et al. [68] for photons, massless particles.

Detection efficiency/fair sampling loophole: Photo-detectors (opposite to detectors for massive particles) have low efficiency, an essential part of the population produced by a source of entangled photons is undetected. Thus a kind of post-selection is in charge. In terms of probability spaces, we can say that each pair of detectors (by cutting a part of population) produces random output described by its own probability space. Thus we again fall to the multi-contextual situation.Footnote 7 Physicists tried to “solve” this problem by proceeding under the assumption of fair sampling, i.e., that the detection selection does not modify the statistical features of the initial population. Recently novel photo-detectors of high efficiency (around 98%) started to be used in Bell’s tests. In 2013 two leading experimental groups closed the detection loophole with the aid of such detectors, in Vienna [69] (Zeilinger’s group) and in Urbana-Champaign [70] (Kwiat’s group).

Experiments in 2013 led to increase of expectations that finally both locality and detection loopholes would be soon closed in a single experiment. However, it seems that the appearance of super-sensitive detectors did not solve the problem completely. Photons disappear not only in the process of detection; they disappear in other parts of the experimental scheme: larger distance - more photons disappear. It seems that the real experimental situation matches with the prediction of A. Khrennikov and I. Volovich that the locality and detection loopholes would be never closed in a single experiment [71] (where by detection efficiency we understand the efficiency of the complete experimental setup): an analog of Heisenberg’s uncertainty relation for these loopholes.

Another problem of the experimental verification of violation of Bell’s inequality is that experiments closing one of previously known loopholes often suffer of new loopholes which were not present in “less advanced experiments”.

In general one has not to overestimate the value of Bell’s test; other test of nonclassicality, for example, violation of LTP, may be less controversial.

Finally, we make a philosophic remark: Bell’s story questions the validity of Popper’s principle of falsification of scientific theories. It seems that it is impossible to falsify the CP-model of statistical physics in any concrete experiment. It seems that one has to accept that the QP-formalism is useful not because it was proven that the CP-formalism is inapplicable, but because the QP-formalism is operationally successful and simple in use. (We remark that QP is based solely on linear algebra which much easier mathematically than the measure theory serving as the basis of CP).

Appendix 2: Exotic Quantum-Like Models: Possible Usefulness for Biological Information Theory

Quantum adaptive dynamics, although essentially deviating from the standard quantum formalism, is still based on complex Hilbert space. Coming back to the question posed in section (“why the quantum formalism?”) we remark that, in principle, there is no reason to expect that the biological information theory should be based on the representation of probabilities by complex amplitudes, normalized vectors of complex Hilbert space. Therefore one may try to explore models which are quite exotic, even comparing with quantum adaptive dynamics.

Some of these models are exotic only from the viewpoint of using of rather special mathematical tools; otherwise they arise very naturally from the probabilistic viewpoint. For example, we point to a novel model, so-called hyperbolic quantum mechanics, which was applied to a series of problems of cognition and decision making [14]. Here, the probability amplitudes are valued not in the field of the complex numbers, but in the algebra of the hyperbolic numbers, numbers of the form \(z=x+jy,\) where xy are real numbers and j is the generator of the algebra satisfying the equation \(j^2=+1.\) Hyperbolic amplitudes describe interference which is stronger than one given by the complex amplitudes used in QM: the interference term is given by the hyperbolic cosine, opposite to the ordinary trigonometric cosine in QM. In [14] there can be found the classification of various types of (probabilistic) interference exhibited in the form of violation of LTP; here the interference term is defined as the magnitude of deviation from LTP. It was found that the mathematical classification leads to only two basic possibilities: either the standard trigonometric interference or the hyperbolic one. Probabilistic data exhibiting the trigonometric interference can be represented in the complex Hilbert space and the data with the hyperbolic interference in the hyperbolic Hilbert space. The third possibility is mixture of the two types of interference terms. This leads to representation of probabilities in a kind of Hilbert space over hyper-complex numbers. We remind that in quantum foundations extended studies were performed to check usefulness of various generalizations of the complex Hilbert space formalism; for example, quaternionic QM or non-Archimedean QM [72]. It seems that such models were not so much useful in physics; in any event foundational studies did not lead to concrete experimental results. However, one might expect that such models, e.g., quanternionic QM, can find applications in quantum-like biological information. Possible usefulness of non-Archimedean, especially p-adic QM, will be discussed later in relation with unconventional probability models.

One of the widely known unconventional probabilistic model is based on relaxation of the assumption that a probability measure has to value in the segment [0, 1] of the real line. So-called “negative probabilities” appear with strange ragularity in a variety of physical problems , see, e.g., [59], for very detailed presentation (we can mention, for examples, the contributions of such leading physicists as P. Dirac, R. Feynman, A. Aspect). Recently negative probabilities started to be used in cognitive psychology and decision making [65, 73].

Negative probabilities appears naturally in p-adic quantum models as limits of relative frequencies with respect to p-adic topology on the set of rational numbers (frequencies are always rational). P-adic statistical stabilization characterizes a new type of randomness which is different from both classical and quantum randomness. P-adic probabilities were used in some biological applications [62]: cognition, population dynamics, genetics.

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Asano, M., Basieva, I., Khrennikov, A. et al. Quantum Information Biology: From Information Interpretation of Quantum Mechanics to Applications in Molecular Biology and Cognitive Psychology. Found Phys 45, 1362–1378 (2015).

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  • Quantum biological information
  • Quantum adaptive dynamics
  • Open quantum systems
  • Information interpretation
  • QBism
  • Molecular biology
  • Genetics
  • Cognition