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== Introduction ==


The standard mathematical methods were originally developed to serve classical physics. The real analysis served as the mathematical basis of Newtonian mechanics (Newton, 1687)<ref>{{cita libro 
 | autore = Newton Isaac
 | titolo = Philosophiae naturalis principia mathematica
 | url = https://archive.org/details/bub_gb_6EqxPav3vIsC
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 | anno = 1687
 | editore = Benjamin Motte
 | città = London UK
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 }}</ref> (and later Hamiltonian formalism); classical statistical mechanics stimulated the measure-theoretic approach to probability theory, formalized in Kolmogorov’s axiomatics (Kolmogorov, 1933)<ref>Kolmogorov A.N.Grundbegriffe Der Wahrscheinlichkeitsrechnung. Springer-Verlag, Berlin (1933)</ref>. However, behavior of biological systems differ essentially from behavior of mechanical systems, say rigid bodies, gas molecules, or fluids. Therefore, although the “classical mathematics” still plays the crucial role in biological modeling, it seems that it cannot fully describe the rich complexity of biosystems and peculiarities of their behavior — as compared with mechanical systems. New mathematical methods for modeling biosystems are on demand.(a,b)

In this paper, we present the applications of the mathematical formalism of quantum mechanics and its methodology to modeling biosystems’ behavior.(c) The recent years were characterized by explosion of interest to applications of quantum theory outside of physics, especially in cognitive psychology, decision making, information processing in the brain, molecular biology, genetics and epigenetics, and evolution theory.4 We call the corresponding models ''quantum-like''. They are not directed to micro-level modeling of real quantum physical processes in biosystems, say in cells or brains (cf. with biological applications of genuine quantum physical theory Penrose 1989,<ref>Penrose R. The Emperor’S New Mind Oxford Univ. Press, New-York (1989)</ref> Umezawa 1993,<ref>Umezawa H. Advanced Field Theory: Micro, Macro and Thermal Concepts AIP, New York (1993)</ref> Hameroff 1994,<ref>Hameroff S. Quantum coherence in microtubules. a neural basis for emergent con- sciousness? J. Cons. Stud., 1 (1994)</ref> Vitiello 1995,<ref>Vitiello G. Dissipation and memory capacity in the quantum brain model Internat. J. Modern Phys. B, 9 (1995), p. 973</ref> Vitiello 2001,<ref>Vitiello G. My Double Unveiled: The Dissipative Quantum Model of Brain, Advances in Consciousness Research, John Benjamins Publishing Company(2001)</ref> Arndt et al., 2009,<ref>Arndt M., Juffmann T., Vedral V. Quantum physics meets biology HFSP J., 3 (6) (2009), pp. 386-400, 10.2976/1.3244985</ref> Bernroider and Summhammer 2012,<ref>Bernroider G., Summhammer J. Can quantum entanglement between ion transition states effect action potential initiation? Cogn. Comput., 4 (2012), pp. 29-37</ref> Bernroider 2017<ref>Bernroider G. Neuroecology: Modeling neural systems and environments, from the quantum to the classical level and the question of consciousness J. Adv. Neurosci. Res., 4 (2017), pp. 1-9</ref>). Quantum-like modeling works from the viewpoint to quantum theory as a measurement theory. This is the original Bohr’s viewpoint that led to ''the Copenhagen interpretation of quantum mechanics'' (see Plotnitsky, 2009<ref>Plotnitsky A. Epistemology and Probability: Bohr, Heisenberg, SchrÖdinger and the Nature of Quantum-Theoretical Thinking Springer, Berlin, Germany; New York, NY, USA (2009</ref> for detailed and clear presentation of Bohr’s views). One of the main bio-specialties is consideration of ''self-measurements that biosystems perform on themselves.'' In our modeling, the ability to perform self-measurements is considered as the basic feature of biological functions (see Section 8.2 and paper Khrennikov et al., 2018<ref name=":0">Khrennikov A., Basieva I., PothosE.M., Yamato I. Quantum Probability in Decision Making from Quantum Information Representation of Neuronal States, Sci. Rep., 8 (2018), Article 16225</ref>).

''Quantum-like models'' (Khrennikov, 2004b<ref>Khrennikov A. On quantum-like probabilistic structure of mental information Open Syst. Inf. Dyn., 11 (3) (2004), pp. 267-275</ref>) reflect the features of biological processes that naturally match the quantum formalism. In such modeling, it is useful to explore ''quantum information theory,'' which can be applied not just to the micro-world of quantum systems. Generally, systems processing information in the quantum-like manner need not be quantum physical systems; in particular, they can be macroscopic biosystems. Surprisingly, the same mathematical theory can be applied at all biological scales: from proteins, cells and brains to humans and ecosystems; we can speak about ''quantum information biology'' (Asano et al., 2015a<ref name=":1">Asano M., Basieva I., Khrennikov A., Ohya M., Tanaka Y., Yamato I. Quantum information biology: from information interpretation of quantum mechanics to applications in molecular biology and cognitive psychology Found. Phys., 45 (10) (2015), pp. 1362-1378</ref>).

In quantum-like modeling, quantum theory is considered as calculus for prediction and transformation of probabilities. Quantum probability (QP) calculus (Section 2) differs essentially from classical probability (CP) calculus based on Kolmogorov’s axiomatics (Kolmogorov, 1933<ref name=":2">Kolmogorov A.N. Grundbegriffe Der Wahrscheinlichkeitsrechnung Springer-Verlag, Berlin (1933)</ref>). In CP, states of random systems are represented by probability measures and observables by random variables; in QP, states of random systems are represented by normalized vectors in a complex Hilbert space (pure states) or generally by density operators (mixed states).5 Superpositions represented by pure states are used to model uncertainty which is yet unresolved by a measurement. The use of superpositions in biology is illustrated by Fig. 1 (see Section 10 and paper Khrennikov et al., 2018<ref name=":0" /> for the corresponding model). The QP-update resulting from an observation is based on the projection postulate or more general transformations of quantum states — in the framework of theory of quantum instruments (Davies and Lewis, 1970<ref name=":3">Davies E.B., Lewis J.T. An operational approach to quantum probability Comm. Math. Phys., 17 (1970), pp. 239-260</ref>, Davies, 1976<ref name=":4">Davies E.B. Quantum Theory of Open Systems. Academic Press, London (1976)</ref>, Ozawa, 1984<ref name=":5">Ozawa M. Quantum measuring processes for continuous observables J. Math. Phys., 25 (1984), pp. 79-87</ref>, Yuen, 1987<ref name=":6">Yuen, H. P., 1987. Characterization and realization of general quantum measurements. M. Namiki and others (ed.) Proc. 2nd Int. Symp. Foundations of Quantum Mechanics, pp. 360–363.</ref>, Ozawa, 1997<ref name=":7">Ozawa M. An operational approach to quantum state reduction Ann. Phys., NY, 259 (1997), pp. 121-137</ref>, Ozawa, 2004<ref name=":8">Ozawa M. Uncertainty relations for noise and disturbance in generalized quantum measurements Ann. Phys., NY, 311 (2004), pp. 350-416</ref>, Okamura and Ozawa, 2016<ref name=":9">Okamura K., Ozawa M. Measurement theory in local quantum physics J. Math. Phys., 57 (2016), Article 015209</ref>) (Section 3).
[[File:Schrodinger 1.jpeg|left|thumb|Fig. 1. Illustration for quantum-like representation of uncertainty generated by neuron’s action potential (originally published in Khrennikov et al. (2018)).]]
We stress that quantum-like modeling elevates the role of convenience and simplicity of quantum representation of states and observables. (We pragmatically ignore the problem of interrelation of CP and QP.) In particular, the quantum state space has the linear structure and linear models are simpler. Transition from classical nonlinear dynamics of electrochemical processes in biosystems to quantum linear dynamics essentially speeds up the state-evolution (Section 8.4). However, in this framework “state” is the quantum information state of a biosystem used for processing of special quantum uncertainty (Section 8.2).

In textbooks on quantum mechanics, it is commonly pointed out that the main distinguishing feature of quantum theory is the presence of ''incompatible observables.'' We recall that two observables {{:F:Khrennikov001}} {{:F:Khrennikov002}} and  are incompatible if it is impossible to assign values to them jointly. In the probabilistic model, this leads to impossibility to determine their joint probability distribution (JPD). The basic examples of incompatible observables are position and momentum of a quantum system, or spin (or polarization) projections onto different axes. In the mathematical formalism, incompatibility is described as noncommutativity of Hermitian operators {{:F:Khrennikov003}} and  {{:F:Khrennikov004}}  representing observables, i.e., {{:F:Khrennikov005}} 

Here we refer to the original and still basic and widely used model of quantum observables, Von Neumann 1955<ref>Von Neumann J. Mathematical Foundations of Quantum Mechanics Princeton Univ. Press, Princeton, NJ, USA (1955)</ref> (Section 3.2).

Incompatibility–noncommutativity is widely used in quantumphysics and the basic physical observables, as say position and momentum, spin and polarization projections, are traditionally represented in this paradigm, by Hermitian operators. We also point to numerous applications of this approach to cognition, psychology, decision making (Khrennikov, 2004a<ref>Khrennikov A. Information Dynamics in Cognitive, Psychological, Social, and Anomalous Phenomena, Ser.: Fundamental Theories of Physics, Kluwer, Dordreht(2004)</ref>, Busemeyer and Bruza, 2012<ref name=":10">Busemeyer J., Bruza P. Quantum Models of Cognition and Decision Cambridge Univ. Press, Cambridge(2012)</ref>, Bagarello, 2019<ref>Bagarello F. Quantum Concepts in the Social, Ecological and Biological Sciences Cambridge University Press, Cambridge (2019)</ref>) (see especially article (Bagarello et al., 2018<ref>Bagarello F., Basieva I., Pothos E.M., Khrennikov A. Quantum like modeling of decision making: Quantifying uncertainty with the aid of heisenberg-robertson inequality J. Math. Psychol., 84 (2018), pp. 49-56</ref>) which is devoted to quantification of the Heisenberg uncertainty relations in decision making). Still, it may be not general enough for our purpose — to quantum-like modeling in biology, not any kind of non-classical bio-statistics can be easily delegated to von Neumann model of observations. For example, even very basic cognitive effects cannot be described in a way consistent with the standard observation model (Khrennikov et al., 2014<ref>Khrennikov A., Basieva I., DzhafarovE.N., Busemeyer J.R. Quantum models for psychological measurements: An unsolved problem. PLoS One, 9 (2014), Article e110909</ref>, Basieva and Khrennikov, 2015<ref>Basieva I., Khrennikov A. On the possibility to combine the order effect with sequential reproducibility for quantum measurements Found. Phys., 45 (10) (2015), pp. 1379-1393</ref>).

We shall explore more general theory of observations based on ''quantum instruments'' (Davies and Lewis, 1970<ref name=":3" />, Davies, 1976<ref name=":4" />, Ozawa, 1984<ref name=":5" />, Yuen, 1987<ref name=":6" />, Ozawa, 1997<ref name=":7" />, Ozawa, 2004<ref name=":8" />, Okamura and Ozawa, 2016<ref name=":9" />) and find useful tools for applications to modeling of cognitive effects (Ozawa and Khrennikov, 2020a<ref>Ozawa M., Khrennikov A. Application of theory of quantum instruments to psychology: Combination of question order effect with response replicability effect Entropy, 22 (1) (2020), pp. 37.1-9436</ref>, Ozawa and Khrennikov, 2020b<ref>Ozawa M., Khrennikov A. Modeling combination of question order effect, response replicability effect, and QQ-equality with quantum instruments (2020) </ref>). We shall discuss this question in Section 3 and illustrate it with examples from cognition and molecular biology in Sections 6, 7. In the framework of the quantum instrument theory, the crucial point is not commutativity vs. noncommutativity of operators symbolically representing observables, but the mathematical form of state’s transformation resulting from the back action of (self-)observation. In the standard approach, this transformation is given by an orthogonal projection on the subspace of eigenvectors corresponding to observation’s output. This is ''the projection postulate.'' In quantum instrument theory, state transformations are more general.

Calculus of quantum instruments is closely coupled with ''theory of open quantum systems'' (Ingarden et al., 1997<ref>Ingarden R.S., Kossakowski A., Ohya M. Information Dynamics and Open Systems: Classical and Quantum Approach Kluwer, Dordrecht (1997)</ref>), quantum systems interacting with environments. We remark that in some situations, quantum physical systems can be considered as (at least approximately) isolated. However, biosystems are fundamentally open. As was stressed by Schrödinger (1944)<ref>Schrödinger E. What Is Life? Cambridge university press, Cambridge (1944)</ref>, a completely isolated biosystem is dead. The latter explains why the theory of open quantum systems and, in particular, the quantum instruments calculus play the basic role in applications to biology, as the mathematical apparatus of quantum information biology (Asano et al., 2015a<ref name=":1" />).

Within theory of open quantum systems, we model epigenetic evolution (Asano et al., 2012b<ref>Asano M., Basieva I., Khrennikov A., Ohya M., Tanaka Y., Yamato I. Towards modeling of epigenetic evolution with the aid of theory of open quantum systems AIP Conf. Proc., 1508 (2012), p. 75 <nowiki>https://aip.scitation.org/doi/abs/10.1063/1.4773118</nowiki></ref>, Asano et al., 2015b<ref name=":11">Asano M., Khrennikov A., Ohya M., Tanaka Y., Yamato I. Quantum Adaptivity in Biology: From Genetics To Cognition Springer, Heidelberg-Berlin-New York(2015)</ref>) (Sections 9, 11.2) and performance of psychological (cognitive) functions realized by the brain (Asano et al., 2011<ref>Asano M., Ohya M., Tanaka Y., BasievaI., Khrennikov A. Quantum-like model of brain’s functioning: decision making from decoherence J. Theor. Biol., 281 (1) (2011), pp. 56-64</ref>, Asano et al., 2015b<ref name=":11" />, Khrennikov et al., 2018<ref name=":0" />) (Sections 10, 11.3).

For mathematically sufficiently well educated biologists, but without knowledge in physics, we can recommend book (Khrennikov, 2016a<ref>Khrennikov A. Probability and Randomness: Quantum Versus Classical Imperial College Press (2016)</ref>) combining the presentations of CP and QP with a brief introduction to the quantum formalism, including the theory of quantum instruments and conditional probabilities.