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''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>). | ''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=": | 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=":Ozawa M. Uncertainty ">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)).]] | [[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). | 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). | ||