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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. | 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=": | 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=":Asano M."> 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> | |||
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="Khrennikov A (2018)."> Khrennikov A., Basieva I., Pothos E.M., Yamato I.> Quantum Probability in Decision Making from Quantum Information Representation of Neuronal States, Sci. Rep., 8 (2018), Article 16225 />) (Sections 10, 11.3).</ref> | 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="Khrennikov A (2018)."> Khrennikov A., Basieva I., Pothos E.M., Yamato I.> Quantum Probability in Decision Making from Quantum Information Representation of Neuronal States, Sci. Rep., 8 (2018), Article 16225 />) (Sections 10, 11.3).</ref> | ||
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. | 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. | ||