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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 name:"Khrennikov25>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>). | 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 name:"Khrennikov25>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=": | 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=":Ozawa M.(2004)"> 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" />) 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" />). | 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" />). | ||