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(Created page with "===Observations=== 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 <math>A</math> <math>B</math> 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 obse...")
 
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===Observations===
===Observations===
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 <math>A</math> <math>B</math> 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 <math>\hat{A}</math> and  <math>\hat{B}</math> representing observables, i.e.,  <math>[\hat{A},\hat{B}]\neq0</math>
Dans les manuels de mécanique quantique, il est communément souligné que la principale caractéristique distinctive de la théorie quantique est la présence d'observables incompatibles. Rappelons que deux observables <math>A</math> <math>B</math> et  sont incompatibles s'il est impossible de leur attribuer des valeurs conjointement. Dans le modèle probabiliste, cela conduit à l'impossibilité de déterminer leur distribution de probabilité conjointe (JPD). Les exemples de base d'observables incompatibles sont la position et la quantité de mouvement d'un système quantique, ou les projections de spin (ou de polarisation) sur différents axes. Dans le formalisme mathématique, l'incompatibilité est décrite comme la non-commutativité des opérateurs hermitiens <math>\hat{A}</math> et <math>\hat{B}</math> représentant les observables, c'est-à-dire <math>[\hat{A},\hat{B}]\neq0</math>
 
Nous nous référons ici au modèle original et toujours fondamental et largement utilisé des observables quantiques, Von Neumann 1955<ref>Von Neumann J. Mathematical Foundations of Quantum Mechanics Princeton Univ. Press, Princeton, NJ, USA (1955)</ref> (Section 3.2).
 
 
L'incompatibilité-non-commutativité est largement utilisée en physique quantique et les observables physiques de base, comme par exemple les projections de position et d'impulsion, de spin et de polarisation, sont traditionnellement représentées dans ce paradigme, par des opérateurs hermitiens. Nous pointons également de nombreuses applications de cette approche à la cognition, à la psychologie, à la prise de décision (Khrennikov, 2004a,<ref>Khrennikov A. Information Dynamics in Cognitive, Psychological, Social, and Anomalous Phenomena, Ser.: Fundamental Theories of Physics, Kluwer, Dordreht(2004)</ref> Busemeyer et 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>) (voir notamment l'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>) qui est consacré à la quantification de la Relations d'incertitude de Heisenberg dans la prise de décision). Pourtant, ce n'est peut-être pas assez général pour notre objectif - à la modélisation de type quantique en biologie, aucun type de biostatistique non classique ne peut être facilement délégué au modèle d'observations de von Neumann. Par exemple, même des effets cognitifs très basiques ne peuvent pas être décrits d'une manière cohérente avec le modèle d'observation standard (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 et 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>).


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.
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.

Revision as of 19:23, 8 April 2023

Observations

Dans les manuels de mécanique quantique, il est communément souligné que la principale caractéristique distinctive de la théorie quantique est la présence d'observables incompatibles. Rappelons que deux observables et  sont incompatibles s'il est impossible de leur attribuer des valeurs conjointement. Dans le modèle probabiliste, cela conduit à l'impossibilité de déterminer leur distribution de probabilité conjointe (JPD). Les exemples de base d'observables incompatibles sont la position et la quantité de mouvement d'un système quantique, ou les projections de spin (ou de polarisation) sur différents axes. Dans le formalisme mathématique, l'incompatibilité est décrite comme la non-commutativité des opérateurs hermitiens et représentant les observables, c'est-à-dire

Nous nous référons ici au modèle original et toujours fondamental et largement utilisé des observables quantiques, Von Neumann 1955[1] (Section 3.2).


L'incompatibilité-non-commutativité est largement utilisée en physique quantique et les observables physiques de base, comme par exemple les projections de position et d'impulsion, de spin et de polarisation, sont traditionnellement représentées dans ce paradigme, par des opérateurs hermitiens. Nous pointons également de nombreuses applications de cette approche à la cognition, à la psychologie, à la prise de décision (Khrennikov, 2004a,[2] Busemeyer et Bruza, 2012,[3] Bagarello, 2019[4]) (voir notamment l'article (Bagarello et al., 2018[5]) qui est consacré à la quantification de la Relations d'incertitude de Heisenberg dans la prise de décision). Pourtant, ce n'est peut-être pas assez général pour notre objectif - à la modélisation de type quantique en biologie, aucun type de biostatistique non classique ne peut être facilement délégué au modèle d'observations de von Neumann. Par exemple, même des effets cognitifs très basiques ne peuvent pas être décrits d'une manière cohérente avec le modèle d'observation standard (Khrennikov et al., 2014,[6] Basieva et Khrennikov, 2015[7]).


We shall explore more general theory of observations based on quantum instruments (Davies and Lewis, 1970[8], Davies, 1976[9], Ozawa, 1984[10], Yuen, 1987[11], Ozawa, 1997[12], Ozawa, 2004[13], Okamura and Ozawa, 2016[14]) and find useful tools for applications to modeling of cognitive effects (Ozawa and Khrennikov, 2020a[15], Ozawa and Khrennikov, 2020b[16]). 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[17]), 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)[18], 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[19]).

Within theory of open quantum systems, we model epigenetic evolution (Asano et al., 2012b[20], Asano et al., 2015b[21]) (Sections 9, 11.2) and performance of psychological (cognitive) functions realized by the brain (Asano et al., 2011[22], Asano et al., 2015b[21], Khrennikov et al., 2018[23]) (Sections 10, 11.3).

For mathematically sufficiently well educated biologists, but without knowledge in physics, we can recommend book (Khrennikov, 2016a[24]) combining the presentations of CP and QP with a brief introduction to the quantum formalism, including the theory of quantum instruments and conditional probabilities.

  1. ↑ Von Neumann J. Mathematical Foundations of Quantum Mechanics Princeton Univ. Press, Princeton, NJ, USA (1955)
  2. ↑ Khrennikov A. Information Dynamics in Cognitive, Psychological, Social, and Anomalous Phenomena, Ser.: Fundamental Theories of Physics, Kluwer, Dordreht(2004)
  3. ↑ Busemeyer J., Bruza P. Quantum Models of Cognition and Decision Cambridge Univ. Press, Cambridge(2012)
  4. ↑ Bagarello F. Quantum Concepts in the Social, Ecological and Biological Sciences Cambridge University Press, Cambridge (2019)
  5. ↑ 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
  6. ↑ Khrennikov A., Basieva I., DzhafarovE.N., Busemeyer J.R. Quantum models for psychological measurements: An unsolved problem. PLoS One, 9 (2014), Article e110909
  7. ↑ 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
  8. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named :3
  9. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named :4
  10. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named :5
  11. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named :6
  12. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named :7
  13. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named :8
  14. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named :9
  15. ↑ 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
  16. ↑ Ozawa M., Khrennikov A. Modeling combination of question order effect, response replicability effect, and QQ-equality with quantum instruments (2020)
  17. ↑ Ingarden R.S., Kossakowski A., Ohya M. Information Dynamics and Open Systems: Classical and Quantum Approach Kluwer, Dordrecht (1997)
  18. ↑ Schrödinger E. What Is Life? Cambridge university press, Cambridge (1944)
  19. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named :1
  20. ↑ 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 https://aip.scitation.org/doi/abs/10.1063/1.4773118
  21. ↑ 21.0 21.1 Asano M., Khrennikov A., Ohya M., Tanaka Y., Yamato I. Quantum Adaptivity in Biology: From Genetics To Cognition Springer, Heidelberg-Berlin-New York(2015)
  22. ↑ 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
  23. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named :0
  24. ↑ Khrennikov A. Probability and Randomness: Quantum Versus Classical Imperial College Press (2016)
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