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Observations

In Lehrbüchern zur Quantenmechanik wird häufig darauf hingewiesen, dass das Hauptunterscheidungsmerkmal der Quantentheorie das Vorhandensein inkompatibler Observablen ist. Wir erinnern uns an diese zwei Observablen 𝐴 𝐵 und  sind inkompatibel, wenn es unmöglich ist, ihnen gemeinsam Werte zuzuordnen. Im probabilistischen Modell führt dies dazu, dass es unmöglich ist, ihre gemeinsame Wahrscheinlichkeitsverteilung (JPD) zu bestimmen. Die grundlegenden Beispiele für inkompatible Observable sind Position und Impuls eines Quantensystems oder Spin- (oder Polarisations-) Projektionen auf verschiedene Achsen. Im mathematischen Formalismus wird Inkompatibilität als Nichtkommutativität hermitescher Operatoren beschrieben Und Observablen darstellen, d.h.

Here we refer to the original and still basic and widely used model of quantum observables, Von Neumann 1955[1] (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[2], Busemeyer and Bruza, 2012[3], Bagarello, 2019[4]) (see especially article (Bagarello et al., 2018[5]) 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[6], Basieva and 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. ↑ 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
  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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