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===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>

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 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>).

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>

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.