Store:QLMen03

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 $A$ $B$ 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 ${\hat {A}}$ and  ${\hat {B}}$ representing observables, i.e., $[{\hat {A}},{\hat {B}}]\neq 0$

Here we refer to the original and still basic and widely used model of quantum observables, Von Neumann 1955 (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, Busemeyer and Bruza, 2012, Bagarello, 2019) (see especially article (Bagarello et al., 2018) 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, Basieva and Khrennikov, 2015).

We shall explore more general theory of observations based on quantum instruments (Davies and Lewis, 1970, Davies, 1976, Ozawa, 1984, Yuen, 1987, Ozawa, 1997, Ozawa, 2004, Okamura and Ozawa, 2016) and find useful tools for applications to modeling of cognitive effects (Ozawa and Khrennikov, 2020a, Ozawa and Khrennikov, 2020b). 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), 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), 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).

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

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