Store:QLMen12

6.4. Mental realism

Since very beginning of quantum mechanics, noncommutativity of operators ${\widehat {A}},{\widehat {B}}$ representing observables $A,B$ was considered as the mathematical representation of their incompatibility. In philosophic terms, this situation is treated as impossibility of the realistic description. In cognitive science, this means that there exist mental states such that an individual cannot assign the definite values to both observables (e.g., questions). The mathematical description of QOE with observables represented by noncommutative operators (in the von Neumann’s scheme) in Wang and Busemeyer (2013) and Wang et al. (2014) made impression that this effect implies rejection of mental realism. The result of Ozawa and Khrennikov (2020a) demonstrates that, in spite of experimentally well documented QOE, the mental realism need not be rejected. QOE can be modeled within the realistic picture mathematically given by the joint probability distribution of observables $A$ and $B$, but with the noncommutative action of quantum instruments updating the mental state:

$[{\mathcal {J_{A}(x)}},{\mathcal {J_{B}(x)}}]={\mathcal {J_{A}(x)}}{\mathcal {J_{B}(x)}}-{\mathcal {J_{B}(x)}},{\mathcal {J_{A}(x)}}\neq 0$

$(20)$

This is the good place to remark that if, for some state $\rho ,[\Im _{A}(x),\Im _{B}(x)]\rho =0$, then QOE disappears, even if $[\Im _{A}(x),\Im _{B}(x)]\neq 0$. This can be considered as the right formulation of Wang–Bussemeyer statement on connection of QOE with noncommutativity. Instead of noncommutativity of operators  ${\widehat {A}}$ and  ${\widehat {B}}$ symbolically representing quantum obseravbles, one has to speak about noncommutativity of corresponding quantum instruments.

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7. Genetics: interference in glucose/lactose metabolism

In paper (Asano et al., 2012a), there was developed a quantum-like model describing the gene regulation of glucose/lactose metabolism in Escherichia coli bacterium.11 There are several types of E. coli characterized by the metabolic system. It was demonstrated that the concrete type of E. coli can be described by the well determined linear operators; we find the invariant operator quantities characterizing each type. Such invariant operator quantities can be calculated from the obtained statistical data. So, the quantum-like representation was reconstructed from experimental data.

Let us consider an event system $\{Q_{+},Q_{-}\}:Q_{+}$ means the event that E. coli activates its lactose operon, that is, the event that -galactosidase is produced through the transcription of mRNA from a gene in lactose operon; $Q_{-}$means the event that E. coli does not activates its lactose operon.

This system of events corresponds to activation observable  that is mathematically represented by a quantum instrument $\Im _{Q}$. Consider now another system of events $\{D_{L},D_{G}\}$ where $D_{L}$ means the event that an E. coli bacterium detects a lactose molecular in cell’s surrounding environment,  means $D_{G}$ detection of a glucose molecular. This system of events corresponds to detection observable $D$ that is represented by a quantum instrument $\Im _{D}$.

In this model, bacterium’s interaction–reaction with glucose/lactose environment is described as sequential action of two quantum instruments, first detection and then activation. As was shown in Asano et al. (2012a), for each concrete type of E. coli bacterium, these quantum instruments can be reconstructed from the experimental data; in Asano et al. (2012a), reconstruction was performed for W3110-type of E. coli bacterium. The classical FTP with observables  $A=D$ and $B=Q$ is violated, the interference term, see (2), was calculated (Asano et al., 2012a).