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2. Classical versus quantum probability

CP was mathematically formalized by Kolmogorov (1933) This is the calculus of probability measures, where a non-negative weight $p(A)$ is assigned to any event $A$. The main property of CP is its additivity: if two events $O_{1},O_{2}$ are disjoint, then the probability of disjunction of these events equals to the sum of probabilities:

$P(O_{1}\lor O_{2})=P(O_{1})+(O_{2})$

QP is the calculus of complex amplitudes or in the abstract formalism complex vectors. Thus, instead of operations on probability measures one operates with vectors. We can say that QP is a vector model of probabilistic reasoning. Each complex amplitude $\psi$ gives the probability by the Born’s rule: Probability is obtained as the square of the absolute value of the complex amplitude.

${\displaystyle P=|\psi |^{2}}$

(for the Hilbert space formalization, see Section 3.2, formula (7)). By operating with complex probability amplitudes, instead of the direct operation with probabilities, one can violate the basic laws of CP.

In CP, the formula of total probability (FTP) is derived by using additivity of probability and the Bayes formula, the definition of conditional probability, $P(O_{2}|O_{1})={\tfrac {P(O_{2})\cap (O_{1})}{PO_{1}}}$, $P(O_{1})>0$

Consider the pair,  and , of discrete classical random variables. Then

$P(B=\beta )=\sum _{\alpha }P(A=\alpha )P(B=\beta |A=\alpha )$

Thus, in CP the $B$-probability distribution can be calculated from the $A$-probability and the conditional probabilities $P(B=\beta |A=\alpha )$

In QP, classical FTP is perturbed by the interference term (Khrennikov, 2010); for dichotomous quantum observables $A$ and $B$ of the von Neumann-type, i.e., given by Hermitian operators ${\hat {A}}$ and ${\hat {B}}$, the quantum version of FTP has the form:

$P(B=\beta )=\sum _{\alpha }P(A=\alpha )P(B=\beta |A=\alpha )+2\sum _{\alpha _{1}<\alpha _{2}}\cos \theta _{\alpha _{1}\alpha _{2}}{\sqrt {P(A=\alpha _{1})P(B=\beta |A=\alpha _{1})}}P(A=\alpha _{2})P(B=\beta |a=\alpha _{2})$

$(1)$

$+2\sum _{\alpha _{1}<\alpha _{2}}\cos \theta _{\alpha _{1}\alpha _{2}}{\sqrt {P(A=\alpha _{1})P(B=\beta |A=\alpha _{1})}}P(A=\alpha _{2})P(B=\beta |a=\alpha _{2})$

$(2)$

If the interference term7 is positive, then the QP-calculus would generate a probability that is larger than its CP-counterpart given by the classical FTP (2). In particular, this probability amplification is the basis of the quantum computing supremacy.

There is a plenty of statistical data from cognitive psychology, decision making, molecular biology, genetics and epigenetics demonstrating that biosystems, from proteins and cells (Asano et al., 2015b) to humans (Khrennikov, 2010, Busemeyer and Bruza, 2012) use this amplification and operate with non-CP updates. We continue our presentation with such examples.