Abstract
In this chapter we present briefly the basic notions of classical and quantum theories of probability and information. This chapter is especially important for biologists, psychologists, experts in cognition, and sociologists who were not trained in quantum theory (but even classical theory is presented in a simple manner). We start with the presentation of the standard measure-theoretic formulation of the modern classical probability theory (Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer, Berlin [1]). Then we turn to fundamentals of quantum formalism, including theory of open quantum systems and its generalizations.
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Notes
- 1.
We remark that, for a discrete random variable, the integral coincides with the sum for the mathematical expectation, see (2.5). And a discrete random variable is integrable if its mathematical expectation is well defined. In general any integrable random variable can be approximated by integrable discrete random variables and its integral is defined as the limit of the integrals for the approximating sequence.
- 2.
Here bar denotes complex conjugation ; for \(z=x+iy,\) \(\bar{z}=x-iy\).
- 3.
See Sect. 1.3 for a further discussion.
- 4.
In particular, by the orthodox Copenhagen interpretation \(\psi \) is interpreted as the physical state of a system. As a consequence, its collapse is a physical event.
- 5.
The set of all the eigenvectors is a linear subspace of \(\fancyscript{H}\).
- 6.
Even the majority of physicists have never read the von Neumann’s book [3] and they have no idea that von Neumann distinguished degenerate and non-degenerate cases. We are aware that this distinction may play an important role in biology.
- 7.
Decoherence is a complicated interpretational issue of quantum mechanics. Some (but not all) researchers treat decoherence as a form of measurement.
- 8.
In quantum information theory this equation is often referred as Lindblad equation.
- 9.
We treat the notion of decision very generally: from decisions made by people to “cell’s decisions”, e.g., to undergo epimutation.
- 10.
In the coordinate form tensor products of vectors and matrices are also known under the name Kronecker product . This structure is widely used in various computational algorithms including computational biology.
- 11.
- 12.
We remark that information features of entanglement can be modeled (mimic) by using coarse graining procedure for classical stochastic processes, even for classical Brownian motion [19]. In the latter paper the entanglement is exhibited at the level of observables corresponding to coarse graining.
- 13.
In computability theory, a decision problem, which has two possible answers, “yes” or “no”, for an input of question is studied well, and its complexity is classified into several complexity classes. NP (Nondeterministic Polynomial time) problem is a decision problem, whose solution is not given in polynomial time on a non-deternimistic Turing machine, and NP-complete problem is a NP problem reduced in polynomial time from any other NP problems. Whether NP-complete problem can be reduced to Polynomial problem is one of the millennium prize problems, it has been discussed for thirty yeas.
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Asano, M., Khrennikov, A., Ohya, M., Tanaka, Y., Yamato, I. (2015). Fundamentals of Classical Probability and Quantum Probability Theory. In: Quantum Adaptivity in Biology: From Genetics to Cognition. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9819-8_2
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