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Statistical maps. II: Operational random variables and the Bell phenomenon. (English) Zbl 1088.81022
This paper is a continuation of the author’s discussion of statistical maps [S. Bugajski, Math. Slovaca 51, No. 3, 321–342 (2001; Zbl 1088.81021)]. It is first proved that any family of operational random variables having independent outcomes can be represented by a single standard random variable. Nevertheless, it is then demonstrated that certain families of operational random variables exhibit the Bell phenomenon which manifests itself in quantum mechanics as the well known Bell inequalities and is impossible in the framework of traditional probability theory. This indicates that operational random variables provide a nontrivial extension of traditional random variables.

MSC:
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
60A99 Foundations of probability theory
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