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Bell-Boole Inequality: Nonlocality or Probabilistic Incompatibility of Random Variables?
International Center for Mathematical Modeling in Physics, Engineering and Cognitive Science, Växjö University, S-35195, Sweden
Received: 10 September 2007; in revised form: 7 March 2008 / Accepted: 7 March 2008 / Published: 19 March 2008
Abstract: The main aim of this report is to inform the quantum information community about investigations on the problem of probabilistic compatibility of a family of random variables: a possibility to realize such a family on the basis of a single probability measure (to construct a single Kolmogorov probability space). These investigations were started hundred of years ago by J. Boole (who invented Boolean algebras). The complete solution of the problem was obtained by Soviet mathematician Vorobjev in 60th. Surprisingly probabilists and statisticians obtained inequalities for probabilities and correlations among which one can find the famous Bell’s inequality and its generalizations. Such inequalities appeared simply as constraints for probabilistic compatibility. In this framework one can not see a priori any link to such problems as nonlocality and “death of reality” which are typically linked to Bell’s type inequalities in physical literature. We analyze the difference between positions of mathematicians and quantum physicists. In particular, we found that one of the most reasonable explanations of probabilistic incompatibility is mixing in Bell’s type inequalities statistical data from a number of experiments performed under different experimental contexts.
Keywords: Bell’s inequality; nonlocality; “death of reality”; probabilistic incompatibility of random variables; Boole’s necessary condition; Vorobjev theorem; contextual description of the EPR-Bohm experiment; parameters of measurement devices; fluctuations of hidden variables of source.
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Khrennikov, A. Bell-Boole Inequality: Nonlocality or Probabilistic Incompatibility of Random Variables? Entropy 2008, 10, 19-32.
Khrennikov A. Bell-Boole Inequality: Nonlocality or Probabilistic Incompatibility of Random Variables? Entropy. 2008; 10(2):19-32.
Khrennikov, Andrei. 2008. "Bell-Boole Inequality: Nonlocality or Probabilistic Incompatibility of Random Variables?" Entropy 10, no. 2: 19-32.