Kolmogorov and von Mises viewpoints to the Greenburger-Horne-Zeilinger paradox

Abstract

We present comparative probabilistic analysis of the Greenburger-Horne-Zeilinger paradox in the frameworks of Kolmogorov's (measure-theoretical) and von Mises' (frequency) models of the probability theory. This analysis demonstrated that the GHZ paradox is merely a consequence of the use of Kolmogorov's probabilistic model. By using von Mises' frequency approach we escape the contradiction between the local realism and quantum formalism. The frequency approach implies automatically contextual interpretation of quantum formalism: different collectives induce different probability distributions. On the other hand, the formal use of Kolmogorov's model implies the identification of such distributions with one abstract Kolmogorov measure. In the measure-theoretical approach we can escape the paradox, if we do not suppose that probability distributions corresponding to different settings of measurement devices are equivalent. We discuss the connection between equivalence/singularity dichotomy in measure theory and the existence of compatible and noncompatible observables.