Bell’s Inequality for Conditional Probabilities

Abstract

In Bell’s “classical probability model” there is no any definition of conditional probabilities. The only thing which is evident from Bell’s works is that conditional probability in his model cannot be defined by Bayes’ formula. We show this in this chapter. To prove this, we use the approach based on Bell-type inequalities in the conventional probabilistic approach, Kolmogorov’s model. We prove an analog of Wigner’s inequality, but for conditional probabilities (under assumption that they are defined by Bayes’ formula as it should be in Kolmogorov’s model). By using this inequality we show that predictions of the conventional (Kolmogorov) and quantum probability models disagree already in the case of non-composite systems.