Mathematical and physical meaning of the Bell inequalities

Abstract: It is shown that the Bell inequalities are closely related to the triangle inequalities involving distance functions amongst pairs of random variables with values $\left{ 0,1\right} $. A hidden variables model may be defined as a mapping between a set of quantum projection operators and a set of random variables. The model is noncontextual if there is a joint probability distribution. The Bell inequalities are necessary conditions for its existence. The inequalities are most relevant when measurements are performed at space-like separation, thus showing a conflict between quantum mechanics and local realism (Bell's theorem). The relations of the Bell inequalities with contextuality, Kochen-Specker theorem, and quantum entanglement are briefly discussed.