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.

Relevant new references added. Section of the physical meaning revised

References in corpus (4)

• Experimental loophole-free violation of a Bell inequality using entangled electron spins separated by 1.3 km
• Significant-loophole-free test of Bell's theorem with entangled photons
• A strong loophole-free test of local realism
• Hidden Variables and the Two Theorems of John Bell

Cited by in corpus (7)

• Local realistic interpretation of entangled photon pairs in the Weyl-Wigner formalism
• Stochastic interpretation of quantum mechanics assuming that vacuum fields are real
• Breaking of Bell inequalities from $S_4$ symmetry: the three orbits case
• The violation of Bell-CHSH inequalities leads to different conclusions depending on the description used
• Correlating Local Quantum Reality with Causally Disconnected Choices
• Violation of Bell Inequalities: Mapping the Conceptual Implications
• The arrow of time and the Bell inequalities