Bell inequalities are derived assuming (i) hidden variables, (ii) positive probabilities for seemingly physical correlations, and (iii) locality. The over-riding role of assumption (ii) has generally not been emphasized. Since results of Bell inequality experiments show a violation of the inequality and agreement with quantum mechanical predictions, one or more of these assumptions is wrong. Thus, in the physical world, we cannot have hidden variables, and/or we must accept negative probabilities, and/or we must accept non-locality. Equivalently, the experiments tell us that any hidden variable theory (with associated non-negative probabilities) must be non-local; on the other hand, if a theory encompasses no hidden variables (e.g. quantum mechanics), the experiments do not make a statement about locality. Of course, the definition of “locality” plays a critical role, and that will be reviewed. In a previous paper (Phys. Lett. A 347, 56–61, 2005), it was shown that the assumption of hidden variables (e.g. seemingly physical correlations) leads directly to negative (non-physical) probabilities in the Wigner—Bell model. In this paper, we provide analyses based both on Bell's derivation of the inequality and on the Clauser—Horne version for inherently stochastic theories. We examine probabilities that must be non-negative in these derivations and show how to evaluate them within the framework of quantum mechanics. We repeatedly show that the assumption of hidden variables in the derivation of a Bell inequality leads to supposedly non-negative probabilities whose quantum mechanical counterparts are, in fact, negative under some conditions.
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Fry, E.S., Qu, X., Scully, M.O. (2009). Do Experimental Violations of Bell Inequalities Require a Nonlocal Interpretation of Quantum Mechanics? II: Analysis à la Bell. In: Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle. The Western Ontario Series in Philosophy of Science, vol 73. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9107-0_10
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