Abstract
Since the experimental observation of the violation of the Bell-CHSH inequalities, much has been said about the non-local and contextual character of the underlying system. However, the hypothesis from which Bell’s inequalities are derived differ according to the probability space used to write them. The violation of Bell’s inequalities can, alternatively, be explained by assuming that the hidden variables do not exist at all, that they exist but their values cannot be simultaneously assigned, that the values can be assigned but joint probabilities cannot be properly defined, or that averages taken in different contexts cannot be combined. All of the above are valid options, selected by different communities to provide support to their particular research program.
Highlights
Quantum mechanics is a probabilistic theory, due to the central role played by the Born rule to relate the calculations with the observations
While some authors have pointed out that non-locality, as well as rejection of realism, are only sufficient conditions for violation of Bell’s inequality [26,27] and that the Bell inequalities only need to be satisfied if all observables can be measured jointly [28], it seems that the analysis of the violation of Bell’s inequality from the probability theory point of view is not fully understood in the physics community
The version of the inequality (16) expressed in probability space 1 is the most commonly found in the literature, and the closest with the original intention of the inequality written by Bell [3,42,44]
Summary
Quantum mechanics is a probabilistic theory, due to the central role played by the Born rule to relate the calculations with the observations. While some authors have pointed out that non-locality, as well as rejection of realism, are only sufficient (but not necessary) conditions for violation of Bell’s inequality [26,27] and that the Bell inequalities only need to be satisfied if all observables can be measured jointly [28], it seems that the analysis of the violation of Bell’s inequality from the probability theory point of view is not fully understood in the physics community In this contribution, our aim is to add more elements, in the common language of probability, to include a variety of interpretations of the violation of Bell’s inequality beyond non-locality. The quantum mechanical description perfectly matches the experimental results, Bell’s inequalities have consequences far beyond quantum physics, and they apply to a variety of generalized probabilistic theories [31] This adds to its relevance, but at the same time, there is no common basis to which all communities adhere for its application.
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