The Bell Inequality, Inviolable by Data Used Consistently with Its Derivation, Is Satisfied by Quantum Correlations Whose Probabilities Satisfy the Wigner Inequality

Abstract

It is not generally known that the inequality that Bell derived using three random variables must be identically satisfied by any three corresponding data sets of ±1's that are writable on paper. This surprising fact is not immediately obvious from Bell's inequality derivation based on causal random variables, but follows immediately if the same mathematical operations are applied to finite data sets. For laboratory data, the inequality is identically satisfied as a fact of pure algebra, and its satisfaction is independent of whether the processes generating the data are local, non-local, deterministic, random, or nonsensical. It follows that if predicted correlations violate the inequality, they represent no three cross-correlated data sets that can exist, or can be generated from valid probability models. Reported data that violate the inequality consist of probabilistically independent data-pairs and are thus inconsistent with inequality derivation. In the case of random variables as Bell assumed, the correlations in the inequality may be expressed in terms of the probabilities that give rise to them. A new inequality is then produced: The Wigner inequality, that must be satisfied by quantum mechanical probabilities in the case of Bell experiments. If that were not the case, predicted quantum probabilities and correlations would be inconsistent with basic algebra.

An Alternative View: Satisfaction of the Four-Variable Bell Inequality Using Quantum Correlations

“…The inequality imposes an algebraic constraint on the correlations leading to its satisfaction. The first three variable pairs occurring in (2.4) determine the fourth variable pair (the third pair is determined by the first two in the three variable case [6]):…”

“…Note that a computer model calculation of a Bell cosine correlation has been reported in [6] [15] where common information is given to two independent computers. Other examples probably exist in a literature of thousands of papers.…”

“…It was originated by Clauser, Horn, Shimony, and Holt, (CHSH) [1], and discussed at length by Bell [5]. It is also considerably more complex to analyze in application than the three variable case treated in [6]. Unfortunately, while the Bell inequality and its experimental violation are widely recognized, the true reason for this violation is only understood by a small but growing number of workers.…”