The Bell inequality, inviolable by data used consistently with its derivation, leads to quantum correlations that satisfy it, and probabilities that satisfy the Wigner inequality

TL;DR

This paper demonstrates that Bell's inequality is always satisfied by actual data sets and introduces the Wigner inequality, which quantum probabilities must satisfy, highlighting the algebraic consistency of quantum correlations.

Contribution

It reveals that Bell's inequality holds for any finite data sets and derives a new Wigner inequality that quantum probabilities must satisfy, ensuring algebraic consistency.

Findings

Bell's inequality is identically satisfied by any finite data set.

Experimental violations of Bell's inequality are inconsistent with valid probability models.

Quantum probabilities must satisfy the derived 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 plus and minus 1s 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, nonlocal, deterministic, random, or nonsensical.It follows that if predicted correlations violate the inequality, they represent no three cross correlated data sets that experimentally 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 results, 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.