Derivations of two Bell's inequalities are given in a form appropriate to the interpretation of experimental data for explicit determination of all the correlations. They are arithmetic identities independent of statistical reasoning and thus cannot be violated by data that meets the conditions for their validity. Two experimentally performable procedures are described to meet these conditions. Once such data are acquired, it follows that the measured correlations cannot all equal a negative cosine of angular differences. The relation between this finding and the predictions of quantum mechanics is discussed in a companion paper.
Derivations of two Bell's inequalities are given in a form appropriate to the interpretation of experimental data for explicit determination of all the correlations. They are arithmetic identities independent of statistical reasoning and thus cannot be violated by data that meets the conditions for their validity. Two experimentally performable procedures are described to meet these conditions. Once such data are acquired, it follows that the measured correlations cannot all equal a negative cosine of angular differences. The relation between this finding and the predictions of quantum mechanics is discussed in a companion paper.
The Bell theorem and inequality were derived as consequences of seemingly reasonable physical and statistical hypotheses. Bell’s assumptions were used to deduce cross-correlations of three spin measurements on two entangled particles neglecting non-commutation. The assumed correlation functions, later confirmed for certain quantum measurements, violate the Bell inequality. The present paper reviews a more general derivation of the Bell inequality showing that it is identically satisfied by finite data sets whether deterministic or random, after assuming merely that they exist. It is thereafter concerned with the consequences of this result for interpretations of the inequality that result in its violation. A primary finding is that correlation functions have differing forms due to quantum commutation, non-commutation, and conditions of measurement, and result in satisfaction of the Bell inequality used consistently with its derivation. A stochastic process having the same correlation function for all variable pairs is shown to be inconsistent with experimentally reported data. The logic of the three and four variable inequalities is shown to be similar. Finally the inequalities in probabilities are shown to follow from those in correlations with quantum mechanical results satisfying either when properly implemented.
The algebraic derivation of the numerical limits of Bell inequalities in either three or four random variables is independent of the assumption of randomness. The limits of the inequalities follow as mathematical consequences of their created algebraic structures independently of application to random or deterministic variables. The inequalities should be called identity-inequalities. A final correlation re-uses data from the previous correlations and thus leads to the inequality limits. It generally has a different functional form from the previous correlations, whether derived as a counterfactual mathematical result, or in a way enabling comparison with experiment. These algebraic facts and their consequences are central to understanding the inequalities’ use, but have not been widely recognized. Logically consistent application of the inequalities to Bell experiments is challenging, given that the number of mathematically assumed random variables is greater than the number of physical variables produced per experimental realization. Given Bell’s rejection of the use of sequential, alternative paths, three experimental runs are here considered to enable acquisition of data to be rearranged for computation of statistical cross-correlations. Predicted quantum mechanical correlations then satisfy the inequality. Since mathematically inconsistent use is sufficient to cause inequality violation, the conclusion that violation implies the nonexistence of underlying variables in the entanglement process does not follow.
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.
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.
Assumed data streams from a delayed choice gedanken experiment must satisfy a Bell's identity independently of locality assumptions. The violation of Bell's inequality by assumed correlations of identical form among these data streams implies that they cannot all result from statistically equivalent variables of a homogeneous process. This is consistent with both the requirements of arithmetic and distinctions between commuting and noncommuting observables in quantum mechanics. Neglect of these distinctions implies a logical loophole in the conventional interpretation of Bell's inequalities.
The Bell inequality is derived under the assumption of three physical data sets, random or deterministic. The data sets represent a laboratory realization of the three probability based variables used by Bell. For physical data as can be written on paper, the derivation of the inequality results only from principles of algebra and is independent of assumptions of locality, hidden variables and even randomness. Cross correlations of thee data sets carried out as Bell correlated three random variables results in the same inequality that is identically satisfied even by deterministic data. However, to obtain three data sets on two particles destroyed by measurement, two experimental runs are required, followed by data matching to reduce four data sets to three. If quantum mechanical probabilities are used to describe the data frequencies, the Bell inequality is satisfied. The situation is analogous to performing two sets of coin flips to compare the effect of two different coin loadings on the probabilities for heads and tails.