Bell’s correlation formula is based on a classical Kolmogorov probability theory. It is tested in experiment with an estimate correlation. The latter is the raw product moment (rpm) correlation that is classical Kolmogorov based as well. We note that the famous Bell inequalities are derived from Bell’s correlation formula. The problem is that the hypothesis: “is it possible that in experiment Bell’s correlation is equal to quantum”, can only found to be false. This is not because the data generating process does not produce the required data. Or that the inequalities sufficiently show that the data cannot violate the inequalities. It is because the statistics of the Bell experiment is flawed. The rpm correlation in experiment cannot be equal to the quantum value in principle. This is because it requires the classical Kolmogorov rpm correlation to be not Kolmogorov.
Bell’s correlation formula is based on a classical Kolmogorov probability theory. It is tested in experiment with an estimate correlation. The latter is the raw product moment (rpm) correlation that is classical Kolmogorov based as well. We note that the famous Bell inequalities are derived from Bell’s correlation formula. The problem is that the hypothesis: “is it possible that in experiment Bell’s correlation is equal to quantum”, can only found to be false. This is not because the data generating process does not produce the required data. Or that the inequalities sufficiently show that the data cannot violate the inequalities. It is because the statistics of the Bell experiment is flawed. The rpm correlation in experiment cannot be equal to the quantum value in principle. This is because it requires the classical Kolmogorov rpm correlation to be not Kolmogorov.
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.
It is often believed that de Finetti's coherence principle naturally leads, in the finite case, to the Kolmogorov's probability theory of random phenomena, which then implies Bell's inequality.Thus, not only a violation of Bell's inequality looks paradoxical in the Kolmogorovian framework, but it should violate also de Finetti's coherence principle.Firstly, we show that this is not the case: the typical theoretical violations of Bell's inequality in quantum physics are in agreement with de Finetti's coherence principle.Secondly, we look for statistical evidence of such violations: we consider the experimental data of measurements of polarization of photons, performed to verify empirically violations of Bell's inequality, and, on the basis of the estimated violation, we test the null hypothesis of Kolmogorovianity for the observed phenomenon.By standard inferential techniques we compute the pvalue for the test and get a clear strong conclusion against the Kolmogorovian hypothesis.
We show that by taking into account randomness of realization of experimental contexts it is possible to construct common Kolmogorov space for data collected for these contexts, although they can be incompatible. We call such a construction "Kolmogorovization" of contextuality. This construction of common probability space is applied to Bell's inequality. It is well known that its violation is a consequence of collecting statistical data in a few incompatible experiments. In experiments performed in quantum optics contexts are determined by selections of pairs of angles ( θ i ,θ ' j ) fixing orientations of polarization beam splitters. Opposite to the common opinion, we show that statistical data corresponding to measurements of polarizations of photons in the singlet state, e.g., in the form of correlations, can be described in the classical probabilistic framework. The crucial point is that in constructing the common probability space one has to take into account not only randomness of the source (as Bell did), but also randomness of context-realizations (in particular, realizations of pairs of angles ( θ i , θ ' j )). One may (but need not) say that randomness of "free will" has to be accounted for.
We show that by taking into account randomness of realization of experimental contexts it is possible to construct common Kolmogorov space for data collected for these contexts, although they can be incompatible. We call such a construction "Kolmogorovization" of contextuality. This construction of common probability space is applied to Bell's inequality. It is well known that its violation is a consequence of collecting statistical data in a few incompatible experiments. In experiments performed in quantum optics contexts are determined by selections of pairs of angles (θi,θ'j) fixing orientations of polarization beam splitters. Opposite to the common opinion, we show that statistical data corresponding to measurements of polarizations of photons in the singlet state, e.g., in the form of correlations, can be described in the classical probabilistic framework. The crucial point is that in constructing the common probability space one has to take into account not only randomness of the source (as Bell did), but also randomness of context-realizations (in particular, realizations of pairs of angles (θi, θ'j)). One may (but need not) say that randomness of "free will" has to be accounted for.
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.
Violation of Bell inequalities is widely regarded as a definitive test for non-locality. However, Bell correlational inequalities must always be satisfied by all jointly present, cross-correlated data. The correlations of variable pairs obtained in repeated runs are not cross-correlated in this way and are not required to satisfy the Bell inequality. In addition, by using information regarded as non-local, proper joint correlations may be computed among counterfactual and measured variables. These correlations satisfy the Bell inequality, but are spatially non-stationary in angle. By using a simple symmetry condition, such considerations may be extended to inequalities in probabilities. The latter may be derived directly from correlational inequalities developed by Clauser, Horne, Shimony and Holt (CHSH). Violation of either correlational or probabilistic Bell inequalities then implies that the Bell correlation cannot be accounted for by a stochastic process that is spatially stationary in angle coordinates. However, other processes may still be allowed.