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arXiv:2006.08066

Wigner and Bell Inequalities relationships and Kolmogorov's Axioms

Felipe Andrade Velozo, José A. C. Nogales

Published Jun 15, 2020• 0 citations
arXivHTMLPDF

Abstract

In this work, we show that Bell's inequality violation of arise from the fact that the condition imposed upon the development of inequality is not respected when it is applied in the idealized experiment. Such a condition is that the quantities taken by probability must be non negative, and such a condition is represented by $|\cal{P(w)}|=\cal{P(w)}$. We will also show that, when trying to define the values of the joint probabilities of $ (Z_1, Z_2, Z_3) $, through the values obtained from the $ (Z_j, Z_k) $ pairs, we find that these values are negative, so not Kolmogorov's axiom is respected: $\cal{P(w)}\geq0$ in cases where Bell's inequality is violated, and we also show that only such violation is possible if Wigner's inequality, in a certain arrangement, is violated, and that both violations are related to the violation of one of Kolmogorov's axioms. At the end of the paper, we suggest a new interpretation of the probabilities involved, in order to avoid the situation of negative probabilities and the violation of Bell's inequality and, consequently, Wigner's inequality.

DOI
10.48550/arxiv.2006.08066

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In this work, we show that Bell's inequality violation of arise from the fact that the condition imposed upon the development of inequality is not respected when it is applied in the idealized experiment. Such a condition is that the quantities taken by probability must be non negative, and such a condition is represented by ∣P(w)∣=P(w)|\cal{P(w)}|=\cal{P(w)}∣P(w)∣=P(w). We will also show that, when trying to define the values of the joint probabilities of (Z1,Z2,Z3)(Z_1, Z_2, Z_3)(Z1​,Z2​,Z3​), through the values obtained from the (Zj,Zk)(Z_j, Z_k)(Zj​,Zk​) pairs, we find that these values are negative, so not Kolmogorov's axiom is respected: P(w)≥0\cal{P(w)}\geq0P(w)≥0 in cases where Bell's inequality is violated, and we also show that only such violation is possible if Wigner's inequality, in a certain arrangement, is violated, and that both violations are related to the violation of one of Kolmogorov's axioms. At the end of the paper, we suggest a new interpretation of the probabilities involved, in order to avoid the situation of negative probabilities and the violation of Bell's inequality and, consequently, Wigner's inequality.
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.
Among the various "no-go theorems" in the foundations of quantum mechanics, Bell inequalities are among the few, which do not implicitly rely on the mathematical structure of quantum mechanics. Instead, Bell inequalities are commonly understood to rest on Kolmogorov's theory of probability and a "locality" assumption. Physical evidence supporting their violation is hence taken as evidence against at least one of these two assumptions. This work provides a comprehensive analysis of the physical debate on Bell inequalities from the point of view of Kolmogorov's theory of probability, focusing on the Bell-Clauser-Horn-Shimony-Holt inequality (BCHSH inequality). The central contribution is the motivation and mathematical study of the notion of contextuality in Bell experiments: in so-called contextual Bell models the detectors themselves behave randomly and the distribution of the random detector parameters depends on the chosen detector settings. It is shown how even "classical" Bell experiments motivate such a description. The need to account for contextuality was pointed out before by de la Peña, Cetto, Brody, Lochak, Bohm, Hiley and others. A broad class of contextual models is proven to satisfy the BCHSH inequality, thereby generalizing a previous result by Gill and Lambare. Nonetheless, even under strong locality requirements, contextual Bell models may violate the BCHSH inequality: based on work by Popescu and Rohrlich, we consider a "proof of concept" example exhibiting the maximum possible violation. Contrary to the much debated "loopholes" in empirical tests of Bell inequalities, contextuality thus reveals a conceptual misunderstanding in the physical conclusions commonly drawn from Bell inequality violations.
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.
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.
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 violation of Bell inequalities is often interpreted as showing that, if hidden variables exist, they must be contextual and non local. But they can also be explained questioning the probability space employed, or the validity of the Kolmogorov axioms. In this article we explore the additional constrains which can be deduced from two widely used objetive probability theories: frequentism and propensities. One of the strongest objections in the deduction of one version of Bell inequalities goes about the probability space, which assumes the existence of values for the output of the experiment in each run, while only two of the four values can be measured each time, making them counterfactual. It is shown that frequentism rejects the possibility of using counterfactual situations, while long-run propensities allow their use. In this case the introduction of locality and contextuality does not help to explain the violation, and an alternative explanation could point to a failure of the probability. Single case propensities were designed to associate probabilities to single events, but they need to be conditional to the whole universe, and do not have a clear link with the observed relative frequencies. It heavily limits their use.