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.
We analyse the proof of Bell's inequality and demonstrate that this inequality is related to one particular model of probability theory, namely Kolmogorov measure-theoretical axiomatics, 1933. We found a (numerical) statistical correction to Bell's inequality. Such an additional term in the right hand side of Bell's inequality can be considered as a probability invariant of a quantum state. This is a measure of nonreproducibility of hidden variables in different runs of experiments. Experiments to verify Bell's inequality can be considered as just experiments to estimate this constant. It seem…
In the mentioned paper we presented results of the estimation of Kolmogorov complexity of sequences of random numbers generated in a famous Bell's experiment, aimed to study the security of QKD. We focused on series of time differences between successive detections of coincidences, and found that randomness cannot be taken for granted. It was then criticized that the theorems that demonstrate the randomness of series produced in Bell's experiments involve series of measurement outcomes, not of measurement times. Here we reply to this objection and present data of series of outcomes, showing th…
We propose a formalism to derive the maximal bound of generalized Bell type inequalities and shows that the formalism can be applied to various form of Bell functions. The generic Bell function is defined to generate the combinations of all the possible correlations whose local realistic bound can be obtained from the series of the constraint equations. The application of the constraints converts the optimization problem into the counting problems whose complexity is dramatically reduced. It is also shown that generic Bell function can be used to generate many other known Bell type functions s…
We perform frequency analysis of the EPR-Bell argumentation. One of the main consequences of our investigation is that the existence of probability distributions of the Kolmogorov-type which was supposed by some authors is a mathematical assumption which may not be supported by actual physical quantum processes. In fact, frequencies for hidden variables for quantum particles and measurement devices may fluctuate from run to run of an experiment. These fluctuations of frequencies for micro-parameters need not contradict to the stabilization of frequencies for physical observables. If, neverthel…
Our aim is to emphasize the role of mathematical models in physics, especially models of geometry and probability. We briefly compare developments of geometry and probability by pointing to similarities and differences: from Euclid to Lobachevsky and from Kolmogorov to Bell. In probability Bell played the same role as Lobachevsky in geometry. In fact, violation of Bell's inequality implies the impossibility to apply the classical probability model of Kolmogorov (1933) to quantum phenomena. Thus quantum probabilistic model (based on Born's rule) is an example of non-Kolmogorovian model of proba…
Quantum systems are the ultimate touchstone for the production of random sequences of numbers. Spatially spread entangled systems allow the generation of identical random sequences in remote locations. The impossibility of observing a quantum system, without disturbing it, ensures that the messages encoded using these sequences cannot be eavesdropped. This is the basis of Quantum Key Distribution. It is then of crucial importance knowing whether the sequences generated in the practice by spatially spread entangled states are truly random, or not. Yet, that knowledge is not immediate. One of th…
We discuss coupling of violation of Bell's inequality and non-Kolmogorovness of statistical data in the EPR-Bohm experiment. We emphasize that nonlocalty and "death of realism" are only sufficient, but not necessary conditions of non-Kolmogorovness. There can be found other sufficient conditions of non-Kolmogorovness and, hence, violation of Bell's inequality. We find one important source of non-Kolmogorovness by analyzing axiomatics of quantum mechanics. We pay attention to the postulate (due to von Neumann and Dirac) on simultaneous measurement of quantum observables given by commuting opera…
Within the Dempster-Schafer theory of evidence a non-Kolmogorovian kind of epistemic uncertainty arises, which is encoded using multi-valued maps. We analyse the possible implications such non-Kolmogorovian epistemic uncertainty may have for Bell-type inequalities relating to the Einstein-Podolsky-Rosen-Bohm (EPRB) thought experiment. Our analysis leads to a notion of contextuality concerning complexes of physical measurement conditions. The use of multi-valued maps reveals an implicit link between this contextuality and counterfactual outcomes, and results in a formulation wherein the states …
We present an alternative approach to modeling Einstein-Podolsky-Rosen-Bohm (EPRB)-type experiments. The basis for our approach will be to replace the conventional Kolmogorov theory of probability, with the more general Dempster-Schafer theory of evidence. This leads to the identification of a distinct type of epistemic uncertainty, which is not present in Kolmogorov theory, but which is relevant in modeling experiments with operationally incompatible observables. Our treatment naturally yields conditional relative frequencies, which are perfectly compatible with classical determinism and loca…
We prove that the locality condition is irrelevant to Bell in equality. We check that the real origin of the Bell's inequality is the assumption of applicability of classical (Kolmogorovian) probability theory to quantum mechanics. We describe the chameleon effect which allows to construct an experiment realizing a local, realistic, classical, deterministic and macroscopic violation of the Bell inequalities.
We remind the viewpoint that violation of Bell's inequality might be interpreted not only as an evidence of the alternative -- either nonlocality or ``death of reality'' (under the assumption the quantum mechanics is incomplete). Violation of Bell's type inequalities is a well known sufficient condition of incompatibility of random variables -- impossibility to realize them on a single probability space. Thus, in fact, we should take into account an additional interpretation of violation of Bell's inequality -- a few pairs of random variables (two dimensional vector variables) involved in the …
Most of the standard proofs of the Bell theorem are based on the Kolmogorov axioms of probability theory. We show that these proofs contain mathematical steps that cannot be reconciled with the Kolmogorov axioms. Specifically we demonstrate that these proofs ignore the conclusion of a theorem of Vorob'ev on the consistency of joint distributions. As a consequence Bell's theorem stated in its full generality remains unproven, in particular, for extended parameter spaces that are still objective local and that include instrument parameters that are correlated by both time and instrument settings…
We perform the analysis of probabilistic assumptions of Bell's approach. We emphasize that J. Bell wrote about probability without to specify the concrete axiomatics of probability theory. The careful analysis demonstrated that (surprisingly) J. Bell did not apply the classical probability model (Kolmogorov) to describe ``classical physical framework.'' In fact, he created his own probabilistic model and compared it with the quantum one. The crucial point is that J. Bell did not pay attention to {\it conditional probabilities.} We show that conditional probability in his model cannot be define…
We derive a new class of time-dependent Bell inequalities in Wigner form under the assumption of locality in the framework of Kolmogorov's probability theory. We consider violation of the obtained inequalities for three cases: spin correlations in an external magnetic field; oscillations of neutral pseudoscalar mesons; and decays of a pseudoscalar into a fermion-antifermion pair.
Kolmogorov's first axiom of probability is probability takes values between 0 and 1; however, in Cox's derivation of probability having a maximum value of unity is arbitrary since he derives probability as a tool to rank degrees of plausibility. Probability can then be used to make inferences in instances of incomplete information, which is the foundation of Baysian probability theory. This article formulates a rule, which if obeyed, allows probability to take complex values and still be consistent with the interpretation of probability theory as being a tool to rank plausibility. It is then s…
We show that the noncontextual inequality proposed by Klyachko et al. [Phys. Rev. Lett. 101, 020403 (2008)] belongs to a broader family of inequalities, one associated to each compatibility structure of a set of events (a graph), and its independence number. These have the surprising property that the maximum quantum violation is given by the Lovasz theta-function of the graph, which was originally proposed as an upper bound on its Shannon capacity. Furthermore, probabilistic theories beyond quantum mechanics may have an even larger violation, which is given by the so-called fractional packing…
We develop an analogue of probability theory for probabilities taking values in topological groups. We generalize Kolmogorov's method of axiomatization of probability theory: main distinguishing features of frequency probabilities are taken as axioms in the measure-theoretic approach. We also present a review of non-Kolmogorovian probabilistic models including models with negative, complex, and $p$-adic valued probabilities. The latter model is discussed in details. The introduction of $p$-adic (as well as more general non-Archimedean) probabilities is one of the main motivations for considera…
We show that paradoxical consequences of violations of Bell's inequality are induced by the use of an unsuitable probabilistic description for the EPR-Bohm-Bell experiment. The conventional description (due to Bell) is based on a combination of statistical data collected for different settings of polarization beam splitters (PBSs). In fact, such data consists of some conditional probabilities which only partially define a probability space. Ignoring this conditioning leads to apparent contradictions in the classical probabilistic model (due to Kolmogorov). We show how to make a completely cons…
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 $(θ_i, θ^\prime_j)$ fix…
Counterfactual definiteness must be used as at least one of the postulates or axioms that are necessary to derive Bell-type inequalities. It is considered by many to be a postulate that is not only commensurate with classical physics (as for example Einstein's special relativity), but also separates and distinguishes classical physics from quantum mechanics. It is the purpose of this paper to show that Bell's choice of mathematical functions and independent variables implicitly includes counterfactual definiteness and reduces the generality of the physics of Bell-type theories so significantly…
Mainstream interpretations of quantum theory maintain that violations of the Bell inequalities deny at least either realism or Einstein locality. Here we investigate the premises of the Bell-type inequalities by returning to earlier inequalities presented by Boole and the findings of Vorob'ev as related to these inequalities. These findings together with a space-time generalization of Boole's elements of logic lead us to a completely transparent Einstein local counterexample from everyday life that violates certain variations of the Bell inequalities. We show that the counterexample suggests a…
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 experimen…
We show the presence of genuine quantum structures in human language. The neo-Darwinian evolutionary scheme is founded on a probability structure that satisfies the Kolmogorovian axioms, and as a consequence cannot incorporate quantum-like evolutionary change. In earlier research we revealed quantum structures in processes taking place in conceptual space. We argue that the presence of quantum structures in language and the earlier detected quantum structures in conceptual change make the neo-Darwinian evolutionary scheme strictly too limited for Evolutionary Epistemology. We sketch how we bel…
The probabilistic structure of quantum mechanics is investigated in the frequency framework. Such an approach can be interpreted as a contextual approach to quantum probabilities. By using rather complicated frequency calculations we reproduce the EPR-Bohm correlation function which is typically derived by using the calculus of probabilities in a Hilbert space. Our frequency probabilistic model of the EPR-Bohm experiment is a realist model -- physical observables are considered as objective properties of physical systems. It is also local -- a measurement over one part of a composite system do…
The celebrating theorem of A. Fine implies that the CHSH inequality is violated if and only if the joint probability distribution for the quadruples of observables involved the EPR-Bohm-Bell experiment does not exist, i.e., it is impossible to use the classical probabilistic model (Kolmogorov, 1933). In this note we demonstrate that, in spite of Fine's theorem, the results of observations in the EPR-Bohm-Bell experiment can be described in the classical probabilistic framework. However, the "quantum probabilities" have to be interpreted as conditional probabilities, where conditioning is with …
In this paper we propose a time-independent \textit{equality} and time-dependent \textit{inequality}, suitable for an experimental test of the hypothesis of realism. The derivation of these relations is based on the concept of conditional probability and on Bayes' theorem in the framework of Kolmogorov's axiomatics of probability theory. The equality obtained is intrinsically different from the well known GHZ-equality and its variants, because violation of the new equality might be tested in experiments with only two microsystems in a maximally entangled Bell state $\big\vert\, Ψ^-\,\big\rangl…
Some authors have raised the question whether the probabilities stemming from a quantum mechanical computation are entitled to enter the Bell and the Clauser-Horne inequalities. They have remarked that if the quantum probabilities are given the status of conditional ones and the statistics for the various settings of the detectors in a given experiment is properly kept into account, the inequalities happen to be no longer violated. In the present paper a classical simile modeled after the quantum mechanical instances is closely scrutinised. It is shown that the neglect of the conditional chara…
We propose a geometric multiparty extension of Clauser-Horne (CH) inequality. The standard CH inequality can be shown to be an implication of the fact that statistical separation between two events, $A$ and $B$, defined as $P(A\oplus B)$, where $A\oplus B=(A-B)\cup(B-A)$, satisfies the axioms of a distance. Our extension for tripartite case is based on triangle inequalities for the statistical separations of three probabilistic events $P(A\oplus B \oplus C)$. We show that Mermin inequality can be retrieved from our extended CH inequality for three subsystems. With our tripartite CH inequality,…
