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arXiv:1108.0922 · Journal of Computational and Theoretical Nanoscience

The Assumption in Bell's Inequalities and Entanglement Problem

M. V. Lokajíček

Published Dec 1, 2012• 4 citations
arXivHTMLPDF

Abstract

Bell derived the given inequalities on the basis of one rather forceful assumption that was supposed to hold in the hidden variable theory. However, this assumption has been so strong that it has corresponded only to the classical physics; any dependence on spin orientations of individual objects (i.e., on quantum characteristics) having been practically excluded. And consequently, the hidden-variable theory has not been refused by corresponding experimental data. On the other side, Einstein's criticism of Copenhagen quantum mechanics based on ontological approach refusing nonlocality has been entitled as a much higher limit value corresponds to Bell's probability combination in such a case. And there is not any reason to believe in direct interaction between matter objects at greater distances as shown recently also by K. Hess on the basis of another approach.

DOI
10.1166/jctn.2012.2608

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Bell derived the given inequalities on the basis of one rather forceful assumption that was supposed to hold in the hidden variable theory. However, this assumption has been so strong that it has corresponded only to the classical physics; any dependence on spin orientations of individual objects (i.e., on quantum characteristics) having been practically excluded. And consequently, the hidden-variable theory has not been refused by corresponding experimental data. On the other side, Einstein's criticism of Copenhagen quantum mechanics based on ontological approach refusing nonlocality has been entitled as a much higher limit value corresponds to Bell's probability combination in such a case. And there is not any reason to believe in direct interaction between matter objects at greater distances as shown recently also by K. Hess on the basis of another approach.
In the early 1960s, John Bell decided to look once again at the hidden variable problem in quantum mechanics, through an examination of the A. Einstein, with B. Podolsky and N. Rosen (EPR) paradox. Proposals for proofs against the possibility of hidden variables had been advanced both before and after EPR. Bell developed both a “theorem” and an inequality, but this may be overstating the case. Most interpretations are that Bell is ruling out any “action-at-a-distance” type of interactions or variables, but would accept that such forces could violate the inequality. By triple of random variables, mentioned by A. Yu. Krennikov, one has to think once again about the experiment suggested by the Bell inequality. Bell has assumed that the set of hidden variables, and more importantly the probability function of the hidden variables, is unchanging. K. Hess and Walter Philipp began to worry about this assumption, because it has some implicit assumptions of its own.
Eight derivations of Bell’s inequality are given. First a simple and concrete derivation is given drawing on the full strength of the two hypotheses of locality and hidden variables. Two attempts at deriving Bell’s inequality without locality are made. They fail, but give valuable insight into the form which nonlocality must take in quantum physics. Arguments are given against this form of nonlocality. Ontological ideas which allow separation, realism, and locality in quantum mechanics (QM) are indicated. In the following five derivations, a number of variants of the assumption of hidden variables are tried. Among the insights which these derivations give rise to are: (1) Particles cannot be assumed to have spin or polarisation values for as much as one fixed direction. (2) The classical picture of electromagnetic radiation is incompatible with QM. (3) Heisenberg’s idea of potentiality in elementary particles is incompatible with QM. (4) There is a weakest form of hidden variables, called realisability, which is sufficient to yield Bell’s inequality. (5) Quantum states cannot be identified with physical states. The mathematical problem of the integration of quantum states into physical states is nontrivial.
Copenhagen quantum mechanics has required the existence of direct interaction between two objects at great distances, which was criticized by Einstein in 1935. However, the scientific community accepted the standpoint of Bohr that the given phenomenon may exist between microscopic objects. The given conclusion was then accepted practically by the whole physical community when the inequalities of Bell have been shown to be violated in corresponding experiments (in 1982). However, it has been shown recently by us that Bell’s inequalities were derived under a very strong assumption and could not be valid in the given experiment. Consequently, only the Schroedinger equation itself may be admitted as phenomenological description of the whole (microscopic and macroscopic) reality.
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 celebrated inequalities of Bell are based on the assumption that local hidden parameters exist. When combined with conflicting experimental results, these inequalities appear to prove that local hidden parameters cannot exist. This contradiction suggests to many that only instantaneous action at a distance can explain the Einstein, Podolsky, and Rosen type of experiments. We show that, in addition to the assumption that hidden parameters exist, Bell tacitly makes a variety of other assumptions that contribute to his being able to obtain the desired contradiction. For instance, Bell assumes that the hidden parameters do not depend on time and are governed by a single probability measure independent of the analyzer settings. We argue that the exclusion of time has neither a physical nor a mathematical basis but is based on Bell's translation of the concept of Einstein locality into the language of probability theory. Our additional set of local hidden variables includes time-like correlated parameters and a generalized probability density. We prove that our extended space of local hidden variables does not permit Bell-type proofs to go forward.
The basic problem we study here is whether the two-spin correlation experiments together with the theoretical Bell inequalities have already excluded the possibility of introducing hidden variables into quantum theory, as is often concluded. This question is answered negatively by explicitly reproducing the quantum-mechanical two-spin correlation function by a classical model, where the spin is associated with a classical dipolemoment vector. We then study the behavior of single events in classical and quantum models and conclude that the detector efficiency may be a fundamental limitation rather than just a technical problem to be overcome by better techniques. We further show that the assumptions underlying the derivation of Bell inequalities involve statements about single events which are consistent with neither the classical nor quantum models. It is important therefore to work with explicit physical situations rather than with abstract assumptions.