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arXiv:quant-ph/0507032 · AIP conference proceedings

On a local hidden-variable model with ‘isolato’ hypothesis of the EPR-Bohm Gedanken experiment

Satoshi Uchiyama

Published Jan 1, 2006• 0 citations
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Abstract

A local hidden‐variable model based on ‘isolato’ hypothesis of the EPR‐Bohm Gedanken experiment is presented. The ‘isolato’ hypothesis states that one of a pair of spin‐1/2 particles can be in ‘isolato’ mode in which the spin‐1/2 particle shuts itself from any external interactions, and hence it can never be detected. As a result of this, data rejection is made; Bell’s inequality is violated, although the model is genuinely local. In this model, 2/π (≃ 63.6%) of the initially prepared ensemble of pairs of spin‐1/2 particles are detected as a pair of particles, and others are detected as a single particle. This does not disagree with the results of the experiments performed before, since these single spin‐1/2 particles were regarded as noise.

DOI
10.1063/1.2158748

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arXiv:0710.3510•AIP conference proceedings•Jan 1, 2007•20 citations
A local hidden‐variable model based on ‘isolato’ hypothesis of the EPR‐Bohm Gedanken experiment is presented. The ‘isolato’ hypothesis states that one of a pair of spin‐1/2 particles can be in ‘isolato’ mode in which the spin‐1/2 particle shuts itself from any external interactions, and hence it can never be detected. As a result of this, data rejection is made; Bell’s inequality is violated, although the model is genuinely local. In this model, 2/π (≃ 63.6%) of the initially prepared ensemble of pairs of spin‐1/2 particles are detected as a pair of particles, and others are detected as a single particle. This does not disagree with the results of the experiments performed before, since these single spin‐1/2 particles were regarded as noise.
A local hidden-variable model based on `isolato' hypothesis of the EPR--Bohm Gedanken experiment is presented. The `isolato' hypothesis states that one of a pair of spin-1/2{1}/{2}1/2 particles can be in `isolato' mode in which the spin-1/2{1}/{2}1/2 particle shuts itself from any external interactions, and hence it can never be detected. As a result of this, data rejection is made; Bell's inequality is violated, although the model is genuinely local. In this model, 2/pi2/\\pi2/pi (simeq\\simeqsimeq 63.6%) of the initially prepared ensemble of pairs of spin-1/2{1}/{2}1/2 particles are detected as a pair of particles, and others are detected as a single particle. This does not disagree with the results of the experiments performed before, since these single spin-1/2{1}/{2}1/2 particles were regarded as noise.
Measuring processes of a single spin-1/2 object and of a pair of spin-1/2 objects in the EPR-Bohm state are modeled by systems of differential equations. These models are based on mainly the following two ideas: (i) as a result of the interaction with a measuring apparatus, the state of the object, changes and approaches an attractor, in which the object possesses the measured value in the phase space of states of objects; (ii) a quantum-mechanical state corresponds to an ensemble of states in the corresponding attractor in the phase space. The latter model is a local model with hidden variables of the EPR-Bohm gedanken experiment. Although there is no dynamical interaction between the pair of spin-1/2 objects, the model reproduces approximately the quantum-mechanical correlations by using coincidence counting. Hence the Bell inequality is violated. This result supports the idea that the coincidence counting is the source of the apparent nonlocality in the EPR-Bohm gedankenexperiment.
John Bell and others used a locality condition to establish inequalities that they believe must be satisfied by any local hidden-variable model for the spin probability distribution for two entangled particles in a Bohm-EPR experiment. We show that this condition is invalid because it contradicts the product rule of probability theory for any model that exhibits the expected property of perfect correlation. This breaks the connection between Bell inequalities and the existence of any local hidden-variable model of interest. As already known, these inequalities give necessary conditions for the existence of third/fourth-order joint probability distributions for the spin outcomes from three/four separate Bohm-EPR experimental set-ups that are consistent with the second-order joint spin distributions for each experiment after marginalization. If a Bell inequality is violated, as quantum mechanics theory predicts and experiments show can happen, then at least one third-order joint probability is negative. However, this does not imply anything about the existence of local hidden-variable models for the second-order joint probability distributions for the spin outcomes of a single experiment. The locality condition does seem reasonable under the widely-applied frequentist interpretation of the spin probability distributions that views them as real properties of a random process that are manifested through their relative frequency of occurrence, which gives conditioning in the probabilities for the spin outcomes a causal role. In contrast, under the Bayesian interpretation of probability, probabilistic conditioning on one particle's spin outcome in the product rule is viewed as information to make probabilistic predictions of the other particle's spin outcome. There is nothing causal and so no reason to develop a locality condition.
Measuring processes of a single spin-1/2 object and of a pair of spin-1/2 objects in the EPR-Bohm state are modeled by systems of differential equations. These models are based on mainly the following two ideas: (i) as a result of the interaction with a measuring apparatus, the state of the object, changes and approaches an attractor, in which the object possesses the measured value in the phase space of states of objects; (ii) a quantum-mechanical state corresponds to an ensemble of states in the corresponding attractor in the phase space. The latter model is a local model with hidden variables of the EPR-Bohm gedanken experiment. Although there is no dynamical interaction between the pair of spin-1/2 objects, the model reproduces approximately the quantum-mechanical correlations by using coincidence counting. Hence the Bell inequality is violated. This result supports the idea that the coincidence counting is the source of the apparent nonlocality in the EPR-Bohm gedankenexperiment.
The EPR-Bohm experiment is studied in a classical model with hidden variables. The main hypothesis of Bell about this experiment must be modified in a non-trivial way to allow for the fact that our model is non-linear and so non fully deterministic. With our classical model, we then arrive at the same angular behaviour than the one that is predicted by Quantum Mechanics for the correlated measurement of spin projections on the polarizer axis. If the spin-spin interaction predicted by our particle model, or more precisely in the present case, the spin-polarizer field axis interaction, is not able to align the spin axis with the polarizer axis, then our classical result differs by a factor 3 from the quantum mechanical result. It is surprising to find that the angular behaviour is easier to reproduce by a classical model than the amount of probability. Naively, one would have expected the reverse.
The quantum theory (QT) and new stochastic approaches have no deterministic prediction for a single measurement or for a single time‐series of events observed for a trapped ion, electron or any other individual physical system. The predictions of QT being of probabilistic character apply to the statistical distribution of the results obtained in various experiments. The Copenhagen interpretation (CI) of QT acknowledged the abstract and statistical character of the predictions of QT but at the same time claimed that a state vector Ψ provided complete description of each individual physical system. The assigning the state vector to an individual physical system together with a postulate of its instantaneous reduction in the measurements was shown by Einstein Podolski and Rosen to lead to so called EPR paradox for the experiments with the entangled pairs of the particles. EPR concluded that a state vector could not provide a complete description of the individual systems and the question arose whether the probabilistic predictions of QT could be derived from some more fundamental spatio‐temporal deterministic description of the invisible sub‐phenomena by introduction of supplementary parameters. The experimental violation of the Bell inequalities in the spin polarization correlation experiments (SPCE) which were the implementations of Bohm and EPR gedanken experiments eliminated so called local and realistic models of the sub‐phenomena. Quite often the violation of these inequalities has been incorrectly interpreted as a proof of the completeness of QT or as the violation of the locality and causality in the micro‐world. The local and realistic models overlooked the fact that an experimental outcome is only the information about a particular system‐system or system‐instrument interaction. Namely the quantum phenomena are described in terms of the probabilities. It is well known that the probability distribution is not an attribute of a dice but it is a characteristic of a whole random experiment: “rolling a dice”. In this paper we will underline: that the quantum probabilities do not describe the lack of knowledge about the physical systems but the lack of knowledge about the experiments and therefore they are “contextual” and that the statistical long range correlations between two random variables X and Y are not a proof of any causal relation between these variables. Moreover any probabilistic model used to described a random experiment is consistent only with a specific protocol telling how the random experiment has to be performed. The probabilistic model used to prove Bell inequalities implied a protocol completely inappropriate and impossible to implement for SPCE . We conclude that the meaningful question about the predictable completeness of QT is still open and it should be answered by a careful and unconventional analysis of the experimental data. The correct understanding of statistical and contextual character of QT is essential for the research in the domain of quantum information and quantum computing.