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

Satoshi Uchiyama

Published Jan 1, 2005• 0 citations
OpenAlex

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, $2/\\pi$ ($\\simeq$ 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.

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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.
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.
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.
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.