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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Hidden-Variable Example Based upon Data Rejection

Philip Pearle

Published Oct 15, 1970• 619 citations
Publisher

Abstract

A deterministic local hidden-variable model is presented which describes the simultaneous measurement of the spins of two spin-\textonehalf{} particles which emerged from the decay of a spin-zero particle. In this model the measurement of the spin of a particle has one of three possible outcomes: spin parallel to the apparatus axis, spin antiparallel to the apparatus axis, or the particle goes undetected. It is shown that agreement with the predictions of quantum theory is obtained provided the experimenter rejects the "anomalous" data in which only one particle is detected. A reasonably model-independent lower bound to the fraction of undetected particles is also computed: It is found that in 14% of the decays or more, one or both of the particles will go undetected.

DOI
10.1103/physrevd.2.1418

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A deterministic local hidden-variable model is presented which describes the simultaneous measurement of the spins of two spin-\textonehalf{} particles which emerged from the decay of a spin-zero particle. In this model the measurement of the spin of a particle has one of three possible outcomes: spin parallel to the apparatus axis, spin antiparallel to the apparatus axis, or the particle goes undetected. It is shown that agreement with the predictions of quantum theory is obtained provided the experimenter rejects the "anomalous" data in which only one particle is detected. A reasonably model-independent lower bound to the fraction of undetected particles is also computed: It is found that in 14% of the decays or more, one or both of the particles will go undetected.
Pearle (1970) gave an example of a local hidden variables model which exactly reproduced the singlet correlations of quantum theory, through the device of data-rejection: particles can fail to be detected in a way which depends on the hidden variables carried by the particles and on the measurement settings. If the experimenter computes correlations between measurement outcomes of particle pairs for which both particles are detected, he or she is actually looking at a subsample of particle pairs, determined by interaction involving both measurement settings and the hidden variables carried in the particles. We correct a mistake in Pearle's formulas (a normalization error) and more importantly show that the model is simpler than first appears. We illustrate with visualizations of the model and with a small simulation experiment, with code in the statistical programming language R included in the paper. Open problems are discussed.
The experimental violation of the Bell inequality establishes necessary but not sufficient conditions that any theory must obey. Namely, a theory compatible with the experimental observations can satisfy at most two of the three hypotheses at the basis of Bell's theorem: free will, no signaling, and outcome independence. Quantum mechanics satisfies the first two hypotheses but not the latter. Experiments not only violate the Bell inequality, but show an excellent agreement with quantum mechanics. This fact restricts further the class of admissible theories. In this work, the author determines the form of the hidden-variable models that reproduce the quantum mechanical predictions for a spin singlet while satisfying both the hypotheses of free will and no signaling. Two classes of hidden-variable models are given as an example, and a general recipe to build infinitely many possible models is provided.
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.
Abstract We propose a novel class of local deterministic models to explain the correlations observed in entangled singlet quantum states. In these models, hidden variables influence both the entangled pair preparation and the measurement settings, thereby relaxing the measurement independence assumption used in Bell’s theorem. Our model accurately reproduces the expected quantum results and demonstrates that freely chosen measurement settings do not necessarily preclude the possibility of hidden variables inducing correlations between the settings and outcomes. This approach reveals an intricate relationship between local determinism and quantum mechanics, showing that local hidden variable theories can be consistent with the predictions of quantum theory. This result calls into question the conventional interpretation, based on Bell’s inequalities, that local theories are fundamentally incompatible with quantum mechanics.