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Applied Mathematics

Bell Correlations without Entanglement: A Local Wave Model Using Gaussian-Poisson Statistics and Single Count-Pair Selection

Louis Sica

Published Jan 1, 2014• 5 citations
Publisher

Abstract

Based on the violation of Bell inequalities, it has been believed that the derivation of Bell correlations requires a quantum description that depends on entanglement. However, the present paper computes Bell correlations among polarization analyzer output intensities from two spatially separated sets of superposed random wave pairs. To obtain proper Bell correlations, the general statistics must be modified to represent single event pair selection. The correlations between analyzer output components are then in one-to-one correspondence with those computed from the entanglement formalism.

DOI
10.4236/am.2014.518276

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Based on the violation of Bell inequalities, it has been believed that the derivation of Bell correlations requires a quantum description that depends on entanglement. However, the present paper computes Bell correlations among polarization analyzer output intensities from two spatially separated sets of superposed random wave pairs. To obtain proper Bell correlations, the general statistics must be modified to represent single event pair selection. The correlations between analyzer output components are then in one-to-one correspondence with those computed from the entanglement formalism.
Based on the Bell theorem and the resulting interpretation of the violation of Bell inequalities by experimental data, it has been believed that a theoretical computation of Bell correlations requires explicit use of the entanglement formalism. Physically this implies a permanent superposition of light waves in spite of the physical separation that occurs in the course of their propagation to spatially separated detectors. Nevertheless, in the present model, Bell correlations are calculated using explicitly separated waves with photon occurrence probabilities given by wave intensities based on nonlinear optics and quantum electrodynamics. The non-locality implied in the conventional calculation is thereby eliminated, but interference effects between photon excited waves and vacuum waves are assumed, as specified to be necessary in the design of Bell-experiment sources.
As discussed below, Bell's inequalities and experimental results rule out commutative hidden variable models as a basis for Bell correlations, but not necessarily non-commutative probability models. A local probability model is constructed for Bell correlations based on non-commutative operations involving polarizers. As in the entanglement model, the Bell correlation is obtained from a probability calculus without explicit use of deterministic hidden variables. The probability calculus used is associated with chaotic light. Joint wave intensity correlations at spatially separated polarization analyzers are computed using common information originating at the source. When interpreted as photon count rates, these yield quantum mechanical joint probabilities after the contribution of indeterminate numbers of photon pairs greater than one is subtracted out. The formalism appears to give a local account of Bell correlations.
In the derivation of Bell's inequalities, probability distribution is supposed to be a function of only hidden variable. We point out that the true implication of the probability distribution of Bell's correlation function is the distribution of the joint measurement outcomes on the two sides. So it is a function of both hidden variable and settings. In this case, Bell's inequalities fail. Our further analysis shows that Bell's locality holds neither for dependent events nor for independent events. We think that the measurements of EPR pairs are dependent events, thus violation of Bell's inequalities cannot rule out the existence of local hidden variable. In order to explain the results of EPR-type experiments, we suppose that polarization entangled photon pair can be composed of two circularly or linearly polarized photons with correlated hidden variables, and a couple of experiments of quantum measurement are proposed. The first uses delayed measurement on one photon of the EPR pair to demonstrate directly whether measurement on the other could have any non-local influence on it. Then several experiments are suggested to reveal the components of polarization entangled photon pair. The last one uses successive polarization measurements on a pair of EPR photons to show that two photons with a same quantum state will behave in the same way under the same measuring condition.
The original Bell inequality was obtained in a statistical derivation assuming three mutually cross-correlated random variables (four in the later version). Given that observations destroy the particles, the physical realization of three variables from an experiment producing two particles per trial requires two separate trial runs. One assumed variable value (for particle 1) occurs at a fixed instrument setting in both trial runs while a second variable (for particle 2) occurs at alternative instrument settings in the two trial runs. Given that measurements on the two particles occurring in each trial are themselves correlated, measurements from independent realizations at mutually exclusive settings on particle 2 are conditionally independent, i.e., conditionally dependent on particle 1, through probability. This situation is realized from variables defined by Bell using entangled particle pairs. Two correlations have the form that Bell computed from entanglement, but a third correlation from conditionally independent measurements has a different form. When the correlations are computed using quantum probabilities, the Bell inequality is satisfied without recourse to assumptions of non-locality, or non-reality.
Violation of Bell inequalities is widely regarded as a definitive test for non-locality. However, Bell correlational inequalities must always be satisfied by all jointly present, cross-correlated data. The correlations of variable pairs obtained in repeated runs are not cross-correlated in this way and are not required to satisfy the Bell inequality. In addition, by using information regarded as non-local, proper joint correlations may be computed among counterfactual and measured variables. These correlations satisfy the Bell inequality, but are spatially non-stationary in angle. By using a simple symmetry condition, such considerations may be extended to inequalities in probabilities. The latter may be derived directly from correlational inequalities developed by Clauser, Horne, Shimony and Holt (CHSH). Violation of either correlational or probabilistic Bell inequalities then implies that the Bell correlation cannot be accounted for by a stochastic process that is spatially stationary in angle coordinates. However, other processes may still be allowed.
Bell's theorem demonstrates that no local deterministic theory can reproduce the complete set of correlations predicted by standard quantum mechanics. The present work accepts Bell's theorem as mathematically correct but questions the physical interpretation of the quantum correlation function used in Bell analyses. It is argued that entangled particles inherit a complete common physical state at their creation and that subsequent measurements reveal selected observables of this state rather than generating new information or requiring superluminal communication. The paper further proposes that the standard quantum correlation function, (E(θ)=−cosθ), incorporates additional angular dependence beyond the information physically established at the source. Under this interpretation, the observed violation of Bell inequalities arises from the mathematical structure of the correlation function rather than from nonlocal information transfer between distant particles. An explicit local common-source model is presented, leading to a piecewise linear correlation function and motivating a decomposition of the standard quantum prediction into a local component and a supplementary angular term.