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arXiv:1008.0667 · Fluctuation and Noise Letters

NOISE AND BELL'S INEQUALITY

David K. Ferry, László B. Kish

Published Aug 23, 2010• 1 citation
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Abstract

To many, the idea of the EPR paradox and the possibility of local hidden variables were dismissed by the Bell inequality, although the central points of this argument have been around since the advent of quantum mechanics. Yet, there remains considerable evidence that this inequality can be violated even by classical systems. The question really is whether or not strongly correlated classical fields will also violate Bell's inequality. In a previous paper, it was shown that this was the case. Here, we ask the question as to just how much correlation in the classical waves is required to violate the inequality.

DOI
10.1142/s0219477510000320

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To many, the idea of the EPR paradox and the possibility of local hidden variables were dismissed by the Bell inequality, although the central points of this argument have been around since the advent of quantum mechanics. Yet, there remains considerable evidence that this inequality can be violated even by classical systems. The question really is whether or not strongly correlated classical fields will also violate Bell's inequality. In a previous paper, it was shown that this was the case. Here, we ask the question as to just how much correlation in the classical waves is required to violate the inequality.
From the early days of quantum mechanics, there has been a discussion on the concept of reality, exemplified by the EPR paradox. To many, the idea of the paradox and the possibility of local hidden variables was dismissed by the Bell inequality. Yet, there remains considerable evidence that this inequality can be violated even by classical systems, so that experiments showing quantum behavior and the violation of the inequality must be questioned. Here, we demonstrate that classical optical polarization experiments can be shown to violate the Bell inequality. Hence, such experiments cannot be used to distinguish between classical and quantum theories.
Abstract One of the Bell’s assumptions in the original derivation of his inequalities was the hypothesis of locality, i.e., the absence of the influence of two remote measuring instruments on one another. That is why violations of these inequalities observed in experiments are often interpreted as a manifestation of the nonlocal nature of quantum mechanics, or a refutation of a local realism. It is well known that the Bell’s inequality was derived in its traditional form, without resorting to the hypothesis of locality and without the introduction of hidden variables, the only assumption being that the probability distributions are nonnegative. This can therefore be regarded as a rigorous proof that the hypothesis of locality and the hypothesis of existence of the hidden variables not relevant to violations of Bell’s inequalities. The physical meaning of the obtained results is examined. Physical nature of the violation of the Bell inequalities is explained under new EPR-B nonlocality postulate. We show that the correlations of the observables involved in the Bohm–Bell type experiments can be expressed as correlations of classical random variables. The revisited Bell type inequality in canonical notatons reads 〈AB〉 + 〈A’B〉 + 〈AB’〉 − 〈A’B’〉 ⩽ 6.
Most physicists agree that the Einstein–Podolsky–Rosen–Bell paradox exemplifies much of the strange behavior of quantum mechanics, but argument persists about what assumptions underlie the paradox. To clarify what the debate is about, we employ a simple and well-known thought experiment involving two correlated photons to help us focus on the logical assumptions needed to construct the EPR and Bell arguments. The view presented in this paper is that the minimal assumptions behind Bell’s inequality are locality and counterfactual definiteness but not scientific realism, determinism, or hidden variables as are often suggested. We further examine the resulting constraints on physical theory with an illustration from the many-worlds interpretation of quantum mechanics—an interpretation that we argue is deterministic, local, and realist but that nonetheless violates the Bell inequality.
By implicitly assuming that all possible Bell-measurements occur simultaneously, all proofs of Bell's Theorem violate Heisenberg's Uncertainty Principle. This assumption is made in the original form of Bell's inequality, in Wigner's probability inequalities, and in the ``nonlocality without inequalities'' arguments. The introduction of time into derivations of these variants of Bell's theorem results in extra terms related to the time order of the measurements used in constructing correlation coefficients. Since the same locality assumptions are made in the Heisenberg-compliant derivations of this paper, only time-independent classical local hidden variable theories are forbidden by violations of the original Bell inequalities; time-dependent quantum local hidden variable theories can satisfy this new bound and agree with experiment. We further point out that factorizable wavefunctions have been used to describe some EPR experiments and can be used to describe others. These will generate local de Broglie-Bohm trajectories in the description of the data. This second, independent, line of argument also shows that violation of Bell's inequality is only evidence that Heisenberg's Uncertainty Principle cannot be ignored.
The EPR paradox was caused by the provision that quantum variables must have pre-existing values. This type of "hidden property realism" was later falsified by Bell's Theorem. Accordingly, the physical basis for action-at-a-distance between entangled quanta was removed. Yet, modern interpretations present Bell's inequality as a Locality Criterion, as if Bell violations can only happen at the quantum level, and only with remote interactions. This is a questionable practice, considering that classical joint measurements also violate such inequalities for mutually exclusive wave properties. In particular, consecutive measurements of polarization produce the same coefficients of correlation as parallel measurements with entangled quanta, yet they are explicitly local. Furthermore, it is possible to combine parallel and consecutive measurements of Type I polarization-entangled photons in a single experiment, conclusively showing that quantum Bell violations can be local. Surprisingly, classical phenomena also require nonlocal interpretations if pre-existing properties are taken for granted. Hence, the solution is to reject the models with pre-existing properties for both classical and quantum wave-like phenomena.
Bell’s inequality is an expression which can be employed in order to test for the existence of hidden variables in a correlated system. The idea of hidden variables was introduced as a potential explanation for the quantum mechanical phenomenon of non-deterministic results of measurements. The existence of hidden variables can be tested experimentally. First, a particular measurement of one component of a correlated system must be associated with a positive or negative integer value. There must be an equal probability density of obtaining either measurement. Following this, an ensemble average of these measurements must be performed and Bell’s inequality must subsequently be employed. This theorem was essential in settling the EPR argument against quantum mechanical indeterminism, experimentally showing through violation of the inequality that there are indeed no hidden variables in a quantum system. For a system exhibiting classical behavior, Bell’s inequality should not be violated, provided the correlated system meets criteria appropriate to the assumptions underlying Bell’s theorem. A Mathematica model of Bell’s inequality was developed utilizing a Monte Carlo approach of randomizing parameters in order to test models of correlated systems. The model appears to agree with classical determinism within one standard deviation. It is possible to test both classical and quantum mechanical systems if properly correlated, and it is of future interest to test whether this extends to chaotic systems.