Bell's paradoxes, due to the fundamental properties of light and the nature of the photon, are discussed within a single framework with a view to checking the hypothesis that a stationary, non-negative, joint probability distribution function exists. This hypothesis, related to the local theory of hidden parameters as a possible interpretation of quantum theory, enables experimentally verifiable Bell's inequalities to be formulated. The dependence of these inequalities on the number of observers V is considered. Quantum theory predicts the breakdown of Bell's inequalities in optical experiments. It is shown that as V increases, the requirements on the quantum effectiveness of the detector, η, are reduced from η > 2/3 at V = 2 to η > 1/2 for V → ∞. Examples of joint probability distribution functions are given for illustrative purposes, and a way to resolve the Greenberg–Horne–Zeilinger (GHZ) paradox is suggested.
Bell's paradoxes, due to the fundamental properties of light and the nature of the photon, are discussed within a single framework with a view to checking the hypothesis that a stationary, non-negative, joint probability distribution function exists. This hypothesis, related to the local theory of hidden parameters as a possible interpretation of quantum theory, enables experimentally verifiable Bell's inequalities to be formulated. The dependence of these inequalities on the number of observers V is considered. Quantum theory predicts the breakdown of Bell's inequalities in optical experiments. It is shown that as V increases, the requirements on the quantum effectiveness of the detector, η, are reduced from η > 2/3 at V = 2 to η > 1/2 for V → ∞. Examples of joint probability distribution functions are given for illustrative purposes, and a way to resolve the Greenberg–Horne–Zeilinger (GHZ) paradox is suggested.
This chapter begins with a review of the ideas of locality and realism which provide the basis for the Einstein–Podolsky–Rosen paradox. The general conditions defining a local, realistic theory are stated, and then used to prove Bell's theorem, which yields — for any local, realistic theory — an inequality satisfied by certain linear combinations of correlations. It is shown that this inequality is violated by quantum theory and by experiment. The chapter ends with a discussion of possible experimental loopholes and the alternative Greenberger–Horne–Zeilinger (GHZ) state, which would, in principle, make the use of statistical correlations unnecessary.
Most of the standard proofs of the Bell theorem are based on the Kolmogorov axioms of probability theory. We show that these proofs contain mathematical steps that cannot be reconciled with the Kolmogorov axioms. Specifically we demonstrate that these proofs ignore the conclusion of a theorem of Vorob'ev on the consistency of joint distributions. As a consequence Bell's theorem stated in its full generality remains unproven, in particular, for extended parameter spaces that are still objective local and that include instrument parameters that are correlated by both time and instrument settings. Although the Bell theorem correctly rules out certain small classes of hidden variables, for these extended parameter spaces the standard proofs come to a halt. The Greenberger-Horne-Zeilinger (GHZ) approach is based on similar fallacious arguments. For this case we are able to present an objective local computer experiment that simulates the experimental test of GHZ performed by Pan, Bouwmeester, Daniell, Weinfurter and Zeilinger and that directly contradicts their claim that Einstein-local elements of reality can neither explain the results of quantum mechanical theory nor their experimental results.
One of the Bell's assumptions in the original derivation of his inequalities was the hypothesis of locality, i.e., of 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 local realism. In this paper, Bell's inequality is derived in its traditional form, without resorting to the hypothesis of locality, the only assumption being that the probability distributions are nonnegative. These probability distributions are calculated, for a specific optical experiment, in the framework of quantum theory and it is shown that they can take on negative values. This can therefore be regarded as a rigorous proof that the hypothesis of locality is not relevant to violations of Bell's inequalities. The physical meaning of the obtained results is examined.
In their original paper, Einstein, Podolsky, and Rosen (EPR) questioned the probabilistic nature of the quantum theory using an argument based on the quantum-mechanical correlations between two distant particles which had once interacted. Bell later proved that the quantum theory and any local theory differ significantly in their predictions for these experimentally measurable correlations. It is shown that the arguments of EPR can be extended to apply equally well to certain properties of a single particle or system and that, in analogy with Bell's theorem, the predictions of the quantum theory for certain single-particle experiments are in disagreement with the predictions of any local theory. For example, based on either the quantum theory or existing experimental observations, it is possible to arrange for a photon to have a negligible probability of not being absorbed by some system of finite dimensions, such as one of a large number of atoms, within a finite time interval. It is shown that any local theory which satisfies this requirement must also require that negligible interference effects occur in an optical interferometer at sufficiently low intensities from an incoherent source in combination with sufficiently large separation L between the two optical paths. Time-reversal invariance is used to estimate the requirement $L\ensuremath{\gg}\frac{3{\ensuremath{\lambda}}^{2}}{4{\ensuremath{\pi}}^{2}{r}_{c}}$, where $\ensuremath{\lambda}$ is the wavelength of the light and rc is the classical radius of the electron; at optical wavelengths this distance is on the order of 10 m. The same separation should also be required between the photon detectors in any optical experiment based on Bell's theorem. A review of the literature indicates that no existing experimental data, either from optical interferometers or experiments based on Bell's theorem, can be used to distinguish between the local and quantum theories. An investigation of low-intensity optical interference at large path separations is proposed for this purpose.
Introduction Since its inception, fifty years ago, Bell's theorem has had a long history not only of experimental tests but also of theoretical developments. Studying pairs of correlated quantummechanical particles separated in space, in a composite “entangled” state, Bell [1] showed that the joint ascription of hidden variables and locality to the system led to an inequality that is violated by the predictions of quantum mechanics. Fifteen years later, experiments confirmed the predictions of quantum mechanics, ruling out a large class of local realist theories. One of the most meaningful theoretical developments that followed Bell's work was the Greenberger–Horne–Zeilinger (GHZ) theorem, also known as Bell's theorem without inequalities. In 1989, the American physicists Daniel Greenberger and Michael Horne, who had been working on Bell's theorem since the late 1960s, together with the Austrian physicist Anton Zeilinger, introduced a novelty into the testing of entanglement, extending Bell's theorem in a different and interesting direction. According to Franck Laloë [2], For many years, everyone thought that Bell had basically exhausted the subject by considering all really interesting situations, and that two-spin systems provided the most spectacular quantum violations of local realism. It therefore came as a surprise to many when in 1989 Greenberger, Horne, and Zeilinger (GHZ) showed that systems containing more than two correlated particles may actually exhibit even more dramatic violations of local realism. The trio analyzed the Einstein, Podolsky, and Rosen 1935 argument once again and were able to write what are now called Greenberger–Horne–Zeilinger (GHZ) entangled states, involving three or four correlated spin-1/2 particles, leading to conflicts between local realist theories and quantum mechanics. However, unlike Bell's theorem, the conflict now was not of a statistical nature, as was the case with Bell's theorem, insofar as measurements on a single GHZ state could lead to conflicting predictions with local realistic models [3–5]. In this paper we present the history of the creation of this theorem and analyze its scope.
A hidden-variable theory is the traditional, but not unique, basis for constructing various types of Bell's theorem. The starting point may also be a recognition of the existence of a positive-definite probability distribution function. This assumption alone is used to formulate and prove Bell's paradoxes of different types. A specific example is used to show that a formal quantum calculation can sometimes give negative values of the joint probabilities that are used in the proof. An attempt is made to identify the physical meaning of this result and an algorithm for determination of negative joint probabilities of this type is proposed.