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Bell-Test Experiments and Probabilities: EPR Paradox and Bell's Paradox Solved

Gerard van der Ham

Published Mar 1, 2021• 0 citations
OpenAlex

Abstract

In Bell-test experiments two kinds of probabilities appear: calculated probabilities and measured probabilities. The calculated probabilities (Bell's) cannot be detected by the detectors, whereas the measured probabilities (QM) are being detected by the detectors indirectly. The two kinds of probabilities are very different but both can be accounted for in a local-real way. They both can be made visible as projections of one and the same vector space in different directions. The differences between the kinds of probabilities are caused by the fact that detectors and observers perceive things like: vectors, vector spaces, particles and pairs of particles, differently. Observers perceive calculated probabilities (Bell) and detectors detect measured probabilities (QM) but only indirectly. In the paper both kinds of probabilities are being defined exactly. The exact definition of the probabilities makes clear why they appear as they do.

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In Bell-test experiments two kinds of probabilities appear: calculated probabilities and measured probabilities. The calculated probabilities (Bell's) cannot be detected by the detectors, whereas the measured probabilities (QM) are being detected by the detectors indirectly. The two kinds of probabilities are very different but both can be accounted for in a local-real way. They both can be made visible as projections of one and the same vector space in different directions. The differences between the kinds of probabilities are caused by the fact that detectors and observers perceive things like: vectors, vector spaces, particles and pairs of particles, differently. Observers perceive calculated probabilities (Bell) and detectors detect measured probabilities (QM) but only indirectly. In the paper both kinds of probabilities are being defined exactly. The exact definition of the probabilities makes clear why they appear as they do.
Bohm version of the EPR paradox Einstein, Podolsky, and Rosen (EPR) were concerned with the following issue. Given two spatially separated quantum systems A and B and an appropriate initial entangled state, a measurement of a property on system A can be an indirect measurement of B in the sense that from the outcome of the A measurement one can infer with probability 1 a property of B , because the two systems are correlated. There are cases in which either of two properties of B represented by noncommuting projectors can be measured indirectly in this manner, and EPR argued that this implied that system B could possess two incompatible properties at the same time, contrary to the principles of quantum theory. In order to understand this argument, it is best to apply it to a specific model system, and we shall do so using Bohm's formulation of the EPR paradox in which the systems A and B are two spin-half particles a and b in two different regions of space, with their spin degrees of freedom initially in a spin singlet state (23.2). As an aid to later discussion, we write the argument in the form of a set of numbered assertions leading to a paradox: a result which seems plausible, but contradicts the basic principles of quantum theory. The assertions E1–E4 are not intended to be exact counterparts of statements in the original EPR paper, even when the latter are translated into the language of spin-half particles. However, the general idea is very similar, and the basic conundrum is the same.
We show that paradoxical consequences of violations of Bell's inequality are induced by the use of an unsuitable probabilistic description for the EPR-Bohm-Bell experiment. The conventional description (due to Bell) is based on a combination of statistical data collected for different settings of polarization beam splitters (PBSs). In fact, such data consists of some conditional probabilities which only partially define a probability space. Ignoring this conditioning leads to apparent contradictions in the classical probabilistic model (due to Kolmogorov). We show how to make a completely consistent probabilistic model by taking into account the probabilities of selecting the settings of the PBSs. Our model matches both the experimental data and is consistent with classical probability theory.
We demonstrate that the EPR-Bohm probabilities can be easily obtained in the classical (but contextual) probabilistic framework by using the formula of interference of probabilities. From this point of view the EPR-Bell experiment is just an experiment on interference of probabilities. We analyse the time structure of contextuality in the EPR-Bohm experiment. The conclusion is that quantum mechanics does not contradict to a local realistic model in which probabilities are calculated as averages over conditionings/measurements for pairs of instances of time t1<t2.t_1< t_2.t1​<t2​. If we restrict our consideration only to simultaneous measurements at the fixed instance of time ttt we would get contradiction with Bell's theorem. One of implications of this fact might be the impossibility to define instances of {\it time with absolute precision} on the level of the contextual microscopic realistic model.
We demonstrate that the EPR-Bohm probabilities can be easily obtained in the classical (but contextual) probabilistic framework by using the formula of interference of probabilities. From this point of view the EPR-Bell experiment is just an experiment on interference of probabilities. We analyze the time structure of contextuality in the EPR-Bohm experiment. The conclusion is that quantum mechanics does not contradict to a local realistic model in which probabilities are calculated as averages over conditionings/measurements for pairs of instances of time t1 < t2. If we restrict our consideration only to simultaneous measurements at the fixed instance of time t we would get contradiction with Bell’s theorem. One of implications of this fact might be the impossibility to define instances of time with absolute precision on the level of the microscopic realistic model. 1. Introduction.In a series of papers, see, e.g., [1] it was demonstrated
The Bell inequality constrains the outcomes of measurements on pairs of distant entangled particles. The Bell contradiction states that the Bell inequality is inconsistent with the calculated outcomes of these quantum experiments. This contradiction led many to question the underlying assumptions, viz. so-called realism and locality. The probability model underlying the Bell inequality is generally left implicit. We propose an explicit probability model for the CHSH version of the Bell experiment. This model has only two simultaneously observable detector settings per measurement, and therefore does not assume realism. The quantum expectation now becomes a conditional expectation, given the two detector settings. This probability model is in full agreement with both quantum mechanics and experiments. As a result, the model satisfies the Bell inequality; there are no so-called violations. We extend this model to include a hidden variable. This extended model is not Bell-separable. This non-separability implies that the model is non-deterministic or non-local (or both).