Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

CRC Press eBooks

Bell's Theorem

F.J. Duarte

Published Jan 15, 2024• 0 citations
Publisher

Abstract

A detailed derivation of Bell’s inequality is given. As usual, the probabilities in the inequality are replaced from the quantum probabilities originating from quantum entanglement probability amplitudes related to polarization rotation. The results from quantum polarization rotation ( Chapter 13 ) are introduced to successfully re-derive the quantum probability of quantum entanglement first calculated by Pryce and Ward in 1947. Examples are provided.

DOI
10.1201/9781003398707-14

Related papers

Quantum entanglement physics and Bell’s theorem

F.J. Duarte

The physics of quantum entanglement is described entirely by the superposition probability amplitude of quantum entanglement. Here, we review the various derivational origins of this iconic probability amplitude and point out that from a purely physics perspective its origin is completely independent of Bell’s theorem. The implications of this observation, towards hidden variables and newer forms of superdeterminism, are critically examined.

•Jun 4, 2024•0 citations

Quantum entanglement physics and Bell’s theorem

F.J. Duarte

The physics of quantum entanglement is described entirely by the superposition probability amplitude of quantum entanglement. Here, we review the various derivational origins of this iconic probability amplitude and point out that from a purely physics perspective its origin is completely independent of Bell’s theorem. The implications of this observation, towards hidden variables and newer forms of superdeterminism, are critically examined.

•Jun 4, 2024•0 citations

Entangled Polarizations: Probability Amplitudes and Experimental Configurations

F.J. Duarte

This chapter provides the ubiquitous equation for the probability amplitude of polarization entanglement of two photons moving away in different directions, from a common source. The first derivation is performed from a Hamiltonian approach using the two-state infrastructure taught to by R. P. Feynman. The second derivation utilizes an interferometric approach, while the third approach is based on the original analysis used by Ward. The initial link between quantum mechanical concepts and the polarization correlation of photons propagating in opposite directions was given by Wheeler: “According to the pair theory, if one of these photons is polarized in one plane, then the photon that goes off in the opposite direction with equal momentum is linearly polarized in the perpendicular plane.” The theory of entangled quantum polarizations, central to A. Einstein, B. Podolsky, and N. Rosen-type optical experiments, was established in the 1947–1949 period by M. H. Pryce and J. C. Ward and independently by H. S. Snyder et al.

CRC Press eBooks•Oct 29, 2013•0 citations

Bell's Theorem

F. J. Duarte

Hidden variable theories, as encouraged by the EPR paper, that proliferated in the 1950s, were taken quite seriously in some academic quarters, despite the existence of the 1932 von Neumann ‘proof’ that supposedly showed that these hidden theories were incompatible with the stochasticity of quantum mechanics. In 1964, John Bell, a particle theorist, reexamined the von Neumann proof, found it ‘wanting’, and offered his own proof that reaffirmed that hidden variable theories were incompatible with quantum mechanics. Bell added a twist: his proof was in the form a neat inequality theorem that was easy to comprehend and became an instant success. Bell's theorem led to a new generation of experimentalists to rediscover quantum entanglement, and to conduct these experiments in the visible spectrum, to test for evidence of hidden variable theories. The results were negative thus reaffirming the validity of quantum mechanics and its indeterminacy. In this chapter the premises of Bell's theorem are explained as well as its link to the probabilities of quantum entanglement.

CRC Press eBooks•Apr 10, 2026•0 citations

A Study on Bell's Theorem

Tan Tian-rong

This paper proves that Bell's inequality is originated from classical probability theory,and thereby is related neither to locality nor to hidden variables.But it is also possible to derive spin correlation formula in quantum mechanics according to the classical probabilistic theory.This fact indicates that on cer- tain conditions,classical probabilistic theory would be inconsistent with quantum mechanics,but under other conditions,classical probabilistic theory and quantum mechanics would,reach the same goal by different means.Still further,this paper confirms that all of the probability operation rules in classical probability theo- ry,containing joint probability operation rules,are suitable for micro processes,but the event operation rules in classical probability theory are unsuitable.The reason obtaining Bell's inequality lies in the steps applying Boolean algebra rules on the non-Boolean micro event space.

Journal of Changzhou Institute of Technology•Jan 1, 2008•0 citations

The Bell Inequalities: Identifying What Is Testable and What Is Not

Louis Sica

The Bell theorem and inequality were derived as consequences of seemingly reasonable physical and statistical hypotheses. Bell’s assumptions were used to deduce cross-correlations of three spin measurements on two entangled particles neglecting non-commutation. The assumed correlation functions, later confirmed for certain quantum measurements, violate the Bell inequality. The present paper reviews a more general derivation of the Bell inequality showing that it is identically satisfied by finite data sets whether deterministic or random, after assuming merely that they exist. It is thereafter concerned with the consequences of this result for interpretations of the inequality that result in its violation. A primary finding is that correlation functions have differing forms due to quantum commutation, non-commutation, and conditions of measurement, and result in satisfaction of the Bell inequality used consistently with its derivation. A stochastic process having the same correlation function for all variable pairs is shown to be inconsistent with experimentally reported data. The logic of the three and four variable inequalities is shown to be similar. Finally the inequalities in probabilities are shown to follow from those in correlations with quantum mechanical results satisfying either when properly implemented.

arXiv:1903.02116•Journal of Modern Physics•Jan 1, 2020•9 citations
A detailed derivation of Bell’s inequality is given. As usual, the probabilities in the inequality are replaced from the quantum probabilities originating from quantum entanglement probability amplitudes related to polarization rotation. The results from quantum polarization rotation ( Chapter 13 ) are introduced to successfully re-derive the quantum probability of quantum entanglement first calculated by Pryce and Ward in 1947. Examples are provided.
This chapter provides the ubiquitous equation for the probability amplitude of polarization entanglement of two photons moving away in different directions, from a common source. The first derivation is performed from a Hamiltonian approach using the two-state infrastructure taught to by R. P. Feynman. The second derivation utilizes an interferometric approach, while the third approach is based on the original analysis used by Ward. The initial link between quantum mechanical concepts and the polarization correlation of photons propagating in opposite directions was given by Wheeler: “According to the pair theory, if one of these photons is polarized in one plane, then the photon that goes off in the opposite direction with equal momentum is linearly polarized in the perpendicular plane.” The theory of entangled quantum polarizations, central to A. Einstein, B. Podolsky, and N. Rosen-type optical experiments, was established in the 1947–1949 period by M. H. Pryce and J. C. Ward and independently by H. S. Snyder et al.
Hidden variable theories, as encouraged by the EPR paper, that proliferated in the 1950s, were taken quite seriously in some academic quarters, despite the existence of the 1932 von Neumann ‘proof’ that supposedly showed that these hidden theories were incompatible with the stochasticity of quantum mechanics. In 1964, John Bell, a particle theorist, reexamined the von Neumann proof, found it ‘wanting’, and offered his own proof that reaffirmed that hidden variable theories were incompatible with quantum mechanics. Bell added a twist: his proof was in the form a neat inequality theorem that was easy to comprehend and became an instant success. Bell's theorem led to a new generation of experimentalists to rediscover quantum entanglement, and to conduct these experiments in the visible spectrum, to test for evidence of hidden variable theories. The results were negative thus reaffirming the validity of quantum mechanics and its indeterminacy. In this chapter the premises of Bell's theorem are explained as well as its link to the probabilities of quantum entanglement.
This paper proves that Bell's inequality is originated from classical probability theory,and thereby is related neither to locality nor to hidden variables.But it is also possible to derive spin correlation formula in quantum mechanics according to the classical probabilistic theory.This fact indicates that on cer- tain conditions,classical probabilistic theory would be inconsistent with quantum mechanics,but under other conditions,classical probabilistic theory and quantum mechanics would,reach the same goal by different means.Still further,this paper confirms that all of the probability operation rules in classical probability theo- ry,containing joint probability operation rules,are suitable for micro processes,but the event operation rules in classical probability theory are unsuitable.The reason obtaining Bell's inequality lies in the steps applying Boolean algebra rules on the non-Boolean micro event space.
The Bell theorem and inequality were derived as consequences of seemingly reasonable physical and statistical hypotheses. Bell’s assumptions were used to deduce cross-correlations of three spin measurements on two entangled particles neglecting non-commutation. The assumed correlation functions, later confirmed for certain quantum measurements, violate the Bell inequality. The present paper reviews a more general derivation of the Bell inequality showing that it is identically satisfied by finite data sets whether deterministic or random, after assuming merely that they exist. It is thereafter concerned with the consequences of this result for interpretations of the inequality that result in its violation. A primary finding is that correlation functions have differing forms due to quantum commutation, non-commutation, and conditions of measurement, and result in satisfaction of the Bell inequality used consistently with its derivation. A stochastic process having the same correlation function for all variable pairs is shown to be inconsistent with experimentally reported data. The logic of the three and four variable inequalities is shown to be similar. Finally the inequalities in probabilities are shown to follow from those in correlations with quantum mechanical results satisfying either when properly implemented.