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How to (properly) strengthen Bell's theorem using counterfactuals

Tomasz Bigaj

Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern PhysicsJan 1, 20103 citations

Arithmetic Loophole in Bell's Theorem: Overlooked Threat to Entangled-State Quantum Cryptography

Marek Czachor

Bell's theorem is supposed to exclude all local hidden-variable models of quantum correlations.However, an explicit counterexample shows that a new class of local realistic models, based on generalized arithmetic and calculus, can exactly reconstruct rotationally symmetric quantum probabilities typical of two-electron singlet states.Observable probabilities are consistent with the usual arithmetic employed by macroscopic observers but counterfactual aspects of Bell's theorem are sensitive to the choice of hidden-variable arithmetic and calculus.The model is classical in the sense of Einstein, Podolsky, Rosen and Bell: elements of reality exist and probabilities are modeled by integrals of hidden-variable probability densities.Probability densities have a Clauser-Horne product form typical of local realistic theories.However, neither the product nor the integral nor the representation of rotations are the usual ones.The integral has all the standard properties but only with respect to the arithmetic that defines the product.Certain formal transformations of integral expressions found in the usual proofs à la Bell do not work, so standard Bell-type inequalities cannot be proved.The system we deal with is deterministic, local-realistic, rotationally invariant, observers have free will, detectors are perfect, hence the system is free of all the canonical loopholes discussed in the literature.Bell's theorem is supposed to exclude all local hidden-variable models of quantum correlations.However, an explicit counterexample shows that a new class of local realistic models, based on generalized arithmetic and calculus, can exactly reconstruct rotationally symmetric quantum probabilities typical of two-electron singlet states.Observable probabilities are consistent with the usual arithmetic employed by macroscopic observers but counterfactual aspects of Bell's theorem are sensitive to the choice of hidden-variable arithmetic and calculus.The model is classical in the sense of Einstein, Podolsky, Rosen and Bell: elements of reality exist and probabilities are modeled by integrals of hidden-variable probability densities.Probability densities have a Clauser-Horne product form typical of local realistic theories.However, neither the product nor the integral nor the representation of rotations are the usual ones.The integral has all the standard properties but only with respect to the arithmetic that defines the product.Certain formal transformations of integral expressions found in the usual proofs à la Bell do not work, so standard Bell-type inequalities cannot be proved.The system we deal with is deterministic, local-realistic, rotationally invariant, observers have free will, detectors are perfect, hence the system is free of all the canonical loopholes discussed in the literature.

arXiv:2004.04097Acta Physica Polonica AJan 1, 202116 citations

Note on ‘‘Bell’s theorem and the foundations of quantum physics’’ [Am. J. Phys. 5 3, 306 (1985)]

Reed A. Guy, Robert J. Deltete

American Journal of PhysicsJun 1, 19883 citations

Bell’s Theorem Tells Us Not What Quantum Mechanics Is, but What Quantum Mechanics Is Not

Marek Żukowski

arXiv:1501.05640The frontiers collectionNov 14, 20165 citations

Logical Difficulty from Combining Counterfactuals in the GHZ-Bell Theorems

Louis Sica

In eliminating the fair sampling assumption, the Greenberger, Horne, Zeilinger (GHZ) theorem is believed to confirm Bell’s historic conclusion that local hidden variables are inconsistent with the results of quantum mechanics. The GHZ theorem depends on predicting the results of sets of measurements of which only one may be performed. In the present paper, the noncommutative aspects of these unperformed measurements are critically examined. Classical examples and the logic of the GHZ construction are analyzed to demonstrate that combined counterfactual results of noncommuting operations are in general logically inconsistent with performed measurement sequences whose results depend on noncommutation. The Bell theorem is also revisited in the light of this result. It is concluded that negative conclusions regarding local hidden variables do not follow from the GHZ and Bell theorems as historically reasoned.In eliminating the fair sampling assumption, the Greenberger, Horne, Zeilinger (GHZ) theorem is believed to confirm Bell’s historic conclusion that local hidden variables are inconsistent with the results of quantum mechanics. The GHZ theorem depends on predicting the results of sets of measurements of which only one may be performed. In the present paper, the noncommutative aspects of these unperformed measurements are critically examined. Classical examples and the logic of the GHZ construction are analyzed to demonstrate that combined counterfactual results of noncommuting operations are in general logically inconsistent with performed measurement sequences whose results depend on noncommutation. The Bell theorem is also revisited in the light of this result. It is concluded that negative conclusions regarding local hidden variables do not follow from the GHZ and Bell theorems as historically reasoned.

arXiv:1305.2948Applied MathematicsJan 1, 20133 citations

Comment on ‘‘Bell’s theorem and the foundations of quantum physics’’ [Am. J. Phys. 5 3, 306 (1985)]

Frederick Fellows

American Journal of PhysicsJun 1, 19883 citations

Revisiting Bell's theorem to better see counterfactuals

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Revisiting Bell's theorem to better see counterfactuals

Abstract

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