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

Marek Czachor

doi:10.12693/aphyspola.139.70

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

Abstract

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 arith-metic and calculus, can exactly reconstruct rotationally symmetric quantum probabilities typical oftwo-electron singlet states. Observable probabilities are consistent with the usual arithmetic employedby macroscopic observers but counterfactual aspects of Bell’s theorem are sensitive to the choice ofhidden-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-variableprobability densities. Probability densities have a Clauser–Horne product form typical of local realistictheories. However, neither the product nor the integral nor the representation of rotations are the usualones. The integral has all the standard properties but only with respect to the arithmetic that definesthe product. Certain formal transformations of integral expressions found in the usual proofs à la Belldo not work, so standard Bell-type inequalities cannot be proved. The system we deal with is de-terministic, local-realistic, rotationally invariant, observers have free will, detectors are perfect, hencethe system is free of all the canonical loopholes discussed in the literature

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Marek Czachor prof. dr hab.

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Category:: Articles
Type:: artykuły w czasopismach
Published in:: ACTA PHYSICA POLONICA A no. 139, pages 70 - 83,
ISSN: 0587-4246
Language:: English
Publication year:: 2021
Bibliographic description:: Czachor M.: Arithmetic Loophole in Bell's Theorem: Overlooked Threat to Entangled-State Quantum Cryptography// ACTA PHYSICA POLONICA A -Vol. 139,iss. 1 (2021), s.70-83
DOI:: Digital Object Identifier (open in new tab) 10.12693/aphyspola.139.70
Verified by:: Gdańsk University of Technology