Arithmetic loophole in Bell's Theorem: Overlooked threat to entangled-state quantum cryptography

Marek Czachor

Research output: Contribution to journalArticle

2 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)70-83
Number of pages14
JournalActa Physica Polonica A
Issue number1
Publication statusPublished - Jan 2021


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