Imitating quantum probabilities: Beyond Bell's theorem and Tsirelson bounds

TL;DR

This paper presents a hierarchy of local hidden-variable models that can reproduce quantum correlations and explains violations of Bell inequalities as a language mixing problem, surpassing Tsirelson bounds.

Contribution

It introduces an infinite hierarchy of calculi-based local hidden-variable models, revealing violations beyond Tsirelson bounds as a result of hierarchical level mixing.

Findings

Bell inequality violations are due to language mixing.

Hierarchical models can reproduce quantum correlations.

Violations beyond Tsirelson bounds occur from non-neighboring level mixing.

Abstract

Local hidden-variable model of singlet-state correlations discussed in M. Czachor, Arithmetic loophole in Bell's Theorem: Overlooked threat to entangled-state quantum cryptography, Acta Phys. Polon. A 139, 70-83 (2021), is shown to be a particular case of an infinite hierarchy of local hidden-variable models based on an infinite hierarchy of calculi. Violation of Bell-type inequalities is shown to be a `confusion of languages' problem, a result of mixing different but neighboring levels of the hierarchy. Mixing of non-neighboring levels results in violations beyond the Tsirelson bounds.