Quantum-Mechanical Correlations and Tsirelson Bound from Geometric Algebra

TL;DR

This paper demonstrates that the Tsirelson bound and quantum correlations are fundamentally geometric in nature, and that local hidden-variable theories can be compatible with quantum mechanics if physical quantities are treated as vectors rather than scalars.

Contribution

It reveals that the derivation of the Tsirelson bound depends on scalar assumptions and shows that vector-valued physical magnitudes can reconcile local hidden-variable theories with quantum mechanics.

Findings

Tsirelson bound has a geometric origin.

Local hidden-variable theories are compatible with QM when using vectors.

Quantum correlations depend on the geometric nature of physical quantities.

Abstract

The Bell-Clauser-Horne-Shimony-Holt inequality can be used to show that no local hidden-variable theory can reproduce the correlations predicted by quantum mechanics (QM). It can be proved that certain QM correlations lead to a violation of the classical bound established by the inequality, while all correlations, QM and classical, respect a QM bound (the Tsirelson bound). Here, we show that these well-known results depend crucially on the assumption that the values of physical magnitudes are scalars. The result implies, first, that the origin of the Tsirelson bound is geometrical, not physical; and, second, that a local hidden-variable theory does not contradict QM if the values of physical magnitudes are vectors.