Bell inequalities from multilinear contractions

TL;DR

This paper introduces a geometric framework for Bell inequalities using multilinear contractions, linking quantum violations to non-commuting operators and extending previous non-linear inequality work.

Contribution

It generalizes Bell inequality derivations through multilinear contractions, providing new insights into quantum violations and extending results on the Peres conjecture.

Findings

Bell inequalities derived from multilinear contractions.

Quantum mechanics can violate these inequalities with three or more parties.

Unbounded quantum violations are possible within this framework.

Abstract

We provide a framework for Bell inequalities which is based on multilinear contractions. The derivation of the inequalities allows for an intuitive geometric depiction and their violation within quantum mechanics can be seen as a direct consequence of non-vanishing commutators. The approach is motivated by generalizing recent work on non-linear inequalities which was based on the moduli of complex numbers, quaternions and octonions. We extend results on Peres conjecture about the validity of Bell inequalities for quantum states with positive partial transposes. Moreover, we show the possibility of obtaining unbounded quantum violations albeit we also prove that quantum mechanics can only violate the derived inequalities if three or more parties are involved.