Homogenization of Bell inequalities

TL;DR

This paper introduces a homogenization technique that transforms lower-order correlation Bell inequalities into higher-order CHSH-type inequalities, preserving their tightness and enabling more measurement settings.

Contribution

The paper presents a novel homogenization method that converts CH-type Bell inequalities into CHSH-type inequalities while maintaining their tightness, expanding the set of Bell inequalities.

Findings

Homogenization preserves the tightness of Bell inequalities.

Generated 3x3x3 CHSH-type inequalities from 2x2x2 CH-type inequalities.

Demonstrated the method with inequalities derived by Sliwa.

Abstract

A technique, which we call homogenization, is applied to transform CH-type Bell inequalities, which contain lower order correlations, into CHSH-type Bell inequalities, which are defined for highest order correlation functions. A homogenization leads to inequalities involving more settings, that is a choice of one more observable is possible for each party. We show that this technique preserves the tightness of Bell inequalities: a homogenization of a tight CH-type Bell inequality is still a tight CHSH-type Bell inequality. As an example we obtain $3\times3\times3$ CHSH-type Bell inequalities by homogenization of $2\times 2\times 2$ CH-type Bell inequalities derived by Sliwa in [Phys. Lett. A {\bf 317}, 165 (2003)].