Geometric extension of Clauser-Horne inequality to more qubits

TL;DR

This paper introduces a geometric multiparty extension of the Clauser-Horne inequality using statistical separations and triangle inequalities, enabling systematic derivation of Bell-type inequalities for multiple qubits and states.

Contribution

It presents a novel geometric framework for extending CH inequalities to more qubits, connecting with Mermin inequalities and allowing systematic derivation of Bell-type inequalities.

Findings

Mermin inequality derived from extended CH inequality for three qubits

Quantum violations observed with GHZ-type and W-type states

Method generalizes to more subsystems and measurement settings

Abstract

We propose a geometric multiparty extension of Clauser-Horne (CH) inequality. The standard CH inequality can be shown to be an implication of the fact that statistical separation between two events, $A$ and $B$, defined as $P(A\oplus B)$, where $A\oplus B=(A-B)\cup(B-A)$, satisfies the axioms of a distance. Our extension for tripartite case is based on triangle inequalities for the statistical separations of three probabilistic events $P(A\oplus B \oplus C)$. We show that Mermin inequality can be retrieved from our extended CH inequality for three subsystems. With our tripartite CH inequality, we investigate quantum violations by GHZ-type and W-type states. Our inequalities are compared to another type, so-called $N$-site CH inequality. In addition we argue how to generalize our method for more subsystems and measurement settings. Our method can be used to write down several Bell-type inequalities in a systematic manner.