Geometric extension of Clauser-Horne inequality to more qubits

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.