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 in a particular scenario. 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.

Keywords

• geometric multiparty extension
• CH inequality
• Kolmogorov theory