Minimum detection efficiency required for a loophole-free violation of the Braunstein-Caves chained Bell inequalities

Abstract

The chained Bell inequalities of Braunstein and Caves involving $N$ settings per observer have some interesting applications. Here we obtain the minimum detection efficiency required for a loophole-free violation of the Braunstein-Caves inequalities for any $N\ensuremath{\ge}2$. We discuss both the case in which both particles are detected with the same efficiency and the case in which the particles are detected with different efficiencies.