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

TL;DR

This paper determines the minimum detection efficiency needed to achieve a loophole-free violation of Braunstein-Caves chained Bell inequalities for any number of settings greater than two, considering symmetric and asymmetric detection efficiencies.

Contribution

It provides the first comprehensive calculation of the minimum detection efficiency for loophole-free Bell inequality violations for arbitrary settings in Braunstein-Caves inequalities.

Findings

Minimum detection efficiency for symmetric detection cases.

Minimum detection efficiency for asymmetric detection cases.

Applicability to any number of measurement settings N > 2.

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 > 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.