Strict detector-efficiency bounds for n-site Clauser-Horne inequalities

An analysis of detector-efficiency in many-site Clauser-Horne inequalities is presented for the case of perfect visibility. It is shown that there is a violation of the presented n-site Clauser-Horne inequalities if and only if the efficiency is greater than $n/\left(2n-1\right).$ Thus, for a two-site two-setting experiment there are no quantum-mechanical predictions that violate local realism unless the efficiency is greater than $\frac{2}{3}.$ Second, there are n-site experiments for which the quantum-mechanical predictions violate local realism whenever the efficiency exceeds $\frac{1}{2}.$