Fig. 1: Maximum loophole-free violation of the CHSH inequality.
From: Self-testing tilted strategies for maximal loophole-free nonlocality
A plot of the maximum loophole-free value of the CHSH functional, \(C(\tilde{{\boldsymbol{p}}})={\eta }_{A}{\eta }_{B}{c}_{{\mathcal{Q}}}({\eta }_{A},{\eta }_{B})+(1-{\eta }_{A})(1-{\eta }_{B})2\), against detection efficiencies \({\eta }_{A},{\eta }_{B}\in [\frac{1}{2},1]\), where we used the analytical expression for maximum quantum violation of the doubly-tilted CHSH inequality (7), \({c}_{{\mathcal{Q}}}({\eta }_{A},{\eta }_{B})\), derived in Section “Two inefficient detectors”. The solid red line represents Bob’s critical detection efficiency \({\eta }_{B}^{* }=\frac{{\eta }_{A}}{3{\eta }_{A}-1}\)(8), below which a loophole-free quantum violation of the CHSH inequality is not possible26,27,28,29.
