The critical detection efficiency for closing the detection loophole of some modified Bell inequalities

As we know, the violations of Bell inequalities reveal nonlocality. On the other hand, loopholes in any Bell tests can cause the issues in the interpretation of the above conclusion, since an apparent violation of Bell inequality may not correspond to a real violation of local realism. The detection loophole, as an important example, arises when the overall detection efficiency is not larger than a certain threshold value. With the detection loophole, the above-mentioned effect can be observed in many experiments where the partial detected events can cause apparent Bell violations, while the entire ensemble cannot violate them. However, the real violations of Bell inequalities are necessary for device independence quantum information processing tasks. So, it is crucial to investigate the critical detection efficiency for closing the detection loophole of Bell inequalities. Here, by considering some novel Bell inequalities (Mironowicz and Pawlowski in Phys Rev A 88:032319, 2013), we give the critical detection efficiency for closing the detection loophole of these Bell inequalities. Furthermore, we prove the tightness of these detection efficiency bounds. That is, if detection efficiency is not larger than the critical one, we construct local hidden variable models to reproduce these violations. To sum up, if the detection efficiency of experiment exceeds the critical one derived here, the detection loophole is eliminated.