Closing the detection loophole in multipartite Bell experiments with a limited number of efficient detectors

The problem of closing the detection loophole in Bell tests is investigated in the presence of a limited number of efficient detectors using emblematic multipartite quantum states. To this end, a family of multipartite Bell inequalities is introduced basing on local projective measurements conducted by $N-k$ parties and applying a $k$-party Bell inequality on the remaining parties. Surprisingly, we find that most of the studied pure multipartite states involving e.g. cluster states, the Dicke states, and the Greenberger-Horne-Zeilinger states can violate our inequalities with only the use of two efficient detectors, whereas the remaining detectors may have arbitrary small efficiencies. We believe that our inequalities are useful in Bell experiments and device-independent applications if only a small number of highly efficient detectors are in our disposal.