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Two-setting multi-site Bell inequalities for loophole-free tests with up to 50% loss

We consider Bell experiments with N spatially separated qubits where loss is present and restrict to two measurement settings per site. We note the Mermin-Ardehali-Belinskii-Klyshko (MABK) Bell inequalities do not present a tight bound for the predictions of local hidden variable (LHV) theories. The Holder-type Bell inequality derived by Cavalcanti, Foster, Reid and Drummond provides a tighter bound, for high losses. We analyse the actual tight bound for the MABK inequalities, given the measure W=\prod_{k=1}^{N}η_{k} of overall detection efficiency, where η_{k} is the efficiency at the site k . Using these inequalities, we confirm that the maximally entangled Greenberger-Horne-Zeilinger state enables loophole-free falsification of LHV theories provided \prod_{k=1}^{N}η_{k}>2^{(2-N)}, which implies a symmetric threshold efficiency of η\rightarrow50%, as N\rightarrow\infty . Furthermore, loophole-free violations remain possible, even when the efficiency at some sites is reduced well below 0.5, provided N>3 .