Two-setting multisite 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 ourselves 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 [ E. G. Cavalcanti, C. J. Foster, M. D. Reid and P. D. Drummond Phys. Rev. Lett. 99 210405 (2007)] provides a tighter bound, for high losses. We analyze the actual tight bound for the MABK inequalities, given the measure W=∏k=1Nηk of overall detection efficiency, where ηk is the efficiency at site k. Using these inequalities, we confirm that the maximally entangled Greenberger-Horne-Zeilinger state enables loophole-free falsification of LHV theories provided ∏k=1Nηk>2(2−N), which implies a symmetric threshold efficiency of η→50%, as N→∞. Furthermore, loophole-free violations remain possible, even when the efficiency at some sites is reduced well below 0.5, provided N>3.