Quantum nonlocality with arbitrary limited detection efficiency

The demonstration and use of nonlocality, as defined by Bell's theorem, rely strongly on dealing with non-detection events due to losses and detector inefficiencies. Otherwise, the so-called detection loophole could be exploited. The only way to avoid this is to have detection efficiencies that are above a certain threshold. We introduce the intermediate assumption of limited detection efficiency, e.g. in each run of the experiment the overall detection efficiency is lower bounded by η_min > 0. Hence, in an adversarial scenario, the adversaries have arbitrary large but not full control over the inefficiencies. We analyze the set of possible correlations that fulfil Limited Detection Locality (LDL) and show that they necessarily satisfy some linear Bell-like inequalities. We prove that quantum theory predicts violation of one of these inequalities for all η_min > 0. Hence, nonlocality can be demonstrated with arbitrarily small limited detection efficiencies. Finally we propose a generalized scheme that uses this characterization to deal with detection inefficiencies, which interpolates between the two usual schemes, postselection and outcome assignment.