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 detectors' 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, that is, in each run of the experiment, the overall detection efficiency is lower bounded by $\eta_{min} > 0$. Hence, in an adversarial scenario, the adversaries have arbitrary large but not full control over the inefficiencies. We analyse the set of possible correlations that fulfill Limited Detection Locality (LDL) and show that they necessarily satisfy some linear Bell-like inequalities. We prove that quantum theory predicts the violation of one of these inequalities for all $\eta_{min} > 0$. Hence, nonlocality can be demonstrated with arbitrarily small limited detection efficiencies. We validate this assumption experimentally via a twin-photon implementation in which two users are provided with one photon each out of a partially entangled pair. We exploit on each side a passive switch followed by two measurement devices with fixed settings. Assuming the switches are not fully controlled by an adversary, nor by hypothetical local variables, we reveal the nonlocality of the established correlations despite a low overall detection efficiency.