Bell inequality violation in the presence of vacancies and incomplete measurements

The characterization of a quantum system can be complicated by non-ideal measurement processes. In many systems, the underlying physical measurement is only sensitive to a single fixed state, complementary outcomes are inferred by non-detection leaving them vulnerable to out-of-Hilbert space errors such as particle loss. It is still possible to directly verify the violation of a Bell inequality, hence witness entanglement of a bipartite state, in the presence of large vacancy rates using such an incomplete measurement by optimizing the measurement settings. The scheme is robust against imperfect a priori state knowledge and also moderate amounts of error in state determination.