Optimal entanglement witnesses in a split spin-squeezed Bose-Einstein condensate

How do we detect quantum correlations in bipartite scenarios using a split many-body system and collective measurements on each party? We address this question by deriving entanglement witnesses using either only first-order or both first-and second-order moments of local collective spin components. In both cases, we derive optimal witnesses for spatially split spin-squeezed states in the presence of local white noise. We then compare the two optimal witnesses with respect to their resistance to various noise sources operating either at the preparation or at the detection level. We finally evaluate the statistics required to estimate the value of these witnesses when measuring a split spin-squeezed Bose-Einstein condensate. Our results can be seen as a step toward Bell tests with many-body systems.