Multi-partite Entanglement Verification, with or without tomography

In this talk, we study this problem of certifying multipartite entanglement without tomography in constrain that only single copy measurements can be applied. This task is formulate as a membership problem related to a dividing quantum state space, therefore, related to the geometric structure of state space. We show that universal entanglement detection among all states can never be accomplished without full state tomography. Moreover, we show that almost any multipartite correlation, include genuine entanglement detection, entanglement depth verification, requires full state tomography. However, universal entanglement detection among pure states can be much more efficient, even we only allow local measurements. Almost optimal local measurement scheme for detecting pure states entanglement is provided.