Entanglement Verification, with or without tomography. (arXiv:1802.02648v1 [quant-ph])

Multipartite entanglement has been widely regarded as key resources in
distributed quantum computing, for instance, multi-party cryptography,
measurement based quantum computing, quantum algorithms. It also plays a
fundamental role in quantum phase transitions, even responsible for transport
efficiency in biological systems.

Certifying multipartite entanglement is generally a fundamental task. Since
an $N$ qubit state is parameterized by $4^N-1$ real numbers, one is interested
to design a measurement setup that reveals multipartite entanglement with as
little effort as possible, at least without fully revealing the whole
information of the state, the so called "tomography", which requires
exponential energy.

In this paper, we study this problem of certifying entanglement without
tomography in the 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 all
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.