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We give an introduction to the theory of multi-partite entanglement. We begin by describing the coordinate system of the field: Are we dealing with pure or mixed states, with single or multiple copies, what notion of locality is being used, do we aim to classify states according to their type of entanglement or to quantify it? Building on the general theory of multi-partite entanglement - to the extent that it has been achieved - we turn to explaining important classes of multi-partite entangled states, including matrix product states, stabilizer and graph states, bosonic and fermionic Gaussian states, addressing applications in condensed matter theory. We end with a brief discussion of various applications that rely on multi-partite entangled states: quantum networks, measurement-based quantum computing, non-locality, and quantum metrology.
We show that spin squeezing criteria commonly used for entanglement detection can be erroneous, if the probe is not symmetric. We then derive a lower bound on squeezing for separable states in spin systems probed asymmetrically. Using this we further
Invariance under local unitary operations is a fundamental property that must be obeyed by every proper measure of quantum entanglement. However, this is not the only aspect of entanglement theory where local unitaries play a relevant role. In the pr
The Mermin inequality provides a criterion for experimentally ruling out local-realistic descriptions of multiparticle systems. A violation of this inequality means that the particles must be entangled, but does not, in general, indicate whether N-pa
The relative entropy of entanglement $E_R$ is defined as the distance of a multi-partite quantum state from the set of separable states as measured by the quantum relative entropy. We show that this optimisation is always achieved, i.e. any state adm
We investigate and quantify various measures of bipartite and tripartite entanglement in the context of two and three flavor neutrino oscillations. The bipartite entanglement is analogous to the entanglement swapping resulting from a beam splitter in