Multipartite entanglement groups

We propose to characterize multipartite entanglement of pure states as local unitary transformations acting on some parts of a system that can be undone by local unitary transformations acting on other parts. This leads to a definition of multipartite entanglement in terms of entanglement groups, constructed as certain quotients of the stabilizer group and its subgroups. We analyze properties of these entanglement groups and show that they imply restrictions which correspond to monogamy of entanglement. We use these groups to propose a coarse-grained classification scheme for entanglement in multi-partite quantum systems and we show that this group theory characterization of entanglement underlies several well-known quantum tasks.