Bipartite entanglement, spherical actions, and geometry of local unitary orbits
Alan Huckleberry, Marek Kuś, Adam Sawicki · Journal of Mathematical Physics · 2013
We use the geometry of the moment map to investigate properties of pure entangled states of composite quantum systems. The orbits of equally entangled states are mapped by the moment map onto coadjoint orbits of local transformations (unitary transformations which do not change entanglement). Thus, the geometry of coadjoint orbits provides a partial classification of different entanglement classes. To achieve the full classification, a further study of fibers of the moment map is needed. We show how this can be done effectively in the case of the bipartite entanglement by employing Brion's theorem. In particular, we presented the exact description of the partial symplectic structure of all local orbits for two bosons, fermions, and distinguishable particles putting a special emphasis on the generality of the approach allowing one to consider all three cases in completely parallel manners.