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Local unitary symmetries and entanglement invariants

We investigate the relation between local unitary symmetries and entanglement invariants of multi-qubit systems. The Hilbert space of such systems can be stratified in terms of states with different types of symmetry. We review the connection between this stratification and the ring of entanglement invariants and the corresponding geometric description in terms of algebraic varieties. On a stratum of a non-trivial symmetry group the invariants of the symmetry preserving operations gives a sufficient description of entanglement. Finding these invariants is often a simpler problem than finding the invariants of the local unitary group. The conditions, as given by the Luna-Richardson theorem, for when the ring of such invariants is isomorphic to the ring of local unitary invariants on the stratum are discussed. As an example we consider symmetry groups that can be diagonalized by local unitary operations and for which the group action on each qubit is non-trivial. On the stratum of such a symmetry the entanglement can be described in terms of a canonical form and the invariants of the symmetry preserving operations. This canonical form and the invariants are directly determined by the symmetry group. Further, we briefly discuss how some recently proposed entanglement classification schemes capture symmetry properties.