Entanglement measures under symmetry

We show how to simplify the computation of the entanglement of formation and the relative entropy of entanglement for states, which are invariant under a group of local symmetries. For several examples of groups we characterize the state spaces, which are invariant under these groups. For specific examples we calculate the entanglement measures. In particular, we derive an explicit formula for the entanglement of formation for $(U\otimes U)$-invariant states, and we find a counterexample of the additivity conjecture for the relative entropy of entanglement.