Asymptotic relative entropy of entanglement for orthogonally invariant states

For a special class of bipartite states we calculate explicitly the asymptotic relative entropy of entanglement ER^∞ with respect to states having a positive partial transpose. This quantity is an upper bound to distillable entanglement. The states considered are invariant under rotations of the form OO, where O is any orthogonal matrix. We show that in this case ER^∞ is equal to another upper bound on distillable entanglement, constructed by Rains. To perform these calculations, we have introduced a number of results that are interesting in their own right: (i) the Rains bound is convex and continuous; (ii) under some weak assumption, the Rains bound is an upper bound to ER^∞; (iii) for states for which the relative entropy of entanglement ER is additive, the Rains bound is equal to ER.