Asymptotic Relative Entropy of Entanglement for Orthogonally Invariant States

Abstract

For a special class of bipartite states we calculate explicitly the asymptotic relative entropy of entanglement $E_R^{\infty }$ with respect to states having a positive partial transpose (PPT). This quantity is an upper bound to distillable entanglement. The states considered are invariant under rotations of the form $O\otimes O$, where $O$ is any orthogonal matrix. We show that in this case $E_R^{\infty }$ is equal to another upper bound on distillable entanglement, constructed by Rains. To perform these calculations, we have introduced a number of new 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 $E_R^{\infty }$; (iii) for states for which the relative entropy of entanglement $E_R$ is additive, the Rains bound is equal to $E_R$.

11 pages, 5 figures

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