  Rains' bound is arguably the best known upper bound of the distillable
entanglement by operations completely preserving positivity of partial
transpose (PPT) and was conjectured to be additive and coincide with the
asymptotic relative entropy of entanglement. We disprove both conjectures by
explicitly constructing a special class of mixed two-qubit states. We then
introduce an additive semidefinite programming lower bound ($E_M$) for the
asymptotic Rains' bound, and it immediately becomes a computable lower bound
for entanglement cost of bipartite states. Furthermore, $E_M$ is also proved to
be the best known upper bound of the PPT-assisted deterministic distillable
entanglement and gives the asymptotic rates for all pure states and some class
of genuinely mixed states.
