  We demonstrate the irreversibility of asymptotic entanglement manipulation
under quantum operations that completely preserve the positivity of partial
transpose (PPT), resolving a major open problem in quantum information theory.
Our key tool is a new efficiently computable additive lower bound for the
asymptotic relative entropy of entanglement with respect to PPT states, which
can be used to evaluate the entanglement cost under local operations and
classical communication (LOCC). We find that for any rank-two mixed state
supporting on the $3\otimes3$ antisymmetric subspace, the amount of distillable
entanglement by PPT operations is strictly smaller than one entanglement bit
(ebit) while its entanglement cost under PPT operations is exactly one ebit. As
byproduct, we find that for this class of states, both the Rains' bound and its
regularization, are strictly less than the asymptotic relative entropy of
entanglement. So, in general, there is no unique entanglement measure for the
manipulation of entanglement by PPT operations. We further show a computable
sufficient condition for the irreversibility of entanglement distillation by
LOCC (or PPT) operations.
