  Recently, entanglement concentration was explicitly shown to be irreversible.
However, it is still not clear what kind of states can be reversibly converted
in the asymptotic setting by LOCC when neither the initial nor the target state
is maximally entangled. We derive the necessary and sufficient condition for
the reversibility of LOCC conversions between two bipartite pure entangled
states in the asymptotic setting. In addition, we show that conversion can be
achieved perfectly with only local unitary operation under such condition
except for special cases. Interestingly, our result implies that an error-free
reversible conversion is asymptotically possible even between states whose
copies can never be locally unitarily equivalent with any finite numbers of
copies, although such a conversion is impossible in the finite setting. In
fact, we show such an example. Moreover, we establish how to overcome the
irreversibility of LOCC conversion in two ways. As for the first method, we
evaluate how many copies of the initial state is to be lost to overcome the
irreversibility of LOCC conversion. The second method is to add a supplementary
state appropriately, which also works for LU conversion unlike the first
method. Especially, for the qubit system, any non-maximally pure entangled
state can be a universal resource for the asymptotic reversibility when copies
of the state is sufficiently many. More interestingly, our analysis implies
that far-from-maximally entangled states can be better than nearly maximally
entangled states as this type of resource. This fact brings new insight to the
resource theory of state conversion.
