  Assume that Alice, Bob, and Charlie share a tripartite pure state
$|\psi_{ABC}\rangle$. We prove that if Alice cannot distill entanglement with
either Bob or Charlie using $|\psi_{ABC}\rangle$ and local operations with any
one of the following configurations for classical communication: $(A\to B,
A\leftrightarrow C), (A\leftrightarrow B, A\to C),$ and $(A\leftrightarrow B,
A\leftrightarrow C)$, then the same is also true for the other two
configurations. Moreover, this happens precisely when the state is such that
both its reductions $\rho_{AB}$ and $\rho_{AC}$ are separable, which is further
equivalent to the reductions being PPT. This, in particular, implies that any
NPT bipartite state is such that either the state itself or its complement is
2-way distillable. In proving these results, we first obtain an explicit lower
bound on the 2-way distillable entanglement of low rank bipartite states.
Furthermore, we show that even though not all low rank states are 1-way
distillable, a randomly sampled low rank state will almost surely be 1-way
distillable.
