Entanglement transformations using separable operations

We study conditions for the deterministic transformation $\mid \psi ⟩\to \mid \varphi ⟩$ of a bipartite entangled state by a separable operation. If the separable operation is a local operation with classical communication (LOCC), Nielsen’s majorization theorem provides necessary and sufficient conditions. For the general case, we derive a necessary condition in terms of products of Schmidt coefficients, which is equivalent to the Nielsen condition when either of the two factor spaces is of dimension 2, but is otherwise weaker. One implication is that no separable operation can reverse a deterministic map produced by another separable operation, if one excludes the case where the Schmidt coefficients of $\mid \psi ⟩$ are the same as those of $\mid \varphi ⟩$. The question of sufficient conditions in the general separable case remains open. When the Schmidt coefficients of $\mid \psi ⟩$ are the same as those of $\mid \varphi ⟩$, we show that the Kraus operators of the separable transformation restricted to the supports of $\mid \psi ⟩$ on the factor spaces are proportional to unitaries. When that proportionality holds and the factor spaces have equal dimension, we find conditions for the deterministic transformation of a collection of several full Schmidt rank pure states $\mid {\psi }_j⟩$ to pure states $\mid {\varphi }_j⟩$.