We investigate the conditions under which a set
S of pure bipartite quantum states on a $D\ifmmode\times\else\texttimes\fi{}D$ system can be locally cloned deterministically by separable operations, when at least one of the states is full Schmidt rank. We allow for the possibility of cloning using a resource state that is less than maximally entangled. Our results include that: (i) all states in
S must be full Schmidt rank and equally entangled under the
G-concurrence measure, and (ii) the set
S can be extended to a larger clonable set generated by a finite group
G of order
∣G∣=N, the number of states in the larger set. It is then shown that any local cloning apparatus is capable of cloning a number of states that divides
D exactly. We provide a complete solution for two central problems in local cloning, giving necessary and sufficient conditions for (i) when a set of maximally entangled states can be locally cloned, valid for all
D; and (ii) local cloning of entangled qubit states with nonvanishing entanglement. In both of these cases, we show that a maximally entangled resource is necessary and sufficient, and the states must be related to each other by local unitary ``shift'' operations. These shifts are determined by the group structure, so need not be simple cyclic permutations. Assuming this shifted form and partially entangled states, then in
D=3 we show that a maximally entangled resource is again necessary and sufficient, while for higher-dimensional systems, we find that the resource state must be strictly more entangled than the states in
S. All of our necessary conditions for separable operations are also necessary conditions for local operations and classical communication (LOCC), since the latter is a proper subset of the former. In fact, all our results hold for LOCC, as our sufficient conditions are demonstrated for LOCC, directly.
Some progress is reported on conditions for convertibility among bipartite 2x2 entangled states: An inconvertibility condition related to the rank of an entangled state is given that it is impossible to convert to an entangled state with lower rank under separable operations; a particular set of local operations and classical communication (LOCC) is used to analyze convertibility of three subclasses of states - Werner states, Bell diagonal states and maximally entangled mixed states (MEMS). It is conjectured that MEMS may lie on the bottom of entangled state ordering for given entanglement of formation. A plausible way is suggested of systematically calculating convertibility in a general subclass of bipartite states whose density matrices are defined to be diagonal in a common basis. The set of LOCC adopted in this work is argued to be generalizable to provide sufficient conditions for convertibility among a large range of general 2x2 entangled states.
In this Rapid Communication we have established the physical character of pure bipartite states with the same amount of entanglement in the same Schmidt rank that either they are local unitarily connected or they are incomparable. There exists an infinite number of deterministically locally inequivalent classes of pure bipartite states in the same Schmidt rank (starting from three) having the same amount of entanglement. Further, if there exists incomparable states with the same entanglement in higher Schmidt ranks (greater than three), then they should differ in at least three Schmidt coefficients.
We study conditions for the deterministic transformation $\ensuremath{\mid}\ensuremath{\psi}⟩\ensuremath{\rightarrow}\ensuremath{\mid}\ensuremath{\phi}⟩$ 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 $\ensuremath{\mid}\ensuremath{\psi}⟩$ are the same as those of $\ensuremath{\mid}\ensuremath{\phi}⟩$. The question of sufficient conditions in the general separable case remains open. When the Schmidt coefficients of $\ensuremath{\mid}\ensuremath{\psi}⟩$ are the same as those of $\ensuremath{\mid}\ensuremath{\phi}⟩$, we show that the Kraus operators of the separable transformation restricted to the supports of $\ensuremath{\mid}\ensuremath{\psi}⟩$ 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 $\ensuremath{\mid}{\ensuremath{\psi}}_{j}⟩$ to pure states $\ensuremath{\mid}{\ensuremath{\phi}}_{j}⟩$.