We derive a sufficient condition for a set of pure states, each entangled in two remote
N-dimensional systems, to be transformable to
k-dimensional-subspace equivalent entangled states (
k≤N) by same local operations and classical communication. If
k=N, the condition is also necessary. This condition reveals the function of the relative marginal density operators of the entangled states in the entanglement manipulation without sufficient information of the initial states.
We investigate the cases where a set
S of states
{∣ψi⟩} cannot be cloned by itself, but is clonable with the help of another system prepared in state
ρ^i. When
S is pair-wise nonorthogonal, it is known that one can generate the copy from
ρ^i alone, with no interaction with the original system. Here we show that a set containing orthogonal pairs exhibits a property forming a striking contrast; For any such set, there is a choice of
ρ^i that enables cloning only when the two systems are interacted in a purely quantum manner that is not achievable via classical communication.
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.
We consider transmission of classical information through a quantum channel from one sender to several receivers. In contrast to quantum information, classical information carried by two non-orthogonal states of a quantum system can be cloned exactly. This restricts a naive extrapolation of the standard no-cloning theorem. The optimal cloning induces entanglement between clones which increases with non-orthogonality of the states. The capacity of the corresponding quantum channel is considered. As a byproduct of our results, we obtain that for an entangled system with an unknown state there is no transformation which removes the entanglement totally without affecting the corresponding marginal density matrices.