Local cloning of entangled states

Abstract

We investigate the conditions under which a set $\SC$ of pure bipartite quantum states on a $D\times D$ 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 $\SC$ must be full Schmidt rank and equally entangled under the $G$-concurrence measure, and (ii) the set $\SC$ 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 non-vanishing entanglement. In both of these cases, 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 $\SC$. All of our necessary conditions for separable operations are also necessary conditions for 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.

REVTEX 15 pages, 1 figure, minor modifications. Same as the published version. Any comments are welcome!

References in corpus (6)

• Local distinguishability with preservation of entanglement
• Efficient implementation of bipartite nonlocal unitary gates using prior entanglement and classical communication
• Separable Operations on Pure States
• Entanglement transformations using separable operations
• Local Cloning of Entangled Qubits
• Local cloning of genuine entangled states of three qubit

Cited by in corpus (5)

• Entanglement requirements for implementing bipartite unitary operations
• General approach to quantum channel impossibility by local operations and classical communication
• Local quantum protocols for separable measurements with many parties
• Strong bounds on required resources for quantum channels by local operations and classical communication
• Separable Operations, Graph Codes and the Location of Quantum Information