Local cloning of entangled states
Vlad Gheorghiu, Li Zhi Yu, Scott M. Cohen · Physical Review A · 2010
We investigate the conditions under which a set $\mathcal{S}$ 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 $\mathcal{S}$ must be full Schmidt rank and equally entangled under the $G$-concurrence measure, and (ii) the set $\mathcal{S}$ 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 $\mathcal{S}$. 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.