  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.

Quantum Physics