We discuss the (im)possibility of the exact cloning of orthogonal but genuinely entangled three qubit states aided with entangled ancilla under local operation and classical communication. Whereas any two orthogonal Greenberger-Horne-Zeilinger (GHZ) states taken from the canonical GHZ basis can be cloned with the help of a known GHZ state, surprisingly we find that no two
W states can be cloned by using any known three qubit (possibly entangled) state as a blank copy.
In classical information theory one can, in principle, produce a perfect copy of any input state. In quantum information theory, the no cloning theorem prohibits exact copying of non-orthogonal states. Moreover, if we wish to copy multiparticle entangled states and can perform only local operations and classical communication (LOCC), then further restrictions apply. We investigate the problem of copying orthogonal, entangled quantum states with an entangled blank state under the restriction to LOCC. Throughout, the subsystems have finite dimension D . We show that if all of the states to be copied are non-maximally entangled, then novel LOCC copying procedures based on entanglement catalysis are possible. We then study in detail the LOCC copying problem where both the blank state and at least one of the states to be copied are maximally entangled. For this to be possible, we find that all the states to be copied must be maximally entangled. We obtain a necessary and sufficient condition for LOCC copying under these conditions. For two orthogonal, maximally entangled states, we provide the general solution to this condition. We use it to show that for D = 2, 3, any pair of orthogonal, maximally entangled states can be locally copied using a maximally entangled blank state. However, we also show that for any D which is not prime, one can construct pairs of such states for which this is impossible.
In classical information theory one can, in principle, produce a perfect copy of any input state. In quantum information theory, the no cloning theorem prohibits exact copying of nonorthogonal states. Moreover, if we wish to copy multiparticle entangled states and can perform only local operations and classical communication (LOCC), then further restrictions apply. We investigate the problem of copying orthogonal quantum states with an entangled blank state under the restriction to LOCC. Throughout, the subsystems have finite dimension D. We show that if some of the states to be copied are non-maximally entangled, then novel LOCC copying procedures based on entanglement catalysis are possible. We then study in detail the LOCC copying problem where both the blank state and at least one of the states to be copied are maximally entangled. For this to be possible, we find that all the states to be copied must be maximally entangled. We obtain a necessary and sufficient condition for LOCC copying under these conditions. For two orthogonal, maximally entangled states, we provide the general solution to this condition. We use it to show that for D=2,3, any pair of orthogonal, maximally entangled states can be locally copied using a maximally entangled blank state. However, we also show that for any D which is not prime, one can construct pairs of such states for which this is impossible.
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.
There exist sets of multipartite orthogonal states which can not be distinguished by local operation and classical communication (LOCC). The simplest example is the set of four Bell states. The problem of characterizing a set in terms of distinguishability is a hard problem in general. For maximally entangled states, some results are known. Local cloning of entangled states by supplying necessary entanglement in the blank state is another idea that was introduced very recently. Interestingly there are sets of orthogonal entangled states which can be distinguished locally but can not be copied by local operation (even if necessary entanglement is supplied in the blank state) contrary to the fact that happens in the case of global operation.