Strategies and Networks for State-Dependent Quantum Cloning

The no-cloning theorem of Wootters, Zurek [1] and Dieks [2] shows that it is impossible to devise a means of perfectly replicating an arbitrary, unknown quantum state. While such an ideal cloning machine cannot be constructed, this important result does not prohibit cloning strategies which have a limited degree of success. Numerous interesting contributions have been made to this subject by many authors, showing that the no-cloning theorem is not the last word on the matter. All quantum cloning machines, except those designed to copy only orthogonal states [1–3], are fallible. Such machines can be divided into two categories: universal and state-dependent cloners. The first kind produce approximate copies of a completely unknown quantum state. This idea was proposed originally by Buz̆ek and Hillery [4] who showed how to construct a device which can copy any quantum state of a qubit equally well, where the figure of merit is the local fidelity. This is the square overlap between the state of one of the approximate copies produced by the machine, and the exact state. The Buz̆ek-Hillery cloning machine was later shown to be optimal by Gisin and Massar [5] and Bruß et al [6], who demonstrated that these clones attain the maximum possible local fidelity. More recently, the optimum universal cloning problem for multi-level quantum systems has been solved by Werner [7] and Keyl and Werner [8]. The second category is that of state-dependent cloning machines. These are designed to reproduce only a finite number of states, and here we can identify two sub-categories. Approximate state-dependent cloning machines were examined first, by Hillery and Buz̆ek [9]. These machines deterministically generate approximate clones of states belonging to a finite set. Such devices have also been considered by Bruß et al [10]. More recently, it was discovered by Duan and Guo that a set of non-orthogonal, but linearly-independent pure states can sometimes be exactly cloned [11,12]. For non-orthogonal states, there is a non-zero probability that the cloning attempt fails. However, it is always possible to tell if it has been a success and under some circumstances, this latter approach may be more useful. Several results have been obtained for optimum cloning of two states. For approximate cloning, Bruß et al [10] have determined the maximum local and global fidelities for making two copies of an unknown state belonging to a known pair having equal a priori probabilities. The global fidelity is the average square-overlap between the complete state of the system of approximate clones and that of the exact clones. Duan and Guo [11] have found the maximum success probability if these copies are required to be exact. More recently, we have found the maximum success probability if we have M initial copies of the state, and wish to obtain N copies, where N > M , and have shown how this result is related to unambiguous state identification [13]. In this paper, we obtain more general results relating to approximate cloning, and investigate properties of cloning machines which are hybrids of both types of state-dependent cloning. In section II, we show that the use of the global fidelity as the figure of merit for approximate cloning establishes an important link between cloning and state identification, at least if the states to be copied are linearly-independent. We then solve the complete optimisation