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arXiv:quant-ph/0402071 · Physical Review A

Quantum cloning in spin networks

Gabriele De Chiara, Rosario Fazio, Chiara Macchiavello, Simone Montangero, G. Massimo Palma

Published Dec 8, 2004• 67 citations
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Abstract

We introduce an approach to quantum cloning based on spin networks and we demonstrate that phase covariant cloning can be realized using no external control but only with a proper design of the Hamiltonian of the system. In the $1\ensuremath{\rightarrow}2$ cloning we find that the $XY$ model saturates the value for the fidelity of the optimal cloner and gives values comparable to it in the general $N\ensuremath{\rightarrow}M$ case. We finally discuss the effect of external noise. Our protocol is much more robust to decoherence than a conventional procedure based on quantum gates.

DOI
10.1103/physreva.70.062308

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We introduce an approach to quantum cloning based on spin networks and we demonstrate that phase covariant cloning can be realized using no external control but only with a proper design of the Hamiltonian of the system. In the 1\ensuremath→21\ensuremath{\rightarrow}21\ensuremath→2 cloning we find that the XYXYXY model saturates the value for the fidelity of the optimal cloner and gives values comparable to it in the general N\ensuremath→MN\ensuremath{\rightarrow}MN\ensuremath→M case. We finally discuss the effect of external noise. Our protocol is much more robust to decoherence than a conventional procedure based on quantum gates.
In this paper we present an approach to quantum cloning with unmodulated spin networks. The cloner is realized by a proper design of the network and a choice of the coupling between the qubits. We show that in the case of phase covariant cloner the XYXYXY coupling gives the best results. In the $1\ensuremath{\rightarrow}2$ cloning we find that the value for the fidelity of the optimal cloner is achieved, and values comparable to the optimal ones in the general $N\ensuremath{\rightarrow}M$ case can be attained. If a suitable set of network symmetries are satisfied, the output fidelity of the clones does not depend on the specific choice of the graph. We show that spin network cloning is robust against the presence of static imperfections. Moreover, in the presence of noise, it outperforms the conventional approach. In this case the fidelity exceeds the corresponding value obtained by quantum gates even for a very small amount of noise. Furthermore, we show how to use this method to clone qutrits and qudits. By means of the Heisenberg coupling it is also possible to implement the universal cloner although in this case the fidelity is 10% off that of the optimal cloner.
In this paper we demonstrate that optimal $1\ensuremath{\rightarrow}M$ phase-covariant cloning quantum cloning is available via free dynamical evolution of spin networks. By properly designing the network and the couplings between spins, we show that optimal $1\ensuremath{\rightarrow}M$ phase-covariant cloning can be achieved if the initial state is prepared as a specific symmetric state. Especially, when MMM is an odd number, the optimal phase-covariant cloning can be achieved without ancillas. Moreover, we demonstrate that the same framework is capable for optimal $1\ensuremath{\rightarrow}2$ universal cloning.
After introducing the no-cloning theorem and the most common forms of approximate quantum cloning, universal quantum cloning is considered in detail. The connections it has with universal NOT-gate, quantum cryptography and state estimation are presented and briefly discussed. The state estimation connection is used to show that the amount of extractable classical information and total Bloch vector length are conserved in universal quantum cloning. The 1  2 qubit cloner is also shown to obey a complementarity relation between local and nonlocal information. These are interpreted to be a consequence of the conservation of total information in cloning. Finally, the performance of the 1  M cloning network discovered by Bužek, Hillery and Knight is studied in the presence of decoherence using the Barenco et al. approach where random phase fluctuations are attached to 2-qubit gates. The expression for average fidelity is calculated for three cases and it is found to depend on the optimal fidelity and the average of the phase fluctuations in a specific way. It is conjectured to be the form of the average fidelity in the general case. While the cloning network is found to be rather robust, it is nevertheless argued that the scalability of the quantum network implementation is poor by studying the effect of decoherence during the preparation of the initial state of the cloning machine in the 1 ! 2 case and observing that the loss in average fidelity can be large. This affirms the result by Maruyama and Knight, who reached the same conclusion in a slightly different manner.
We propose a quantum cloning machine, which clones a qubit into two clones assuming known modulus of expectation value of Pauli ${\ensuremath{\sigma}}_{z}$ matrix. The process is referred to as the mirror phase-covariant cloning for which the input state is a priori less known than that for the standard phase-covariant cloning. Analytical expressions describing the cloning transformation and fidelity of the clones are found. Extremal equations for the optimal cloning are derived and analytically solved by generalizing a method of Fiur\'a\ifmmode \check{s}\else \v{s}\fi{}ek [Phys. Rev. A 64, 062310 (2001)]. Quantum circuits implementing the optimal cloning transformation and their physical realization in a quantum-dot system are described.