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arXiv:quant-ph/0410211 · Physical Review A

Cloning transformations in spin networks without external control

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

Published Jul 22, 2005• 28 citations
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Abstract

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 $XY$ 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.

DOI
10.1103/physreva.72.012328

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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→21\ensuremath{\rightarrow}21\ensuremath→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→MN\ensuremath{\rightarrow}MN\ensuremath→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.
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 XYXYXY 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.
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.
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.
Abstract Copying quantum information is forbidden by the no-cloning theorem, yet approximate copies with the highest fidelity allowed by quantum mechanics are a basic resource for quantum networks and for assessing the security of quantum key distribution. Such cloners are usually gate sequences, but a spin network with fixed couplings can perform the transformation autonomously: the unknown qubit and blank qubits sit on the vertices of a graph and the free evolution does the rest. Until now, the only spin graph known to reach the optimal phase-covariant fidelity for every number of clones MMM was the star with a tuned anisotropy. Which other graphs can clone optimally, and why, has remained open. Here we show that the complete bipartite graph K2,M−1K_{2,M-1}K2,M−1​---the unknown qubit and one target on one side, the remaining targets on the other---reaches the optimal 1→M1\to M1→M phase-covariant fidelity for every M≥3M\ge 3M≥3, with all couplings given in closed form. For odd MMM a staggered Ising anisotropy suffices, with the parameters fixed by two integers; for even MMM an additional Ising coupling among the middle targets is required, with its value an explicit algebraic number. Both families reduce, in the permutation-symmetric sectors, to two four-level problems: a perfect-state-transfer ladder on a four-cycle and a freeze block whose characteristic polynomial we obtain analytically. We verify the formulas with no free parameter up to M=10M=10M=10 and classify exactness for all four-vertex graphs with a least-squares test of the exact revival and transfer equations. The results connect the fractional-revival engineering of spin chains to optimal cloning and identify the hidden involution behind both. Beyond cloning, optimal quantum machines can thus be read off from the spectrum of small symmetric blocks, suggesting autonomous implementations of other covariant transformations on spin and Rydberg arrays.
We establish the best possible approximation to a perfect quantum cloning machine that produces two clones out of a single input. We analyze both universal and state-dependent cloners. The maximal fidelity of cloning is shown to be 5/6 for universal cloners. It can be achieved either by a special unitary evolution or by a teleportation scheme. We construct the optimal state-dependent cloners operating on any prescribed two nonorthogonal states and discuss their fidelities and the use of auxiliary physical resources in the process of cloning. The optimal universal cloners permit us to derive an upper bound on the quantum capacity of the depolarizing quantum channel.
We present a scheme which transforms a one-input qubit to MMM identical copies with optimal fidelity via free dynamical evolution of spin star networks. We demonstrate that the Heisenberg XXZXXZXXZ coupling can meet the challenge. The initial state of the copying machine and the parameters of the spin Hamiltonian are discussed in detail. Furthermore we have proposed a feasible method to prepare the initial state of the copying machine.