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arXiv:0705.2694 · Physical Review A

Optimal 1→M universal quantum cloning via spin networks

Jiang Zhang, Qing Chen, Shaolong Wan

Published Sep 11, 2007• 10 citations
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Abstract

We present a scheme which transforms a one-input qubit to $M$ identical copies with optimal fidelity via free dynamical evolution of spin star networks. We demonstrate that the Heisenberg $XXZ$ 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.

DOI
10.1103/physreva.76.034302

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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.
We review our recent work on the universal (i.e. input state independent) optimal quantum copying (cloning) of qubits. We present unitary transformations which describe the optimal cloning of a qubit and we present the corresponding quantum logical network. We also present network for an optimal quantum copying ``machine'' (transformation) which produces N+1 identical copies from the original qubit. Here again the quality (fidelity) of the copies does not depend on the state of the original and is only a function of the number of copies, N. In addition, we present the machine which universaly and optimally clones states of quantum objects in arbitrary-dimensional Hilbert spaces. In particular, we discuss universal cloning of quantum registers.
We present quantum cloning machines that transform NNN identical qubits into M>NM>NM>N identical copies and we prove that the fidelity (quality) of these copies is optimal. The connection between cloning and measurement is discussed in detail. When the number of clones MMM tends towards infinity, the fidelity of each clone tends towards the optimal fidelity that can be obtained by a measurement on the input qubits. More generally, quantum cloning machines are universal devices to translate quantum information into classical information.
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.
We analyze how information encoded in quantum systems can be optimally processed. In particular, we investigate copying (cloning) of quantum information (represented as states of 2-level quantum systems--qubits). We present unitary transformations which describe the optimal universal cloning of a qubit. Universality of the transformation guarantees that the fidelity of the cloning does not depend on the input state of the qubit, i.e. all states are cloned equally well. We present network for the optimal universal quantum cloning `machine' (transformation) which produces N + 1 copies from the original qubit. Here again the quality (fidelity) of the copies does not depend on the state of the original and is only a function of the number of copes, N. We also present the `machine' which universally and optimally clones states of quantum objects in arbitrary- dimensional Hilbert spaces. In particular, we discuss universal cloning of quantum registers. In addition to cloning of qubits we analyzed another universal operation-- the Universal NOT. We present the optimal transformation and the corresponding logical network which optimally complements an arbitrary input state of a qubit. We show that the fidelity of the performance of the Universal NOT operation increases as a function of the number of input qubits prepared in the same state.
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.
While the no-cloning theorem, which forbids the perfect copying of quantum states, is well-known as one of the defining features of quantum mechanics, the question of how well the theory allows a state to be cloned is yet to be completely solved. In this paper, rigorous solutions to the problem of M→NM\rightarrow NM→N asymmetric cloning of qudits are obtained in a number of interesting cases. The central result is the solution to the 1→N1 \rightarrow N1→N universal asymmetric qudit cloning problem for which the exact trade-off in the fidelities of the clones for every NNN and ddd is derived. Analogous results are proven for qubits when M=N−1M=N-1M=N−1. We also consider state-dependent 1→N1 \rightarrow N1→N qubit cloning, providing a general parametrization in terms of a Heisenberg star Hamiltonian. In all instances, we determine the feasibility of implementing the cloning economically, i.e., without an ancilla, and determine the dimension of the ancilla when an economic implementation is not possible.