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Chinese Physics Letters

Universal Quantum Cloning Machines for Two Identical Mixed Qubits

Yang Shuai, Meisheng Zhao, Nai-Le Liu, Zeng‐Bing Chen

Published Oct 17, 2007• 5 citations
Publisher

Abstract

We present a series of universal quantum cloning machines for two identical mixed qubits. Every machine is optimal in the sense that it achieves the optimal bound of the single copy shrinking factor. Unlike in the case of pure state cloning, the single copy shrinking factor does not uniquely determine the cloning map in the case of mixed state cloning.

DOI
10.1088/0256-307x/24/11/009

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We present a series of universal quantum cloning machines for two identical mixed qubits. Every machine is optimal in the sense that it achieves the optimal bound of the single copy shrinking factor. Unlike in the case of pure state cloning, the single copy shrinking factor does not uniquely determine the cloning map in the case of mixed state cloning.
Quantum cloning of two identical mixed qubits ρ⊗ρ\rho \otimes \rhoρ⊗ρ is studied. We propose the quantum cloning transformations not only for the triplet (symmetric) states but also for the singlet (antisymmetric) state. We can copy these two identical mixed qubits to MMM (M≥2M\ge 2M≥2) copies. This quantum cloning machine is optimal in the sense that the shrinking factor between the input and the output single qubit achieves the upper bound. The result shows that we can copy two identical mixed qubits with the same quality as that of two identical pure states.
The NNN to MMM $(M\ensuremath{\geqslant}N)$ universal quantum broadcasting of mixed states ${\ensuremath{\rho}}^{\ensuremath{\bigotimes}N}$ is proposed for a qubit system. The broadcasting of mixed states is universal and optimal in the sense that the shrinking factor is independent of the input state and achieves the upper bound. The quantum broadcasting of mixed qubits is a generalization of the universal quantum cloning machine for identical pure input states. A pure state decomposition of the identical mixed qubits ${\ensuremath{\rho}}^{\ensuremath{\bigotimes}N}$ is obtained.
We present a unified universal quantum cloning machine, which combines several different existing universal cloning machines together, including the asymmetric case. In this unified framework, the identical pure states are projected equally into each copy initially constituted by input and one half of the maximally entangled states. We show explicitly that the output states of those universal cloning machines are the same. One importance of this unified cloning machine is that the cloning procession is always the symmetric projection, which reduces dramatically the difficulties for implementation. Also, it is found that this unified cloning machine can be directly modified to the general asymmetric case. Besides the global fidelity and the single-copy fidelity, we also present all possible arbitrary-copy fidelities.
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 show that no universal quantum cloning machine exists that can broadcast an arbitrary mixed qubit with a constant fidelity. Based on this result, we investigate the dependent quantum cloner in the sense that some parameter of the input qubit ${\ensuremath{\rho}}_{s}(\ensuremath{\theta},\ensuremath{\omega},\ensuremath{\lambda})$ is regarded as constant in the fidelity. For the case of constant $\ensuremath{\omega}$, we establish the $1\ensuremath{\rightarrow}2$ optimal symmetric dependent cloner with a fidelity 1∕21∕21∕2. It is also shown that the $1\ensuremath{\rightarrow}M$ optimal quantum cloning machine for pure qubits is also optimal for mixed qubits, when $\ensuremath{\lambda}$ is the unique parameter in the fidelity. For general $N\ensuremath{\rightarrow}M$ broadcasting of mixed qubits, the situation is very different.