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arXiv:quant-ph/0612205 · Physical Review A

Mixed qubits cannot be universally broadcast

Lin Chen, Yi-Xin Chen

Published Jun 19, 2007• 14 citations
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Abstract

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 $1∕2$. 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.

DOI
10.1103/physreva.75.062322

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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 \ensuremathρs(\ensuremathθ,\ensuremathω,\ensuremathλ){\ensuremath{\rho}}_{s}(\ensuremath{\theta},\ensuremath{\omega},\ensuremath{\lambda})\ensuremathρs​(\ensuremathθ,\ensuremathω,\ensuremathλ) is regarded as constant in the fidelity. For the case of constant \ensuremathω\ensuremath{\omega}\ensuremathω, we establish the 1\ensuremath→21\ensuremath{\rightarrow}21\ensuremath→2 optimal symmetric dependent cloner with a fidelity 1∕21∕21∕2. It is also shown that the 1\ensuremath→M1\ensuremath{\rightarrow}M1\ensuremath→M optimal quantum cloning machine for pure qubits is also optimal for mixed qubits, when \ensuremathλ\ensuremath{\lambda}\ensuremathλ is the unique parameter in the fidelity. For general N\ensuremath→MN\ensuremath{\rightarrow}MN\ensuremath→M broadcasting of mixed qubits, the situation is very different.
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.
Due to the axioms of quantum mechanics, perfect cloning of an unknown quantum state is impossible. But since imperfect cloning is still possible, a question arises: ``Is there an optimal quantum cloning machine?'' Buzek and Hillery answered this question and constructed their famous B-H quantum cloning machine. The B-H machine clones the state of an arbitrary single qubit in an optimal manner and hence it is universal. Generalizing this machine for a two-qubit system is straightforward, but during this procedure, except for product states, this machine loses its universality and becomes a state-dependent cloning machine. In this paper, we propose some classes of optimal universal local quantum state cloners for a particular class of two-qubit systems, more precisely, for a class of states with known Schmidt basis. We then extend our machine to the case that the Schmidt basis of the input state is deviated from the local computational basis of the machine. We show that more local quantum coherence existing in the input state corresponds to less fidelity between the input and output states. Also we present two classes of a state-dependent local quantum copying machine. Furthermore, we investigate local broadcasting of two aspects of quantum correlations, i.e., quantum entanglement and quantum discord, defined, respectively, within the entanglement-separability paradigm and from an information-theoretic perspective. The results show that although quantum correlation is, in general, very fragile during the broadcasting procedure, quantum discord is broadcasted more robustly than quantum entanglement.
We derive a tight upper bound for the fidelity of a universal $N\ensuremath{\rightarrow}M$ qubit cloner, valid for any $M\ensuremath{\ge}N$, where the output of the cloner is required to be supported on the symmetric subspace. Our proof is based on the concatenation of two cloners and the connection between quantum cloning and quantum state estimation. We generalize the operation of a quantum cloner to mixed and/or entangled input qubits described by a density matrix supported on the symmetric subspace of the constituent qubits. We also extend the validity of optimal state estimation methods to inputs of this kind.
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.