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arXiv:quant-ph/0609182 · Physical Review A

Optimal broadcasting of mixed states

Gui-Fang Dang, Heng Fan

Published Aug 22, 2007• 13 citations
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Abstract

The $N$ to $M$ $(M\ensuremath{\geqslant}N)$ 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.

DOI
10.1103/physreva.76.022323

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The NNN to MMM (M\ensuremath⩾N)(M\ensuremath{\geqslant}N)(M\ensuremath⩾N) universal quantum broadcasting of mixed states \ensuremathρ\ensuremath⨂N{\ensuremath{\rho}}^{\ensuremath{\bigotimes}N}\ensuremathρ\ensuremath⨂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 \ensuremathρ\ensuremath⨂N{\ensuremath{\rho}}^{\ensuremath{\bigotimes}N}\ensuremathρ\ensuremath⨂N is obtained.
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.
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.
"Broadcasting", namely distributing information over many users, suffers in-principle limitations when the information is quantum. This poses a critical issue in quantum information theory, for distributed processing and networked communications. For pure states ideal broadcasting coincides with the so-called "quantum cloning", describing an hypothetical ideal device capable of producing from a finite number N of copies of a state (drawn from a set) a larger number M>N of output copies of the same state. Since such a transformation is not isometric, it cannot be achieved by any physical machine for a quantum state drawn from a non orthogonal set: this is essentially the content of the "no-cloning" theorem. For mixed states the situation is quite different, since from the point of view of each single user a local marginal mixed state is indistinguishable from the partial trace of an entangled state, and there are infinitely many joint output states that correspond to ideal broadcasting. Indeed, for sufficiently large number NNN of input copies, not only ideal broadcasting of noncommuting mixed states is possible, but one can even purify the state in the process. Such state purification with an increasing number of copies has been named "superbroadcasting". In this paper we will review some recent results on superbroadcasting of qubits, for two different sets of input states, corresponding to universally covariant broadcasting and to phase-covariant broadcasting of equatorial states.