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Chinese Physics Letters

Optimal Broadcasting of Mixed Equatorial Qubits

Zong‐Wen Yu

Published Aug 1, 2009• 2 citations
Publisher

Abstract

We derive an optimal 2 → M phase-covariant quantum broadcasting of mixed equatorial qubits. This quantum broadcasting 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 equatorial qubits with the same quality as those of two identical pure equatorial states.

DOI
10.1088/0256-307x/26/8/080304

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Chinese Physics Letters•Oct 17, 2007•5 citations
We derive an optimal 2 → M phase-covariant quantum broadcasting of mixed equatorial qubits. This quantum broadcasting 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 equatorial qubits with the same quality as those of two identical pure equatorial states.
"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.
We consider cloning transformations of equatorial qubits $|{\ensuremath{\psi}}_{\ensuremath{\varphi}}〉=1/\sqrt{2}(|0〉{+e}^{i\ensuremath{\varphi}}|1〉)$ and qutrits $|{\ensuremath{\psi}}_{\ensuremath{\varphi},\ensuremath{\theta}}〉=1/\sqrt{3}(|0〉{+e}^{i\ensuremath{\varphi}}|1〉{+e}^{i\ensuremath{\theta}}|2〉),$ with the transformation covariant for rotation of the phases $\ensuremath{\varphi}$ and $\ensuremath{\theta}.$ The optimal cloning maps are derived without simplifying assumptions from first principles, for any number of input and output qubits, and for a single-input qutrit and any number of output qutrits. We also compare the cloning maps for global and single-particle fidelities, and we show that the two criteria lead to different optimal maps.
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.