New Journal of Physics

Optimal asymmetric phase-covariant and real state cloning inddimensions

Wen‐Hai Zhang, Liu Ye

Published Sep 10, 2007• 46 citations

Abstract

We generalize three symmetric cloners to the asymmetric cases and present three explicit cloning transformations as well as their corresponding fidelity distributions in d dimensions. The three asymmetric cloners, including the optimal asymmetric phase-covariant cloning (APCC) and the suboptimal asymmetric economical phase-covariant cloning (AEPCC) working without ancilla and the optimal asymmetric real state cloning (ARSC), together with the optimal asymmetric universal quantum cloning (AUQC) construct a generic cloning, where the quantum information of initial systems of different pure input states in d dimensions with their information not completely known can be optimally distributed to different final systems. By comparison of the fidelity distributions of the four asymmetric cloners, some interesting results can be obtained.

DOI: 10.1088/1367-2630/9/9/318

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New Journal of Physics

Optimal asymmetric phase-covariant and real state cloning inddimensions

Wen‐Hai Zhang, Liu Ye

Published Sep 10, 2007• 46 citations

Publisher

Abstract

DOI: 10.1088/1367-2630/9/9/318

Related papers

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We investigate the optimal distribution of quantum information over multipartite systems in asymmetric settings. We introduce cloning transformations that take $N$ identical replicas of a pure state in any dimension as input and yield a collection of clones with nonidentical fidelities. As an example, if the clones are partitioned into a set of ${M}_{A}$ clones with fidelity ${F}^{A}$ and another set of ${M}_{B}$ clones with fidelity ${F}^{B}$ , the trade-off between these fidelities is analyzed, and particular cases of optimal $N\ensuremath{\rightarrow}{M}_{A}+{M}_{B}$ cloning machines are exhibited. We also present an optimal $1\ensuremath{\rightarrow}1+1+1$ cloning machine, which is an example of a tripartite fully asymmetric cloner. Finally, it is shown how these cloning machines can be optically realized.We investigate the optimal distribution of quantum information over multipartite systems in asymmetric settings. We introduce cloning transformations that take $N$ identical replicas of a pure state in any dimension as input and yield a collection of clones with nonidentical fidelities. As an example, if the clones are partitioned into a set of ${M}_{A}$ clones with fidelity ${F}^{A}$ and another set of ${M}_{B}$ clones with fidelity ${F}^{B}$, the trade-off between these fidelities is analyzed, and particular cases of optimal $N\ensuremath{\rightarrow}{M}_{A}+{M}_{B}$ cloning machines are exhibited. We also present an optimal $1\ensuremath{\rightarrow}1+1+1$ cloning machine, which is an example of a tripartite fully asymmetric cloner. Finally, it is shown how these cloning machines can be optically realized.

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arXiv:quant-ph/0505212Quantum Information and ComputationMay 27, 200521 citations

d

-dimensional case and derive the explicit transformation of the optimal symmetric $1\ensuremath{\rightarrow}2$ real state cloning in the

d

-dimensional system. The maximal fidelity of two clones obtained coincides with the theoretical value [P. Vavez and N. J. Cerf, Phys. Rev. A 68, 032313 (2003)].

We consider cloning transformations of d-dimensional states of the form

e^{i\phi_0}|0> + e^{i\phi_1}|1> +...+ e^{i\phi_{d-1}}|d-1>

that are covariant with respect to rotations of the phases

\phi_i

's. The optimal cloning maps are easily obtained using a well-defined general characterization of state-dependent

1 \rightarrow 2

cloning transformations in arbitrary dimensions. Our results apply to symmetric as well as asymmetric cloners, so that the balance between the fidelity of the two clones can be analyzed.

The possibility of cloning a

d

-dimensional quantum system without an ancilla is explored, extending on the economical phase-covariant cloning machine for qubits found in Phys. Rev. A 60, 2764 (1999). We prove the impossibility of constructing an economical version of the optimal universal $1\ensuremath{\rightarrow}2$ cloning machine in any dimension. We also show, using an ansatz on the generic form of cloning machines, that the

d

-dimensional $1\ensuremath{\rightarrow}2$ phase-covariant cloner, which optimally clones all balanced superpositions with arbitrary phases, can be realized economically only in dimension

d=2

. The used ansatz is supported by numerical evidence up to

d=7

. An economical phase-covariant cloner can nevertheless be constructed for

d>2

, albeit with a slightly lower fidelity than that of the optimal cloner requiring an ancilla. Finally, using again an ansatz on cloning machines, we show that an economical version of the $1\ensuremath{\rightarrow}2$ Fourier-covariant cloner, which optimally clones the computational basis and its Fourier transform, is also possible only in dimension

d=2

We investigate the optimal distribution of quantum information over multipartite systems in asymmetric settings. We introduce cloning transformations that take

N

identical replicas of a pure state in any dimension as input and yield a collection of clones with nonidentical fidelities. As an example, if the clones are partitioned into a set of

{M}_{A}

clones with fidelity

{F}^{A}

and another set of

{M}_{B}

clones with fidelity

{F}^{B}

, the trade-off between these fidelities is analyzed, and particular cases of optimal $N\ensuremath{\rightarrow}{M}_{A}+{M}_{B}$ cloning machines are exhibited. We also present an optimal $1\ensuremath{\rightarrow}1+1+1$ cloning machine, which is an example of a tripartite fully asymmetric cloner. Finally, it is shown how these cloning machines can be optically realized.