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Physical Review A

Characterization of low-cost one-to-two qubit cloning

Thomas Durt, Jiangfeng Du

Published Jun 16, 2004• 58 citations
Publisher

Abstract

In this paper we establish a deep connection between the 3-qubit one-to-two phase-covariant quantum cloning network of Fuchs et al. [Phys. Rev. A 56, 1163 (1997)], and its low-cost two-qubit counterpart due to Niu and Griffiths [Phys. Rev. A 60, 2764 (1999)]. A general, necessary, and sufficient criterion is derived in order to characterize the reducibility of 3-qubit one-to-two cloners to two-qubit one-to-two cloners. When this criterion is fulfilled, low-cost cloning is possible. We show that the optimal isotropic or universal 3-qubit one-to-two cloning machine is not reducible to a 2-qubit one-to-two cloner.

DOI
10.1103/physreva.69.062316

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Following the work of Niu and Griffiths, in \emph{Phys.Rev.A 58, 4377(1998)}, we shall investigate the problem, how to design the optimal quantum cloning machines (QCMs) for qubit system, with the help of Bloch-sphere representation. In stead of the quality factor there, the Fiuráusek's optimal condition, where the optimal cloning machine should maximize a convex mixture of the average fidelity, is used as the optimality criterion in present protocol. Almost all of the known optimal QCMs in previous works, the cloning for states with fixed polar angle, the phase-covariant cloning, the universal QCMs, the cloning for two arbitrary pure states, and the mirror phase-covariant cloning, should be discussed in a systematic way. The known results, the optimal fidelities for various input ensembles according to different optimality criteria, are recovered here. Our present scheme also offers a general way of constructing the unitary transformation to realize the optimal cloning.

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In this paper we establish a deep connection between the 3-qubit one-to-two phase-covariant quantum cloning network of Fuchs et al. [Phys. Rev. A 56, 1163 (1997)], and its low-cost two-qubit counterpart due to Niu and Griffiths [Phys. Rev. A 60, 2764 (1999)]. A general, necessary, and sufficient criterion is derived in order to characterize the reducibility of 3-qubit one-to-two cloners to two-qubit one-to-two cloners. When this criterion is fulfilled, low-cost cloning is possible. We show that the optimal isotropic or universal 3-qubit one-to-two cloning machine is not reducible to a 2-qubit one-to-two cloner.
In this paper we establish a deep connection between the 3 qubit one-to-two phase-covariant quantum cloning network of Fuchs et al. [C. Fuchs, N. Gisin, R.B. Griffiths, C.S. Niu, and A. Peres, Phys. Rev. A 56 no 4, 1163 (1997)], and its economic 2 qubit counterpart due to Niu and Griffiths [Phys.Rev. A 60 no 4, 2764 (1999)]. A general, necessary and sufficient criterion is derived in order to characterize the reducibility of 3 qubit cloners to 2 qubit cloners. When this criterion is fulfilled, economic cloning is possible. We show that the optimal isotropic or universal 3 qubit cloning machine is not reducible to a 2 qubit cloner.
We propose a quantum cloning machine, which clones a qubit into two clones assuming known modulus of expectation value of Pauli ${\ensuremath{\sigma}}_{z}$ matrix. The process is referred to as the mirror phase-covariant cloning for which the input state is a priori less known than that for the standard phase-covariant cloning. Analytical expressions describing the cloning transformation and fidelity of the clones are found. Extremal equations for the optimal cloning are derived and analytically solved by generalizing a method of Fiur\'a\ifmmode \check{s}\else \v{s}\fi{}ek [Phys. Rev. A 64, 062310 (2001)]. Quantum circuits implementing the optimal cloning transformation and their physical realization in a quantum-dot system are described.
The possibility of cloning a ddd-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 ddd-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=2d=2d=2. The used ansatz is supported by numerical evidence up to d=7d=7d=7. An economical phase-covariant cloner can nevertheless be constructed for d>2d>2d>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=2d=2d=2.
Following the work of Niu and Griffiths, in \emph{Phys.Rev.A 58, 4377(1998)}, we shall investigate the problem, how to design the optimal quantum cloning machines (QCMs) for qubit system, with the help of Bloch-sphere representation. In stead of the quality factor there, the Fiuráusek's optimal condition, where the optimal cloning machine should maximize a convex mixture of the average fidelity, is used as the optimality criterion in present protocol. Almost all of the known optimal QCMs in previous works, the cloning for states with fixed polar angle, the phase-covariant cloning, the universal QCMs, the cloning for two arbitrary pure states, and the mirror phase-covariant cloning, should be discussed in a systematic way. The known results, the optimal fidelities for various input ensembles according to different optimality criteria, are recovered here. Our present scheme also offers a general way of constructing the unitary transformation to realize the optimal cloning.