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Physical Review A

Local and nonlocal cloning of coherent states with known phases using linear optics

Yuli Dong, Xu‐Bo Zou, Guang‐Can Guo

Published Mar 25, 2008• 3 citations
Publisher

Abstract

First, we present a scheme for implementation of$N\ensuremath{\rightarrow}M$ local cloning of coherent states with known phases, based on linear optics, single-mode squeezed sources, and homodyne detection. The amplification of such cloning is accomplished with the help of a strongly single-mode squeezed source. Then we propose a scheme for $N\ensuremath{\rightarrow}M$ nonlocal cloning of such input states by employing an additional two-mode squeezed source. In the case of ideal squeezed sources, the fidelities of the two cloning machines can reach a high level.

DOI
10.1103/physreva.77.034304

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First, we present a scheme for implementation ofN\ensuremath→MN\ensuremath{\rightarrow}MN\ensuremath→M local cloning of coherent states with known phases, based on linear optics, single-mode squeezed sources, and homodyne detection. The amplification of such cloning is accomplished with the help of a strongly single-mode squeezed source. Then we propose a scheme for N\ensuremath→MN\ensuremath{\rightarrow}MN\ensuremath→M nonlocal cloning of such input states by employing an additional two-mode squeezed source. In the case of ideal squeezed sources, the fidelities of the two cloning machines can reach a high level.
We propose a scheme for continuous-variable quantum cloning of coherent states with phase-conjugate input modes using linear optics. The quantum cloning machine yields MMM identical optimal clones from NNN replicas of a coherent state and NNN replicas of its phase conjugate. This scheme can be straightforwardly implemented with the setups accessible at present since its optical implementation only employs simple linear optical elements and homodyne detection. Compared with the original scheme for continuous-variable quantum cloning with phase-conjugate input modes proposed by Cerf and Iblisdir [Phys. Rev. Lett. 87, 247903 (2001)], which utilized a nondegenerate optical parametric amplifier, our scheme loses the output of phase-conjugate clones and is regarded as irreversible quantum cloning.
Optimal Gaussian NNN-to-MMM cloning with linear optics and Gaussian cloning of known-phase coherent states
We show how to implement the optimal Gaussian NNN-to-MMM cloning with linear optics and homodyne detection. We also show that the Gaussian NNN-to-MMM cloning of known-phase coherent states can be performed with the fidelity 2MN2MN+M−N\sqrt \frac{2 M N}{2M N+M -N}2MN+M−N2MN​​ by linear optics and homodyne detection, and with 21+1N+1−1M\frac{2}{\sqrt{1+\frac{1}{N}}+\sqrt {1-\frac{1}{M}}}1+N1​​+1−M1​​2​ by utilizing quadrature squeezing. From the classical limit of the cloning (1-to-∞\infty∞ cloning), a necessary condition of continuous variable quantum key distribution using known-phase coherent states is provided.
We analyze in details a scheme for cloning of Gaussian states based on linear optical components and homodyne detection recently demonstrated by Andersen et al. [Phys. Rev. Lett. 94, 240503 (2005)]. The input-output fidelity is evaluated for a generic (pure or mixed) Gaussian state taking into account the effect of nonunit quantum efficiency and unbalanced mode mixing. In addition, since in most quantum information protocols the covariance matrix of the set of input states is not perfectly known, we evaluate the average cloning fidelity for classes of Gaussian states with the degree of squeezing and the number of thermal photons being only partially known.
We investigate the economical Gaussian cloning of coherent states with the known phase, which produces MMM copies from NNN input replica and can be implemented with degenerate parametric amplifiers and beam splitters.The achievable fidelity of single copy is given by $2M\sqrt{N}∕[\sqrt{N}(M\ensuremath{-}1)+\sqrt{(1+N)({M}^{2}+N)}]$, which is bigger than the optimal fidelity of the universal Gaussian cloning. The cloning machine presented here works without ancillary optical modes and can be regarded as the continuous variable generalization of the economical cloning machine for qudits.