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arXiv:quant-ph/0601164 · Physical Review A

Cloning of Gaussian states by linear optics

Stefano Olivares, Matteo G. A. Paris, Ulrik L. Andersen

Published Jun 22, 2006• 46 citations
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Abstract

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.

DOI
10.1103/physreva.73.062330

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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 performance of a selective cloning machine based on linear optical elements and Gaussian measurements, which allows one to clone at will one of the two incoming input states. This machine is a complete generalization of a $1\ensuremath{\rightarrow}2$ cloning scheme demonstrated by Andersen et al. [Phys. Rev. Lett. 94, 240503 (2005)]. The input-output fidelity is studied for a generic Gaussian input state, and the effect of nonunit quantum efficiency is also taken into account. We show that, if the states to be cloned are squeezed states with known squeezing parameter, then the fidelity can be enhanced using a third suitable squeezed state during the final stage of the cloning process. A binary communication protocol based on the selective cloning machine is also discussed.
We propose an asymmetric quantum cloning scheme. Based on the proposal and experiment by Andersen et al. [Phys. Rev. Lett. 94, 240503 (2005)], we generalize it to two asymmetric cases: quantum cloning with asymmetry between output clones and between quadrature variables. These optical implementations also employ linear elements and homodyne detection only. Finally, we also compare the utility of symmetric and asymmetric cloning in an analysis of a squeezed-state quantum key distribution protocol and find that the asymmetric one is more advantageous.
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 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.
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.