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Physical Review A

Gaussian cloning of coherent states with known phases

Moorad Alexanian

Published Apr 18, 2006• 8 citations
Publisher

Abstract

The fidelity for cloning coherent states is improved over that provided by optimal Gaussian and non-Gaussian cloners for the subset of coherent states that are prepared with known phases. Gaussian quantum cloning duplicates all coherent states with an optimal fidelity of $2∕3$. Non-Gaussian cloners give optimal single-clone fidelity for a symmetric 1-to-2 cloner of 0.6826. Coherent states that have known phases can be cloned with a fidelity of $4∕5$. The latter is realized by a combination of two beam splitters and a four-wave mixer operated in the nonlinear regime, all of which are realized by interaction Hamiltonians that are quadratic in the photon operators. Therefore, the known Gaussian devices for cloning coherent states are extended when cloning coherent states with known phases by considering a nonbalanced beam splitter at the input side of the amplifier.

DOI
10.1103/physreva.73.045801

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The fidelity for cloning coherent states is improved over that provided by optimal Gaussian and non-Gaussian cloners for the subset of coherent states that are prepared with known phases. Gaussian quantum cloning duplicates all coherent states with an optimal fidelity of 2∕32∕32∕3. Non-Gaussian cloners give optimal single-clone fidelity for a symmetric 1-to-2 cloner of 0.6826. Coherent states that have known phases can be cloned with a fidelity of 4∕54∕54∕5. The latter is realized by a combination of two beam splitters and a four-wave mixer operated in the nonlinear regime, all of which are realized by interaction Hamiltonians that are quadratic in the photon operators. Therefore, the known Gaussian devices for cloning coherent states are extended when cloning coherent states with known phases by considering a nonbalanced beam splitter at the input side of the amplifier.
We consider the optimal cloning of quantum coherent states with single-clone and joint fidelity as figures of merit. While the latter is maximized by a Gaussian cloner, the former is not: the optimal single-clone fidelity for a symmetric 1-to-2 cloner is 0.6826, compared to 2/3 in a Gaussian setting. This cloner can be realized with an optical parametric amplifier and certain non-Gaussian bimodal states. Finally, we show that the single-clone fidelity of the optimal 1-to-infinity cloner is 1/2. It is achieved by a Gaussian scheme and cannot be surpassed even with supplemental bound entangled states.
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.
We investigate the asymmetric Gaussian cloning of coherent states which produces MMM copies from NNN input replicas in such a way that the fidelity of each copy may be different. We show that the optimal asymmetric Gaussian cloning can be performed with a single phase-insensitive amplifier and an array of beam splitters. We obtain a simple analytical expression characterizing the set of optimal asymmetric Gaussian cloning machines and prove the optimality of these cloners using the formalism of Gaussian completely positive maps and semidefinite programming techniques. We also present an alternative implementation of the asymmetric cloning machine where the phase-insensitive amplifier is replaced with a beam splitter, heterodyne detector, and feedforward.
The laws of quantum mechanics forbid the perfect copying of an unknown quantum state, known as the no-cloning theorem. In spite of this, approximate cloning with imperfect fidelity is possible, which opens up the field of quantum cloning. In general, quantum cloning can be divided into discrete variable and continuous variable (CV) categories. In the CV regime, all-optical implementation of the optimal N→M quantum cloning has been proposed in two original parallel works, which involves a parametric amplifier and a set of beam splitters and thus avoids the optic-electro and electro-optic conversions in the current CV quantum cloning technologies. However, such original proposal of all-optical CV optimal N→M quantum cloning scheme has never been experimentally implemented. Here, we show that optimal N→M quantum cloning of coherent states can be realized by utilizing a parametric amplifier based on four-wave mixing process in a hot atomic vapor and a set of beam splitters. In particular, we realize 1→M, 2→M, and 4→M quantum cloning. We find that the fidelity of N→M quantum cloning increases with the decrease of clone number M and the increase of original replica number N. The best cloning fidelity achieved in our experiment is about 93.3% ±1.0% in the 4→5 case. Our results may find potential applications in realizing all-optical high-fidelity quantum state transfer and all-optical high-compatibility eavesdropping attack in quantum communication networks.