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Kluwer Academic Publishers eBooks

Universal Copying of Coherent States: A Gaussian Cloning Machine

Nicolas J. Cerf, S. Iblisdir

Published Feb 24, 2006• 4 citations
Publisher

Abstract

A Gaussian cloning machine is presented that duplicates with a same fidelity two canonically conjugate continuous variables such as the quadratures of a mode of the electromagnetic field.This cloner is displacement and rotation covariant in phase space, hence it duplicates all coherent states with afidelity of 2/3. The optimality of this cloner is discussed, as well as the extension to N→M continuous cloners.

DOI
10.1007/0-306-47114-0_2

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A Gaussian cloning machine is presented that duplicates with a same fidelity two canonically conjugate continuous variables such as the quadratures of a mode of the electromagnetic field.This cloner is displacement and rotation covariant in phase space, hence it duplicates all coherent states with afidelity of 2/3. The optimality of this cloner is discussed, as well as the extension to N→M continuous cloners.
Cloning machines, that is, transformations that achieve the best approximate copying of a quantum state compatible with the no-cloning theorem, have been a fundamental research topic over the last five years. This study is of particular significance given the close connection between quantum cloning and quantum cryptography: using an optimal cloner generally makes it possible to obtain a tight bound on the best individual eavesdropping strategy in a quantum cryptosystem. In this Chapter, the issue of cloning a continuous-variable quantum system will be analyzed, and a Gaussian cloning transformation will be introduced. This cloner, which copies equally well any two canonically conjugate continuous variables, duplicates all coherent states with a same fidelity (F=2/3). The optical implementation of this cloner and its extension to N-to-M cloners will also be discussed. Finally, the use of this cloner for the security assessment of a quantum key distribution scheme relying on continuous (Gaussian) key carriers will be sketched.
The cloning of conjugate continuous quantum variables is analyzed based on the concept of Gaussian cloning machines, i.e., transformations that yield copies that are Gaussian mixtures centered on the state to be copied. The optimality of Gaussian cloning machines that transform N identical input states into M output states is investigated, and bounds on the fidelity of the process are derived via a connection with quantum estimation theory. In particular, the optimal N-to-MMM cloning fidelity for coherent states is found to be equal to $MN/(MN+M\ensuremath{-}N).$
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.
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.
A quantum cloning machine is introduced that yields MMM identical optimal clones from NNN replicas of a coherent state and ${N}^{\ensuremath{'}}$ replicas of its phase conjugate. It also optimally produces ${M}^{\ensuremath{'}}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}M+N}^{\ensuremath{'}}\ensuremath{-}N$ phase-conjugate clones at no cost. For well chosen ratios ${N}^{\ensuremath{'}}/N$, this machine is shown to provide better cloning fidelities than the standard $({N+N}^{\ensuremath{'}})\ensuremath{\rightarrow}M$ cloner. The special cases of the optimal balanced cloner $({N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}N}^{\ensuremath{'}})$ and optimal measurement $(M\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\infty})$ are investigated.