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arXiv:quant-ph/0102077 · Physical Review Letters

Quantum Cloning Machines with Phase-Conjugate Input Modes

Nicolas J. Cerf, S. Iblisdir

Published Nov 27, 2001• 37 citations
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Abstract

A quantum cloning machine is introduced that yields $M$ identical optimal clones from $N$ 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.

DOI
10.1103/physrevlett.87.247903

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A quantum cloning machine is introduced that yields MMM identical optimal clones from NNN replicas of a coherent state and N\ensuremath′{N}^{\ensuremath{'}}N\ensuremath′ replicas of its phase conjugate. It also optimally produces M\ensuremath′=M+N\ensuremath′\ensuremath−N{M}^{\ensuremath{'}}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}M+N}^{\ensuremath{'}}\ensuremath{-}NM\ensuremath′=M+N\ensuremath′\ensuremath−N phase-conjugate clones at no cost. For well chosen ratios N\ensuremath′/N{N}^{\ensuremath{'}}/NN\ensuremath′/N, this machine is shown to provide better cloning fidelities than the standard (N+N\ensuremath′)\ensuremath→M({N+N}^{\ensuremath{'}})\ensuremath{\rightarrow}M(N+N\ensuremath′)\ensuremath→M cloner. The special cases of the optimal balanced cloner (N=N\ensuremath′)({N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}N}^{\ensuremath{'}})(N=N\ensuremath′) and optimal measurement (M=\ensuremath∞)(M\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\infty})(M=\ensuremath∞) are investigated.
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.
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 $MN/(MN+M\ensuremath{-}N).$
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 investigate the optimal distribution of quantum information over multipartite systems in asymmetric settings. We introduce cloning transformations that take NNN identical replicas of a pure state in any dimension as input and yield a collection of clones with nonidentical fidelities. As an example, if the clones are partitioned into a set of MA{M}_{A}MA​ clones with fidelity FA{F}^{A}FA and another set of MB{M}_{B}MB​ clones with fidelity FB{F}^{B}FB, the trade-off between these fidelities is analyzed, and particular cases of optimal $N\ensuremath{\rightarrow}{M}_{A}+{M}_{B}$ cloning machines are exhibited. We also present an optimal $1\ensuremath{\rightarrow}1+1+1$ cloning machine, which is an example of a tripartite fully asymmetric cloner. Finally, it is shown how these cloning machines can be optically realized.
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.