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arXiv:quant-ph/0005044 · Physical Review A

OptimalN-to- M cloning of conjugate quantum variables

Nicolas J. Cerf, S. Iblisdir

Published Sep 18, 2000• 104 citations
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Abstract

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-$M$ cloning fidelity for coherent states is found to be equal to $MN/(MN+M\ensuremath{-}N).$

DOI
10.1103/physreva.62.040301

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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).MN/(MN+M\ensuremath{-}N).MN/(MN+M\ensuremath−N).
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.
Quantum state cloning is the task of approximately producing additional copies of an unknown quantum state from a finite number of input copies. The optimal cloning fidelity is known exactly for pure states, but no comparable characterization is known for general mixed states. We determine the optimal asymptotic copy complexity for ddd-dimensional states of rank at most rrr: producing MMM additional copies with worst-case fidelity at least 1−ε1-\varepsilon1−ε requires and is achievable with N=Θ(Mrd/ε)N=Θ(Mrd/\varepsilon)N=Θ(Mrd/ε) input copies. Remarkably, the lower bound already holds for a family of states with a fixed flat spectrum, while the matching upper bound is achieved by random purification followed by optimal pure-state cloning. For M=1M=1M=1, we further show that high-fidelity tomography can be coherently converted into cloning with comparable error, revealing an operational origin of the matching cloning and tomography complexities.
We present quantum cloning machines that transform NNN identical qubits into M>NM>NM>N identical copies and we prove that the fidelity (quality) of these copies is optimal. The connection between cloning and measurement is discussed in detail. When the number of clones MMM tends towards infinity, the fidelity of each clone tends towards the optimal fidelity that can be obtained by a measurement on the input qubits. More generally, quantum cloning machines are universal devices to translate quantum information into classical information.
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.