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arXiv:quant-ph/9909037 · Physical Review Letters

Cloning of Continuous Quantum Variables

Nicolas J. Cerf, A. Ipe, X. Rottenberg

Published Aug 21, 2000• 224 citations
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Abstract

The cloning of quantum variables with continuous spectra is analyzed. A Gaussian quantum cloning machine is exhibited that copies equally well the states of two conjugate variables such as position and momentum. It also duplicates all coherent states with a fidelity of 2/3. More generally, the copies are shown to obey a no-cloning Heisenberg-like uncertainty relation.

DOI
10.1103/physrevlett.85.1754

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arXiv:quant-ph/0012048•Physical Review Letters•May 21, 2001•104 citations
The cloning of quantum variables with continuous spectra is analyzed. A Gaussian quantum cloning machine is exhibited that copies equally well the states of two conjugate variables such as position and momentum. It also duplicates all coherent states with a fidelity of 2/3. More generally, the copies are shown to obey a no-cloning Heisenberg-like uncertainty relation.
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 $MN/(MN+M\ensuremath{-}N).$
A family of asymmetric cloning machines for N-dimensional quantum states is introduced. These machines produce two imperfect copies of a single state that emerge from non-identical Heisenberg channels. The trade-off between the quality of the copies imposed by quantum mechanics is shown to result from a complementarity akin to the Heisenberg uncertainty principle. More specifically, the probability distributions of the error operators affecting the two copies are the square modulus of two functions related by a Fourier transform. A no-cloning inequality is derived for the special case of isotropic cloners, quantifying the impossibility of perfect cloning: if π a and π b are the depolarizing fractions associated with the two copies, the domain in (π1/2 a , π1/2 b )-space located inside a particular ellipse representing close-to-perfect cloning is forbidden. More generally, an entropic no-cloning uncertainty relation is also discussed. Finally, the class of asymmetric cloning machines for quantum bits is investigated in detail, and a connection with the capacity of the Pauli channel is displayed.
Abstrac t. A family of asymmetric cloning machines for N-dimensional quantum states is introduced. These machines produce two imperfect copies of a single state that emerge from non-identical Heisenberg channels. The trade-oA between the quality of the copies imposed by quantum mechanics is shown to result from a complementarity akin to the Heisenberg uncertainty principle. More speci® cally, the probability distributions of the error operators aA ecting the two copies are the square modulus of two functions related by a Fourier transform. A no-cloning inequality is derived for the special case of isotropic cloners, quantifying the impossibility of perfect cloning: if oa and ob are the depolarizing fractions associated with the two copies, the domain in …o 1=2 a ;o 1=2 b †-space located inside a particular ellipse representing close-to-perfect cloning is forbidden. More generally, an entropic no-cloning uncertainty relation is also discussed. Finally, the class of asymmetric cloning machines for quantum bits is investigated in detail, and a connection with the capacity of the Pauli channel is displayed.