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arXiv:0711.0039 · Physical Review A

Quantum cloning of continuous-variable entangled states

Christian Weedbrook, Nicolai B. Grosse, Thomas Symul, Ping Koy Lam, Timothy C. Ralph

Published May 13, 2008• 19 citations

Abstract

We consider the quantum cloning of continuous variable entangled states. This is achieved by introducing two symmetric entanglement cloning machines (or $e$-cloners): a local $e$-cloner and a global $e$-cloner; where we look at the preservation of entanglement in the clones under the condition that the fidelity of the clones is maximized. These cloning machines are implemented using simple linear optical elements such as beam splitters and homodyne detection along with squeeze gates. We show that the global $e$-cloner out-performs the local $e$-cloner both in terms of the fidelity of the cloned states as well as the strength of the entanglement of the clones. There is a minimum strength of entanglement (3 dB using the inseparability criterion) of the input state of the global $e$-cloner that is required to preserve the entanglement in the clones.

DOI: 10.1103/physreva.77.052313

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We consider the quantum cloning of continuous variable entangled states. This is achieved by introducing two symmetric entanglement cloning machines (or

e

-cloners): a local

e

-cloner and a global

e

-cloner; where we look at the preservation of entanglement in the clones under the condition that the fidelity of the clones is maximized. These cloning machines are implemented using simple linear optical elements such as beam splitters and homodyne detection along with squeeze gates. We show that the global

e

-cloner out-performs the local

e

-cloner both in terms of the fidelity of the cloned states as well as the strength of the entanglement of the clones. There is a minimum strength of entanglement (3 dB using the inseparability criterion) of the input state of the global

e

-cloner that is required to preserve the entanglement in the clones.

We extend the technique of conditional preparation to a quantum cloning machine, and present a protocol of 1 → 2 conditional cloning of squeezed state and entanglement states. It is shown that the entanglement degree of the cloned entangled states can be well preserved even when the fidelity between the input and output states is beyond the limit of 4/9. This scheme is practicable since only the linear elements of beam splitters, homodyne detections, optical modulations and electrical trigger system, are involved.

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.

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

M

identical optimal clones from

N

replicas of a coherent state and

N

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.