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arXiv:0912.4132 · Physical Review A

Simple proof that Gaussian attacks are optimal among collective attacks against continuous-variable quantum key distribution with a Gaussian modulation

Anthony Leverrier, Philippe Grangier

Published Jun 15, 2010• 70 citations
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Abstract

In this article, we give a simple proof of the fact that the optimal collective attacks against continuous-variable quantum key distribution with a Gaussian modulation are Gaussian attacks. Our proof, which makes use of symmetry properties of the protocol in phase space, is particularly relevant for the finite-key analysis of the protocol and therefore for practical applications.

DOI
10.1103/physreva.81.062314

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In this article, we give a simple proof of the fact that the optimal collective attacks against continuous-variable quantum key distribution with a Gaussian modulation are Gaussian attacks. Our proof, which makes use of symmetry properties of the protocol in phase space, is particularly relevant for the finite-key analysis of the protocol and therefore for practical applications.
We introduce a symmetrization technique that can be used as an extra step in some continuous-variable quantum key distribution protocols. By randomizing the data in phase space, one can dramatically simplify the security analysis of the protocols, in particular in the case of collective attacks. The main application of this procedure concerns protocols with postselection, for which security was established only against Gaussian attacks until now. Here, we prove that under some experimentally verifiable conditions, Gaussian attacks are optimal among all collective attacks.
We prove the security of Gaussian continuous-variable quantum key distribution with coherent states against arbitrary attacks in the finite-size regime. In contrast to previously known proofs of principle (based on the de Finetti theorem), our result is applicable in the practically relevant finite-size regime. This is achieved using a novel proof approach, which exploits phase-space symmetries of the protocols as well as the postselection technique introduced by Christandl, Koenig, and Renner [Phys. Rev. Lett. 102, 020504 (2009)].
We present here an information theoretic study of Gaussian collective attacks on the continuous variable key distribution protocols based on Gaussian modulation of coherent states. These attacks, overlooked in previous security studies, give a finite advantage to the eavesdropper in the experimentally relevant lossy channel, but are not powerful enough to reduce the range of the reverse reconciliation protocols. Secret key rates are given for the ideal case where Bob performs optimal collective measurements, as well as for the realistic cases where he performs homodyne or heterodyne measurements. We also apply the generic security proof of Christiandl et al. to obtain unconditionally secure rates for these protocols.
A fully general approach to the security analysis of continuous-variable quantum key distribution (CV-QKD) is presented. Provided that the quantum channel is estimated via the covariance matrix of the quadratures, Gaussian attacks are shown to be optimal against all collective eavesdropping strategies. The proof is made strikingly simple by combining a physical model of measurement, an entanglement-based description of CV-QKD, and a recent powerful result on the extremality of Gaussian states [M. M. Wolf, Phys. Rev. Lett. 96, 080502 (2006)10.1103/PhysRevLett.96.080502].
We provide a security analysis of continuous-variable quantum key distribution (CVQKD) with discrete modulation against general attacks in a realistic finite-size regime. To realize this goal, we first prove security of the continuous-variable quantum key distribution protocol with discrete modulation against collective attacks by using the reliable tomography of the covariance matrix, leading to the reliable and tight error bounds in the derived confidence regions. Combining the proof with de Finetti reduction, the discrete-modulation-based continuous-variable quantum key distribution (DM-CVQKD) is proved to be secure even exposing to general attacks. Specially, we use an energy test to truncate the Hilbert space globally to provide security.