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arXiv:1806.11356 · Physical Review A

Composable security of two-way continuous-variable quantum key distribution without active symmetrization

Shouvik Ghorai, Eleni Diamanti, Anthony Leverrier

Published Jan 9, 2019• 34 citations
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Abstract

We present a general framework encompassing a number of continuous-variable quantum key distribution protocols, including standard one-way protocols, measurement-device-independent protocols, as well as some two-way protocols, or any other continuous-variable protocol involving only a Gaussian modulation of coherent states and heterodyne detection. The main interest of this framework is that the corresponding protocols are all covariant with respect to the action of the unitary group $\text{U}(n)$, implying that their security can be established thanks to a Gaussian de Finetti reduction. In particular, we give a composable security proof of two-way continuous-variable quantum key distribution against general attacks. We also prove that no active symmetrization procedure is required for these protocols, which would otherwise make them prohibitively costly to implement.

DOI
10.1103/physreva.99.012311

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We present a general framework encompassing a number of continuous-variable quantum key distribution protocols, including standard one-way protocols, measurement-device-independent protocols, as well as some two-way protocols, or any other continuous-variable protocol involving only a Gaussian modulation of coherent states and heterodyne detection. The main interest of this framework is that the corresponding protocols are all covariant with respect to the action of the unitary group U(n)\text{U}(n)U(n), implying that their security can be established thanks to a Gaussian de Finetti reduction. In particular, we give a composable security proof of two-way continuous-variable quantum key distribution against general attacks. We also prove that no active symmetrization procedure is required for these protocols, which would otherwise make them prohibitively costly to implement.
We present a general framework encompassing a number of continuous-variable quantum key distribution protocols, including standard one-way protocols, measurement-device-independent protocols as well as some two-way protocols, or any other continuous-variable protocol involving only a Gaussian modulation of coherent states and heterodyne detection. The main interest of this framework is that the corresponding protocols are all covariant with respect to the action of the unitary group U(n)U(n)U(n), implying that their security can be established thanks to a Gaussian de Finetti reduction. In particular, we give a composable security proof of two-way continuous-variable quantum key distribution against general attacks. We also prove that no active symmetrization procedure is required for these protocols, which would otherwise make them prohibitively costly to implement.
Establishing the security of continuous-variable quantum key distribution against general attacks in a realistic finite-size regime is an outstanding open problem in the field of theoretical quantum cryptography if we restrict our attention to protocols that rely on the exchange of coherent states. Indeed, techniques based on the uncertainty principle are not known to work for such protocols, and the usual tools based on de Finetti reductions only provide security for unrealistically large block lengths. We address this problem here by considering a new type of Gaussian de Finetti reduction, that exploits the invariance of some continuous-variable protocols under the action of the unitary group U(n) (instead of the symmetric group S_{n} as in usual de Finetti theorems), and by introducing generalized SU(2,2) coherent states. Crucially, combined with an energy test, this allows us to truncate the Hilbert space globally instead as at the single-mode level as in previous approaches that failed to provide security in realistic conditions. Our reduction shows that it is sufficient to prove the security of these protocols against Gaussian collective attacks in order to obtain security against general attacks, thereby confirming rigorously the widely held belief that Gaussian attacks are indeed optimal against such protocols.
We give the first composable security proof for continuous-variable quantum key distribution with coherent states against collective attacks. Crucially, in the limit of large blocks the secret key rate converges to the usual value computed from the Holevo bound. Combining our proof with either the de Finetti theorem or the postselection technique then shows the security of the protocol against general attacks, thereby confirming the long-standing conjecture that Gaussian attacks are optimal asymptotically in the composable security framework. We expect that our parameter estimation procedure, which does not rely on any assumption about the quantum state being measured, will find applications elsewhere, for instance, for the reliable quantification of continuous-variable entanglement in finite-size settings.
We present a rigorous security analysis of continuous-variable measurement-device-independent quantum key distribution (CV MDI QKD) in a finite-size scenario. The security proof is obtained in two steps: by first assessing the security against collective Gaussian attacks, and then extending to the most general class of coherent attacks via the Gaussian de Finetti reduction. Our result combines recent state-of-the-art security proofs for CV QKD with findings about min-entropy calculus and parameter estimation. In doing so, we improve the finite-size estimate of the secret key rate. Our conclusions confirm that CV MDI protocols allow for high rates on the metropolitan scale, and may achieve a nonzero secret key rate against the most general class of coherent attacks after 107{10}^{7}107--109{10}^{9}109 quantum signal transmissions, depending on loss and noise, and on the required level of security.
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)].