Composable Security Proof for Continuous-Variable Quantum Key Distribution with Coherent States

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.

Figure 1

Expected secret key rate $r=(1-{\epsilon }_{\text{rob}})l/2n$ secure against collective attacks, as a function of $2n$, the number of exchanged signals. From top to bottom, the transmittance of the quantum channel corresponds to distances of 1, 10, 50, and 100 km for assumed losses of 0.2 dB per kilometer. For each distance, the expected secret key rate reaches the asymptotic value for large enough $n$. The modulation variance is optimized; the reconciliation efficiency is set to $\beta =0.95$, the discretization parameter to $d=5$ (the value of $d$ should be optimized depending on the error correcting codes used in the reconciliation; see, e.g., [46]), the excess noise to $\xi =0.01$, the robustness parameter to ${\epsilon }_{\text{rob}}\le {10}^{-2}$, and the security parameter to $\epsilon ={10}^{-20}$. Dashed lines correspond to the respective asymptotic expected secret key rates. Refer to the Supplemental Material for a detailed derivation of the value of the expected secret key rate.