Composable finite-size effects in free-space continuous-variable quantum-key-distribution systems

Free-space channels provide the possibility of establishing continuous-variable quantum key distribution in global communication networks. However, the fluctuating nature of transmissivity in these channels introduces an extra noise which reduces the achievable secret key rate. We consider two classical postprocessing strategies, postselection of high-transmissivity data and data clusterization, to reduce the fluctuation-induced noise of the channel. We undertake the investigation of such strategies utilizing a composable security proof in a realistic finite-size regime against both collective and individual attacks. We also present an efficient parameter estimation approach to estimate the effective Gaussian parameters over the postselected data or the clustered data. Although the composable finite-size effects become more significant with the postselection and clusterization both reducing the size of the data, our results show that these strategies are still able to enhance the finite-size key rate against both individual and collective attacks with a remarkable improvement against collective attacks, even moving the protocol from an insecure regime to a secure regime under certain conditions.

Figure 1

Postselected key rate (in bits per mode) in the asymptotic (dashed lines) and composable finite-size (solid and dotted lines) regime as a function of the postselection threshold ${\eta }_{\mathrm{th}}$, secure against individual (blue top lines), collective (red middle lines), and general (black dotted bottom lines) attacks. Note that in the asymptotic regime collective attacks are as powerful as general attacks. Note also that for the free-space channels in the two right plots the protocol is not secure against general attacks in the composable finite-size regime. The numerical values for the finite-size regime are the security parameter $\epsilon ={10}^{-9}$ ($\tilde{\epsilon }={10}^{-9}$ for general attacks) and the discretization parameter $d=5$. The other parameters are chosen from the most recent CV QKD experiment [17] as follows. Bob's detector has efficiency ${\eta }_B=0.6$ and electronic noise ${\nu }_B=0.25$. The reconciliation efficiency is considered to be $\beta =0.98$. The expected excess noise of each subchannel is assumed to be fixed as $\xi =0.01$. (Note that, although in our numerical simulations we have assumed a fixed excess noise for each subchannel, our parameter estimation presented in Sec. 4c also works for the case of fluctuating excess noise.) The block size is chosen to be $N={10}^{10}$, half of which (i.e., $k=0.5N$) is used in total for parameter estimation with ${\epsilon }_{\mathrm{PE}}={10}^{-10}$ (${\epsilon }_{\mathrm{PE}}={10}^{-50}$ for general attacks). Note that we also use a subset of data of size $m=0.1N$ for the energy test required for security analysis against general attacks. Since the non-Gaussian noise [i.e., the term $\mathrm{Var}\left(\sqrt{\eta }\right)(\mathrm{V}-1)$ in Eq. (8)] depends on the modulation variance, the modulation variance is optimized for each postselection threshold to maximize the key rate against a given attack. We consider a probability distribution for the free-space channel given by the elliptic-beam model (see Appendix pp3 for more details on the model). From left to right we have the average $\langle \eta \rangle =0.32,0.19,0.12,0.08, \langle \sqrt{\eta }\rangle =0.56,0.43,0.34,0.27$, the variance $\mathrm{Var}\left(\sqrt{\eta }\right)=0.005,0.004,0.003,0.002$, and the maximum transmissivity ${\eta }_{\max }=0.46,0.30,0.20,0.13$ [see Fig. 3 in Appendix pp3 for the corresponding probability distributions $p\left(\eta \right)$].

Figure 2

Key rate (in bits per mode) in the asymptotic regime (circular points on dotted lines) secure against individual (blue top lines) and collective (red bottom lines) attacks and composable finite-size regime (circular points on solid lines) secure against individual (blue top lines) and collective (red bottom lines) attacks, as a function of the number of clusters $n$. The numerical values are the same as in Fig. 1. From left to right we have the average $\langle \eta \rangle =0.12,0.08, \langle \sqrt{\eta }\rangle =0.34,0.27$, the variance $\mathrm{Var}\left(\sqrt{\eta }\right)=0.003,0.002$, and the maximum transmissivity ${\eta }_{\max }=0.20,0.13$.

Figure 3

Probability distribution function $p\left(\eta \right)$ obtained for the generated sampled data ${\eta }_i, n={10}^4$. For the sample generations the parameters are chosen based on an experimentally implemented free-space experiment [18]. The elliptic-beam model [28] shows good agreement with the experimental distribution of the transmissivity [28]. The following parameter values are used: the wavelength $\lambda =809$ nm, the initial beam-spot radius $W_0=20$ mm, the deterministic attenuation $1.25$ dB, and the radius of the receiver aperture $a=40$ mm. The Rytov parameter is given by ${\sigma }_R^2=1.23C_n^2{k}^{7/6}{L}^{11/6}$, where we choose $C_n^2=1.5\times {10}^{-14}{\text{m}}^{-2/3}$, and the propagation distance $L=2,2.5,3,3.5$ km from left to right.