Finite-size analysis of measurement-device-independent quantum cryptography with continuous variables

We study the impact of finite-size effects on the key rate of continuous-variable (CV) measurement-device-independent (MDI) quantum key distribution (QKD), considering two-mode Gaussian attacks. Inspired by the parameter estimation technique developed in by Ruppert et al. [Phys. Rev. A 90, 062310 (2014)], we adapt it to study CV-MDI-QKD and, assuming realistic experimental conditions, we analyze the impact of finite-size effects on the key rate. We find that the performance of the protocol approaches the ideal one, increasing the block size, and, most importantly, that blocks between ${10}^6$ and ${10}^9$ data points may provide key rates $\sim {10}^{-2}$ bit/use over metropolitan distances.

Figure 1

The figure shows the EB representation of CV-MDI QKD. Alice and Bob have TMSV states with modes $(a,A)$ and $(b,B)$. Local modes $a$ and $b$ are kept by the parties, while $A$ and $B$ are sent to the relay through two links with transmittance ${\tau }_A$ and ${\tau }_B$. When Alice and Bob heterodyne the local modes, the traveling ones $A$ and $B$ are projected onto coherent states $|\alpha \rangle$ and $|\beta \rangle$. The relay performs a Bell measurement and broadcasts the outcomes $\gamma$, creating correlation between the parties: For instance, Bob recovers Alice variable $\beta$ by subtracting his variable $\alpha$ from the relay's outputs $\gamma$. The Gaussian attack on the links is simulated by Eve using ancillas $E_1$ and $E_2$ and thermal noise ${\omega }_A\ge 1$ and ${\omega }_B\ge 1$, respectively. These ancillary modes are, in general, two-mode correlated (see text for more details). The ancillary outputs are stored in a quantum memory for a later measurement.

Figure 2

The figure summarizes the impact of finite-size effects on the performance of CV-MDI QKD for both asymmetric (a) and symmetric (b) configurations of the relay, in the presence of optimal two-mode attack. In panel (a), the key rate is plotted as a function of the dB of attenuation on Bob's channel, with the relay placed near Alice ${\tau }_A=0.98$. From top to bottom, the black curves describe the rate for $\overline{N}\gg 1$ with $\xi =0.98$ and optimizing over $V_M$ (solid line). Then we have the cases with finite block size. The dashed line is for $\overline{N}={10}^9$ while the dot-dashed curve is obtained for $\overline{N}={10}^6$. In all cases, the excess noise is about 0.01 SNU. The red curves (*) describe the case obtained for pure loss and assuming $\xi =1,V_M\to \infty ,\overline{N}\to \infty$ (solid line) and $\overline{N}={10}^9$ $$(dashed line). Panel (b) focuses on the symmetric configuration of the relay. The curves are obtained using the same parameters as in panel (a), but setting ${\tau }_A={\tau }_B=\tau$.

Figure 3

This figure shows the impact of the reconciliation efficiency on the key rate. When $\xi =0.95$ (dashed line), the Gaussian modulation maximizing the key rate is $V_M<\infty$. While when $\xi =1$ then the optimal key rate is obtained for $V_M\to \infty$ (solid line). The lines are obtained for pure loss attack, ${\tau }_A=0.98$ and ${\tau }_B=0.7$, and block size $\overline{N}={10}^6$.