Finite-size analysis of continuous-variable measurement-device-independent quantum key distribution

We study the impact of the finite-size effect on the continuous-variable measurement-device-independent quantum key distribution (CV-MDI QKD) protocol, mainly considering the finite-size effect on the parameter estimation procedure. The central-limit theorem and maximum likelihood estimation theorem are used to estimate the parameters. We also analyze the relationship between the number of exchanged signals and the optimal modulation variance in the protocol. It is proved that when Charlie's position is close to Bob, the CV-MDI QKD protocol has the farthest transmission distance in the finite-size scenario. Finally, we discuss the impact of finite-size effects related to the practical detection in the CV-MDI QKD protocol. The overall results indicate that the finite-size effect has a great influence on the secret-key rate of the CV-MDI QKD protocol and should not be ignored.

Figure 1

A schematic diagram of CV-MDI QKD. BS stands for $50:50$ beam splitter, Hom stands for homodyne detection.

Figure 2

Entanglement-based scheme of CV-MDI QKD with practical detector. $T_1\left(T_2\right)$ is the channel transmittance for Alice-Charlie (Bob-Charlie), ${\varepsilon }_1\left({\varepsilon }_2\right)$ is the channel excess noise for Alice-Charlie (Bob-Charlie). $\eta$ is the efficiency of the detection, $v$ is the variance of the thermal state, $v_{el}$ is the variance of electronic noise. EPR is the two-mode squeezed state, P-Hom is practical homodyne detection, Hom is ideal homodyne detection, Het is heterodyne detection.

Figure 3

Comparison between the maximal transmission distance for the CV-MDI QKD protocol under the finite-size effect. The CV-MDI QKD protocol with ideal reconciliation efficiency $\beta =1$, where the secret-key rate $k$ is positive. The block lengths from left to right curves correspond to $N={10}^6,{10}^7,{10}^8,{10}^9,{10}^{10}$. Here we use the ideal modulation variance $V_A=V_B={10}^5$ and excess noises ${\varepsilon }_1={\varepsilon }_2=0.002$ [16].

Figure 4

Optimal modulation variance for the CV-MDI QKD protocol in the asymmetric case with the finite-size effect. CV-MDI QKD protocol with ideal reconciliation efficiency $\beta =1$. From top to bottom, the block length $N$ is equal to ${10}^6$, ${10}^7$, ${10}^8$, ${10}^9$, and ${10}^{10}$. Here we use the excess noises ${\varepsilon }_1={\varepsilon }_2=0.002$ [16].

Figure 5

Secret-key rate for the CV-MDI QKD protocol in the asymmetric case with the finite-size effect. The CV-MDI QKD protocol with ideal reconciliation efficiency $\beta =1$. The block lengths from left to right curves correspond to $N={10}^6,{10}^7,{10}^8,{10}^9,{10}^{10}$ and the asymptotic regime. Here we use the ideal modulation variance $V_A=V_B={10}^5$, and excess noises ${\varepsilon }_1={\varepsilon }_2=0.002$ [16].

Figure 6

Optimal modulation variance for the CV-MDI QKD protocol in asymmetric case with the finite-size effect. The CV-MDI QKD protocol with imperfect reconciliation efficiency $\beta =96.9\%$. From top to bottom, the block length $N$ is equal to ${10}^6$, ${10}^7$, ${10}^8$, ${10}^9$, and ${10}^{10}$. Here we use the excess noises ${\varepsilon }_1={\varepsilon }_2=0.002$ [16].

Figure 7

Comparison among the maximal transmission distance for the CV-MDI QKD protocol under the finite-size effect. The CV-MDI QKD protocol with imperfect reconciliation efficiency $\beta =96.9\%$, within which the secret-key rate $k$ is positive. The block lengths from left to right curves correspond to $N={10}^6,{10}^7,{10}^8,{10}^9,{10}^{10}$. Here we use the optimal modulation variance and excess noises ${\varepsilon }_1={\varepsilon }_2=0.002$ [16].

Figure 8

Secret-key rate for the CV-MDI QKD protocol in the asymmetric case with the finite-size effect. The CV-MDI QKD protocol with imperfect reconciliation efficiency $\beta =96.9\%$. The block lengths from left to right curves correspond to $N={10}^6,{10}^7,{10}^8,{10}^9,{10}^{10}$, and the asymptotic regime. Here we use the optimal modulation variance and excess noises ${\varepsilon }_1={\varepsilon }_2=0.002$ [16].

Figure 9

Secret-key rate for CV-MDI QKD protocol in asymmetric case with finite-size effect. The CV-MDI QKD protocol with perfect reconciliation efficiency $\beta =1$ and imperfect homodyne detectors $\eta =96\%$, ${\upsilon }_{el}=0.015$. The block lengths from left to right curves correspond to $N={10}^6,{10}^7,{10}^8,{10}^9,{10}^{10}$, and the asymptotic regime. Here we use the ideal modulation variance $V_A=V_B={10}^5$, and excess noises ${\varepsilon }_1={\varepsilon }_2=0.002$ [16].