Biased decoy-state measurement-device-independent quantum key distribution with finite resources

Measurement-device-independent quantum key distribution (MDI-QKD) can remove all the side-channel attacks from imperfections in the detection side. However, finite-size resources undoubtedly influence its performance and the achievable finite secret key rates of MDI-QKD are typically lower than that of standard decoy-state QKD. In this paper, we introduce the efficient decoy-state method with biased basis choice into the finite-key analysis and propose a decoy-state protocol for MDI-QKD. By applying vacuum + weak decoy-state method, we analytically derive concise formulas for estimating the lower bound of single-photon yield and the upper bound of phase error rate in the case of finite resources. The simulations show that proper basis choice combined with deliberate intensity choice can substantially enhance the performance of decoy-state MDI-QKD and, without a full optimization program, our protocol can bring a long-distance implementation (168 km on standard optical fiber) of MDI-QKD with a reasonable data size of total transmitting signals $\left(N={10}^{15}\right)$.

Figure 1

Key rate comparison with different probabilities of choosing $X$ basis. A realistic finite size of data $N$ is fixed to be ${10}^{13}$. For different $q_X=0.05,0.25,0.45,0.65,0.85$, the curves from left to right are all plotted by Eq. (1) averaged on the total number of signals $N$ sent by Alice and Bob. The single-photon contribution $n_1^Z$, single-photon errors ${\upsilon }_{11}^X$, and the phase error rate ${\phi }_Z$ are all analytically obtained by Eq. (14), Eq. (22), and Eq. (28), respectively. The intensities of signal states and decoy states are assigned specific values of 0.36 and 0.18, respectively.

Figure 2

Key rate comparison with different finite data sets. The dashed curve denotes the ideal secret key rate when infinite decoy states are used and the intensity is optimized for each distance. The dash-dotted curve denotes the asymptotic secret key rate of the three-intensity decoy-state method with the intensity of signal states and decoy states being 0.5 and 0.1, respectively. The solid curves from left to right are plotted by our protocol for finite number of total pulses from the weak coherent source $N={10}^j$ with $j=12,12.5,13,13.5,14,15$. The intensities of signal states and decoy states are fixed to be 0.36 and 0.18, respectively. But the probability of choosing $X$ basis $q_X$ is optimally chosen for each $N$.