Measurement-device-independent quantum key distribution with a passive decoy-state method

Measurement-device-independent quantum key distribution (MDI-QKD) can remove all detector loopholes. When it is combined with the decoy-state method, the final key is unconditionally secure, even if Alice and Bob do not have strict single-photon sources. However, active modulation of source intensity, which is used to generate the decoy state, may leave side channels and leak additional information to Eve. In this paper, we consider the MDI-QKD with a passive decoy state, in which both Alice and Bob send pulses to an untrusted third party, Charlie. Then, in order to estimate the key generation rate, we derive two tight formulas to estimate the lower bound of the yield and the upper bound of the error rate that both Alice and Bob send a single-photon pulse to Charlie. Furthermore, the statistical fluctuation due to the finite length of data is also taken into account based on the standard statistical analysis.

Figure 1

Schematic diagram of the MDI-QKD protocol with a passive decoy state using time-bin phase encoding and the BB84 protocol. BS stands for a 50:50 beam splitter, and PM stands for a phase modulator. Alice and Bob each prepare phase-randomized weak coherent pulses to generate the joint-distribution states. Specifically, if detector $a_0$ or $b_0$ does not click, the pulse that Alice or Bob sends to Charlie is a signal state; otherwise, the pulse is a decoy state. In each run of the protocol, Alice and Bob each randomly choose one of four states, two phase states in the $X$ basis (0 and $\pi$) modulated by phase modulators (PM) and two time-bin states (short path $r$ and long path $s$) in the $Z$ basis. Then they send all of their pulses to Charlie. After Charlie performs a partial Bell-state measurement (BSM) on the received modes and announces his results, Alice and Bob can distill a secret key by performing error correction and private amplification.

Figure 2

Key generation rate vs the total transmission loss with the passive decoy-state method (solid line) compared to the active decoy-state method using two decoy states (dot-dashed line; cf. Ref. [36]). The key generation rate is maximized by optimizing the intensity of sources. The parameters used in our method are $e_d=1.5\%$, $P_d=3\times {10}^{-6}$, ${\eta }_d=0.12$, and $f=1.16$, which are the same as those in Refs. [36, 37].

Figure 3

(top) $Y_{11}^Z$ and (bottom) $e_{11}^X$ for the passive decoy-state method compared with those for an active one. The dot-dashed lines are obtained for the active decoy-state method with two decoy states and the solid lines are obtained for our passive decoy-state method. The parameters used are the same as those in Fig. 2. The fluctuation of $e_{11}^X$ is caused by the unsmoothness of the optimal intensity of sources.

Figure 4

Key generation rate of passive decoy-state MDI-QKD with statistical fluctuations. From left to right, the length of the data is ${10}^9$, ${10}^{10}$, ${10}^{11}$, ${10}^{12}$, and infinite (corresponding to the asymptotic key rate).