## 2 Three-Intensity Decoy-State MDI-QKD  
In this paper, we adopt a typical decoy-state MDI-QKD with polarization encoding [36], which mainly consists of three steps. (i) [PERSON] generates phase-randomized pulses from WCPs and randomly selects the basis \(W\in\{Z,X\}\). That is, \(P_{Z}=P_{X}=1/2\), where \(P_{Z}\) and \(P_{X}\) are the probabilities of choosing the \(Z\) basis and \(X\) basis, respectively. Then [PERSON] uses an intensity modulator to modulate the pulses with three different intensities and sends them to [PERSON] located in the middle. This three intensities are the intensity of signal state \(\mu_{2}\), the intensity of decoy state \(\mu_{1}\), and the intensity of vacuum state \(\mu_{0}\), respectively. Furthermore, the corresponding percentages being emitted are \(P_{\mu_{2}}\), \(P_{\mu_{1}}\), and \(P_{\mu_{0}}\), respectively. Obviously, \(P_{\mu_{2}}+P_{\mu_{1}}+P_{\mu_{0}}=1\). At the same time, [PERSON] performs the same procedures as [PERSON], and the intensities of [PERSON]'s pulses are noted as \(\
u_{2}\), \(\
u_{1}\), and \(\
u_{0}\) for the signal state, decoy state, and vacuum state, respectively. Similarly, the corresponding percentages being emitted are \(P_{\
u_{12}}\), \(P_{\
u_{1}}\), and \(P_{\
u_{0}}\), respectively, where \(P_{\
u_{2}}+P_{\
u_{1}}+P_{\
u_{0}}=1\). (ii) The pulses from [PERSON] and [PERSON] interfere when they reach [PERSON]. Then [PERSON] performs a Bell state measurement (BSM) on the interference outcomes and announces the measurement results to [PERSON] and [PERSON]. (iii) [PERSON] and [PERSON] compare their bases, and determine the secure keys through [PERSON]'s measurement results. Specifically, if [PERSON] and [PERSON] choose the same basis and [PERSON] has a successful BSM event at the same time, then this part of the pulses can generate keys. It is important to emphasize that the secure keys are only generated from the signal state with \(Z\) basis, and the others are used for parameter estimation. The secure key rate of the decoy-state MDI-QKD [27; 36] is given by  
\[R\geq q\{P_{11}^{\mu_{2}\
u_{2}}Y_{11}^{Z}[1-H(e_{11}^{X})]-Q_{\mu_{2}\
u_{2}}^ {Z}f_{e}H(E_{\mu_{2}\
u_{2}}^{Z})\}. \tag{1}\]  
In the above equation, \(q=P_{Z}^{2}P_{\mu_{2}}P_{\
u_{2}}\) is the probability that [PERSON] and [PERSON] both select the \(Z\) basis and both modulate the pulse as signal state. \(P_{11}^{\mu_{2}\
u_{2}}=\mu_{2}\
u_{2}e^{-\mu_{2}-\
u_{2}}\) is the probability that the pulses from [PERSON]'s signal state and [PERSON]'s signal state are both single-photon pulses. \(Y_{11}^{Z}\) and \(e_{11}^{X}\) are the yield of single-photon state with \(Z\) basis and the quantum bit error rate (QBER) of single-photon state with \(X\) basis. \(H(x)=-x\log_{2}(x)-(1-x)\log_{2}(1-x)\) is the binary Shannon entropy function. \(Q_{\mu_{2}\
u_{2}}^{Z}\) and \(E_{\mu_{2}\
u_{2}}^{Z}\) are the overall gain and overall QBER of signal state with \(Z\) basis, respectively. \(f_{e}>1\) is the error correction efficiency. According to [37; 38], the overall gain \(Q_{\mu_{2}\
u_{1}}^{W}\left(W\in\{X,Z\}\right)\) and the overall QBER \(E_{\mu_{2}\
u_{1}}^{W}\left(W\in\{X,Z\}\right)\) can be obtained by the following equations,  
\[Q_{\mu_{2}\
u_{1}}^{X} =2y^{2}[1+2y^{2}-4 yI_{0}(x)+I_{0}(2x)], \tag{2}\] \[E_{\mu_{2}\
u_{1}}^{X}Q_{\mu_{2}\
u_{2}}^{X} =\epsilon_{0}Q_{\mu_{2}\
u_{1}}^{X}-2(\epsilon_{0}-e_{d})y^{2}[I_ {0}(2x)-1],\] \[Q_{\mu_{2}\
u_{2}}^{Z} =Q_{C}+Q_{E},\] \[E_{\mu_{2}\
u_{1}}^{Z}Q_{\mu_{2}\
u_{1}}^{Z} =\epsilon_{d}Q_{C}+(1-e_{d})Q_{E}.\]  
where  
\[Q_{C} =2(1-p_{d})^{2}e^{-\mu^{\prime}/2}[1-(1-p_{d})e^{-\eta_{d}\mu_{k }/2}]\times[1-(1-p_{d})e^{-\eta_{d}\
u_{1}/2}], \tag{3}\] \[Q_{E} =2p_{d}(1-p_{d})^{2}e^{-\mu^{\prime}/2}[I_{0}(2x)-(1-p_{d})e^{- \mu^{\prime}/2}].\]  
In the above equations, \(\mu_{k}\) and \(\
u_{l}\), \(k,l\in\{0,1,2\}\), are the intensities of pulses emitted by [PERSON] and [PERSON], respectively. \(I_{0}(x)\) is the modified Bessel function of the first kind. \(\epsilon_{0}\) is the error rate of background. \(e_{d}\) is the misalignment-error probability.