## 2 Protocol Description  
In the practical QKD system, we usually choose the weak coherent state source instead of the single photon source. Here, we consider the four-intensity decoy-state SNS protocol [16], and the description of the protocol is presented as follows:  
1. Preparation. At any times window \(i\), [PERSON] ([PERSON]) independently determines whether it is a decoy window or a signal window with probabilities \(p_{x}\) and \(p_{z}\). If it is a decoy window, [PERSON] ([PERSON]) sends out to [PERSON] a decoy pulse in a phase-randomized coherent state \(\left|\sqrt{\mu_{e}}e^{i\delta_{A}}\right\rangle\), \(\left|\sqrt{\mu_{b}}e^{i\delta^{\prime}_{A}}\right\rangle\) or \(\left|0\right\rangle\) (\(\left|\sqrt{\mu_{e}}e^{i\delta_{B}}\right\rangle\), \(\left|\sqrt{\mu_{b}}e^{i\delta^{\prime}_{B}}\right\rangle\) or \(\left|0\right\rangle\)) with probabilities of \(p_{u_{x}}\), \(p_{u_{y}}\), \(p_{0}\). We suppose \(\mu_{a}<\mu_{b}\). If it is a signal window, [PERSON] ([PERSON]) decides to send out to [PERSON] a signal pulse in phase-randomized coherent states \(\left|\sqrt{\mu_{e}}e^{i\delta_{A}}\right\rangle\) or a vacuum state \(\left|0\right\rangle\) (\(\left|\sqrt{\mu_{e}}e^{i\delta_{B}}\right\rangle\) or \(\left|0\right\rangle\)) with probabilities of \(p_{z0}\) and \(1-p_{z0}\), where \(\delta_{A(B)}\) is random in \([0,2\pi)\). Here, we assume that consecutive photons are well-separated in the decoy and the signal time windows. Note that a coherent state of intensity \(\mu\) and global phase \(\delta\) is a linear superposition of photon-numberstates \(\big{|}\sqrt{\mu}e^{i\delta}\big{\rangle}=\sum\limits_{k=0}^{\infty}\frac{e^{- \mu/2}(\sqrt{\mu}e^{i\delta})^{k}}{\sqrt{k!}}|k\rangle\). Whenever [PERSON] or [PERSON] sends a coherent state of intensity \(\mu\), it can be equivalently regarded as a probabilistic mixture of different photon-number states \(\int_{0}^{2\pi}\big{|}\sqrt{\mu}e^{i\delta}\big{\rangle}\big{\langle}\sqrt{\mu }e^{i\delta}\big{|}d\delta/2\pi=\sum\limits_{k=0}^{\infty}\frac{e^{-\mu}\mu^{ k}}{k! }|k\rangle\langle k|\). 2. Measurement. [PERSON] and [PERSON] send the chosen states to [PERSON]. [PERSON] then performs interferometric measurements on the incoming quantum signals after taking phase compensation and announces the measurement results of which detector clicks to [PERSON] and [PERSON]. An effective event is defined as follows: (1) if only one detector clicks corresponding to a time window \(i\) when both [PERSON] and [PERSON] have determined the signal window, it is defined as an effective event. (2) If only one detector clicks corresponding to a time window \(i\) when both [PERSON] and [PERSON] have determined a decoy window and sent the coherent states with the same intensity, and in that time window, the pre-chosen values \(\delta_{A}\) and \(\delta_{B}\) satisfy post-selection criterion, which is: \[1-|\cos(\delta_{A}-\delta_{B}-\psi_{AB})|\leq|\lambda|,\] (1) where \(\delta_{A}\) and \(\delta_{B}\) are the random phases of coherent states prepared by [PERSON] and [PERSON], respectively. \(\psi_{AB}\) could take an arbitrary value and it is set properly to acquire a satisfactory key rate, which will be different from time to time due to phase drift. The value of \(\lambda\) is determined by the size of the phase slice \(\Delta\), which is chosen by [PERSON] and [PERSON]. In fact, Equation (1) is equivalent to: \[|\theta_{A}-\theta_{B}-\psi_{AB}|\leq\frac{\Delta}{2},|\theta_{A}-\theta_{B}- \psi_{AB}-\pi|\leq\frac{\Delta}{2}.\] (2) where \(|x|\) represents the minor angle enclosed by two rays, which enclose the rotational angle. 3. Sifting. [PERSON] and [PERSON] announce decoy windows and signal windows of each other. If both [PERSON] and [PERSON] choose the decoy window, it is defined to be an \(\tilde{X}\) window. If both [PERSON] and [PERSON] choose the signal window, it is defined to be a \(Z\) window. In an \(\tilde{X}\) window, it is an \(X_{1}\) window, which is a subset of \(\tilde{X}\) windows, when they choose the same intensity \(\mu_{a(b)}\).