## 2 SNS Protocol with UB  
### Introduction of SNS Protocol  
Although the structure of the SNS protocol is similar to that of the TF protocol, it optimizes the signal preparation and post-processing steps. The steps of the SNS protocol are as follows [25]:  
1. At each time window \(i\), [PERSON] ([PERSON]) determines whether it is a signal or decoy window. If it is a decoy window, [PERSON] ([PERSON]) prepares a coherent state \(|\sqrt{\mu_{d}}e^{i\delta+i\gamma}\rangle\) and sends it to [PERSON]. If it is a signal window, [PERSON] ([PERSON]) prepares a coherent state \(\left|\sqrt{\mu_{s}}e^{i\delta+i\gamma}\right\rangle\) with probability \(\epsilon\) and sends it to [PERSON]. \(\mu_{d}\) and \(\mu_{s}\) represent the photon intensity of the decoy state and the signal state, \(\delta\in\{\delta_{A},\delta_{B}\}\) represents the random phase, and \(\gamma\in\{\gamma_{A},\gamma_{B}\}\) represents the phase offset of the channel. 2. [PERSON] receives the states sent by [PERSON] and [PERSON] and publishes all the measurement results for the effective events. Effective events are defined as follows: (1) When [PERSON] and [PERSON] simultaneously decide on the signal window, [PERSON] ([PERSON]) decides to send the signal, while [PERSON] ([PERSON]) decides not to send it, corresponding to [PERSON] announcing only detector \(D_{0(1)}\) clicks. (2) When [PERSON] and [PERSON] simultaneously decide on the decoy window, they prepare coherent states with the same intensity, and in this window, the random phases \(\delta_{A}\) and \(\delta_{B}\) satisfy [25]  
\[1-|\cos(\delta_{A}-\delta_{B})|\leqslant|\lambda|, \tag{1}\]  
corresponding to [PERSON] announcing only detector \(D_{1(0)}\) clicks. The value of \(\lambda\) is determined by the size of the phase slice chosen by [PERSON] and [PERSON], as described in Ref. [24]. 3. After [PERSON] publishes all the measurement results, [PERSON] and [PERSON] announce all the windows and the details of the decoy windows to classify the data accordingly and determine the parameters of the security key formula. 4. The secret key rate of the SNS protocol has been given as [25; 26] \[R=2\epsilon(1-\epsilon)P_{1}^{L}(\mu_{s})\mathrm{s}_{1}^{L}[1-H(\mathrm{e}_{1 }^{U})]-fS_{Z}H(E_{Z}).\] (2)  
\(\epsilon\) represents the probability that [PERSON] ([PERSON]) sends the signal state to [PERSON] during the signal window. \(P_{1}^{L}(\mu_{s})\) refers to the lower bound of the probability of sending a signal that contains a single photon. We define \(Z\)-data [26] as events in which [PERSON] and [PERSON] simultaneously select the signal window. \(Z_{1}\)-data refers to effective events where one party chooses to send the signal while the other does not. \(s_{1}^{L}\) and \(\mathrm{e}_{1}^{U}\) represent the lower bound of the count rate and the upper bound of the phase error rate of single-photon events in the \(Z_{1}\)-data. \(S_{Z}\) and \(E_{Z}\) represent the count rate and bit error rate of the signal, \(f\) is the error correction efficiency, and \(H(x)=-xlog_{2}x-(1-x)log_{2}(1-x)\) refers to the binary Shannon entropy function. Under asymptotic conditions, Equation (2) remains valid and can accurately estimate the parameters \(s_{1}^{L}\) and \(\mathrm{e}_{1}^{U}\)[25]. \[s_{1}\geq s_{1}^{L}=\frac{p_{2}^{L}(\mu_{2})\big{[}S_{\mu_{1}}-p_{0}^{U}(\mu_{ 1})S_{\mu_{0}}\big{]}-p_{2}^{U}(\mu_{1})\big{[}S_{\mu_{2}}-p_{0}^{L}(\mu_{2})S_ {\mu_{0}}\big{]}}{p_{2}^{U}(\mu_{2})p_{1}^{U}(\mu_{1})-p_{2}^{L}(\mu_{1})p_{1}^ {L}(\mu_{2})}, \tag{3}\]  
\[e_{1}\leq e_{1}^{U}=\frac{S_{\mu_{1}}E_{\mu_{1}}-p_{0}^{L}(\mu_{1})S_{\mu_{0}} E_{\mu_{0}}}{p_{1}^{L}(\mu_{1})s_{1}^{L}}. \tag{4}\]  
\(\mu_{0}=0<\mu_{1}<\mu_{2}\) represents the three intensities of the decoy windows.