## 4 Numerical Simulations  
Here, we simulate the performance of effects of weak randomness on SNS TF-QKD in the asymptotic case and with the finite-key size. We use the linear model to numerically simulate the performance of the protocol. Firstly, we set the experimental parameters that we may exploit. Then, we set the results of the final secret key rate and the analysis of results. We define \(\eta=10^{-aL/10}\) as the fiber transmittance, where \(\alpha=0.2\) (dB/km) is the fiber loss coefficient and \(L\) is the length of fiber between [PERSON] and [PERSON] ([PERSON]). \(\eta_{d}=80\%\) is the detection efficiency of the relay [PERSON], and \(p_{d}=10^{-10}\) is the dark count of [PERSON]'s detectors. The failure probability of statistical fluctuations analysis is fixed to \(\varepsilon=10^{-10}\), and \(f=1.1\) is the efficiency of error correction. \(R=\ell/N\) is the final secret key rate, where \(N\) is the total number of transmitting signals sent by [PERSON] and [PERSON]. The numerical parameters are listed in Table 1. Here, we set \(\varepsilon_{cor}=\xi=\varepsilon_{PA}=\varepsilon\), \(\xi=3\varepsilon\), and \(\varepsilon_{n_{1}}=4\varepsilon\). Firstly, we analyze the results of weak randomness existing in only one party ([PERSON] or [PERSON]) in the asymptotic and the finite-key size cases in Figure 1. Here, \(p_{1}\
eq 0\), \(p_{2}=0\) means that [PERSON] just masters the randomness information of [PERSON], and \(p_{1}=0,10^{-x}(x=6,5,4,3)\) means that [PERSON] has different abilities for controlling the randomness information. We then analyze the results of weak randomness existing in both parties in the asymptotic case and the finite-key size cases in Figure 2, where [PERSON] masters the randomness information of both [PERSON] and [PERSON]. As illustrated in Figures 1 and 2, the dashed lines from right to left are acquired for different weak randomness parameters \(p_{1}=0,10^{-6},10^{-5},10^{-4},10^{-3}\) with the infinite number of total pulses, and the solid lines from right to left are acquired for different weak randomness parameters \(p_{1}=0,10^{-x}(x=6,5,4,3)\) with the fixed finite number of total pulses \(N=10^{15}\). In the Figure 1, compared with the perfect randomness \(p_{1}=p_{2}=0\), we can calculate that the achievable transmission distance declines \(11.96\%\), \(26.91\%\), \(43.52\%\), \(60.46\%\) in asymptotic cases and declines \(14.39\%\), \(30.93\%\), \(48.56\%\), \(66.19\%\) with the finite-key size when \(p_{1}=10^{-6},10^{-5},10^{-4},10^{-3}\). In the Figure 2, compared with the perfect randomness \(p_{1}=p_{2}=0\), the achievable transmission distance declines \(14.39\%\), \(30.93\%\), \(48.56\%\), \(66.19\%\) in asymptotic cases and declines \(15.95\%\), \(31.89\%\), \(48.50\%\), \(65.12\%\) with the finite-key size when \(p_{1}=0,10^{-6},10^{-5},10^{-4},10^{-3}\). Nevertheless, we find that the secret key rate can exceed the PLOB bound [11] when \(p_{1}(p_{2})\geq 10^{-6}\) with the finite-key size \(N=10^{15}\), and it can still exceed the PLOB bound [11] when \(p_{1}(p_{2})\geq 10^{-5}\) in the asymptotic case. From the above calculation data, we can deduce that the fluctuation \begin{table}
\begin{tabular}{c c c c c c} \hline \(\mathbf{\alpha}\) & \(\mathbf{f}\) & \(\mathbf{\eta_{d}}\) & \(\mathbf{e_{d}}\) & \(\mathbf{p_{d}}\) & \(\varepsilon\) \\ \hline
0.2 & 1.1 & 80\% & 0.15 & 1.0 \(\times\) 10\({}^{-10}\) & 1.0 \(\times\) 10\({}^{-10}\) \\ \hline \end{tabular}
\end{table} 
Table 1: List of experimental parameters applied in the numerical simulation in the following table: \(\alpha\) is the the fiber loss coefficient (dB/km), \(f\) is the efficiency of error correction, \(\eta_{d}\) is the efficiency of the detectors, \(e_{d}\) is the probability of the optical misalignment error, \(p_{d}\) is the dark count rate, and \(\varepsilon\) is the failure probability of statistical fluctuation analysis. of the finite-key size is greater than asymptotic cases for the fixed weak randomness parameters.