## 4 Numerical Simulations
Moreover, comparing with two simulation results, we can find that although the randomness information mastered by [PERSON] of the Figure 2 is twice as much as the randomness information mastered by [PERSON] of the Figure 1, it does not decrease exponentially, which means that once [PERSON] obtains part of the information, it can seriously affect the practical system security. Figure 1: (Color online) The optimal key rate (bits per pulse) in logarithmic scale versus transmission distance between [PERSON] and [PERSON] when the weak randomness exists for only one party ([PERSON] or [PERSON]) \(p_{1}=10^{-x}\ (x=6,5,4,3),p_{2}=0\) (curves from right to left). The dashed lines are results of the asymptotic case, and the solid lines are the results of the finite-key size \(N=10^{15}\). The gray dotted line is the PLOB bound. Figure 2: (Color online)The optimal key rate (bits per pulse) in logarithmic scale versus transmission distance between [PERSON] and [PERSON] when the weak randomness exists for two parties \(p_{1}=p_{2}=10^{-x}(x=6,5,4,3)\)(curves from right to left). The dashed lines are results of asymptotic cases and the solid lines are the results of the finite-key size \(N=10^{15}\). The gray dotted line is the PLOB bound. In order to perform a detailed simulation, we then research the results of the weak randomness for the different total numbers of transmitting signals \(N=10^{x}\)\((x=12,13,15)\). Corresponding simulation results are illustrated in Figure 3, the dashed lines from left to right are acquired for \(N=10^{x}\)\((x=12,13,15)\) with the fixed \(p_{1}=p_{2}=10^{-6}\) and the solid lines from left to right are acquired for \(N=10^{x}\)\((x=12,13,15)\) with the fixed \(p_{1}=p_{2}=0\). We can notice that the generation of the security key rate will be significantly affected, even though the weak randomness parameter is small as \(10^{-6}\), which means that [PERSON] will obtain amounts of information even with small proportions of weak randomness. As shown in Figure 3, the achievable transmission distance declines \(104\) km when \(N=10^{15}\), \(60\) km when \(N=10^{13}\), and \(10\) km when \(N=10^{12}\), so we deduce that the greater the number of total pulses, the more the secure transmission distance decreases. We find that the secret key rate cannot exceed the PLOB bound [11] when \(N\leq 10^{13}\) with the fixed \(p_{1}=p_{2}=10^{-6}\). The number of total pulse increases, so does the number of quantum states which may be attenuated. [PERSON] may obtain more information due to the relation between the expected values and the observed values for the case with different modulated states in the practical QKD system. In this case, the number of modulated states distinguished by [PERSON] may increase which leads to more leakage of the security key information so we are supposed to control the number of total pulses within a rational range rather than arbitrarily choosing. To further study the impacts of the weak randomness for different total numbers of transmitting signals, we then research the secret key rate for \(N=10^{13}\), \(10^{15}\) with \(p_{1}=p_{2}=0\), \(10^{-x}\)\((x=6,5,4,3)\) in Figure 4. As illustrated in Figure 4, the dashed lines from right to left are acquired for different weak randomness parameters \(p_{1}=p_{2}=0\), \(10^{-x}\)\((x=6,5,4,3)\) with the fixed \(N=10^{13}\) and the solid lines from right to left are acquired for different weak randomness parameters \(p_{1}=p_{2}=0\), \(10^{-x}\)\((x=6,5,4,3)\) with the fixed \(N=10^{15}\). We can find that the impact of the weak randomness on final security key rate is greater than the finite total numbers of transmitting signals when \(p_{1}=p_{2}\geq 10^{-4}\) and the security key rate lines of two different \(N\) are approximately asymptotic. The impacts of the weak randomness on final security key rate is weaker than the finite total numbers of transmitting signals when \(p_{1}=p_{2}\leq 10^{-5}\) and the security key rate lines of two different \(N\) are not asymptotic.