## 3 Lower Bound on the Secret Key Rate
With this configuration, it turns out that the optimal value of the parameter \(\mu t\) decreases with increasing distance, while the optimal value of the parameter \(\Omega\) increases with the distance. A similar behavior was also observed in the passive BB84 transmitter (without decoy states) proposed in [42]. In particular, \(\mu t\) diminishes from \(\approx 0.175\) to \(\approx 0.125\), while \(\Omega\) augments from \(\approx 0.393\) to \(\approx 0.7\). At long distances the gain of the protocol is very low and, therefore, it is important to keep both the  
Figure 3: Lower bound on the secret key rate \(R\) given by Equation (12) in logarithmic scale for the passive transmitter with two intensity settings illustrated in Figure 1 (green line). For simulation purposes, we consider the following experimental parameters: the dark count rate of [PERSON]’s detectors is \(\epsilon_{\rm B}=3.2\times 10^{-7}\), the overall transmittance of [PERSON]’s detection apparatus is \(\eta_{\rm B}=0.045\), the loss coefficient of the channel is \(\alpha=0.2\) dB/km, \(q=1/2\), and the efficiency of the error correction protocol is \(f(E^{i})=1.22\). We further assume the channel model described in [6, 59], where we neglect any misalignment effect. Otherwise, the actual secure distance will be smaller. The inset figure shows the value for the optimized parameters \(\mu t\) (dashed line) and \(\Omega\) (solid line) in the passive setup. The optimal value for the threshold parameter \(\Lambda\) turns out to be constant with the distance and equal to \(2\mu(1-t)\), _i.e._, the threshold angle \(\theta_{\Lambda}\) satisfies \(\theta_{\Lambda}=\pi/2\). The black line represents a lower bound on \(R\) for an active asymptotic decoy-state BB84 system with infinite decoy settings [6], while the red line shows the case of a passive transmitter with infinite intensity intervals \(\xi_{i}\) (see Appendix D). multi-photon probability of the source (related with the parameter \(\mu t\)) and the intrinsic error rate of the signals sent by Alice (related with the parameter \(\Omega\)) also low. Figure 3 includes as well an inset plot with the optimized parameters \(\mu t\) (dashed line) and \(\Omega\) (solid line). The optimal value for the parameter \(\Lambda\) turns out to be constant with the distance; it is given by \(\Lambda=2\mu(1-t)\), _i.e._, the threshold angle \(\theta_{\Lambda}\) is equal to \(\pi/2\). This figure also shows a lower bound on the secret key rate for the cases of an active decoy-state BB84 system with infinite decoy settings (black line) [6], and a passive transmitter with infinite intensity intervals \(\xi_{i}\) (red line). The cutoff points where the secret key rate drops down to zero are \(\approx 181\) km (passive setup with two intensity settings), \(\approx 183\) km (passive setup with infinite intensity settings), and \(\approx 192\) km (active transmitter with infinite decoy settings). From the results shown in Figure 3 we see that the performance of the passive transmitter presented in Section 2, with only two intensity settings, is similar to that of an active asymptotic setup, thus showing the practical interest of the passive scheme. The relatively small difference between the achievable secret key rates in both scenarios is due to two main factors: (a) the intrinsic error rate of the signals accepted by Alice, which is zero only in the case of an active source; and (b) the probability \(p_{acc}\) to accept a pulse emitted by the source, which is \(p_{acc}<1\) in the passive setup and \(p_{acc}=1\) in the active scheme. For instance, we have that for most distances \(\Omega\approx 0.393\), which implies \(p_{acc}\approx 0.5\). This fact reduces the key rate on logarithmic scale of the passive transmitter by a factor of \(\log_{10}p_{acc}\approx 0.3\). The additional factor of \(\approx 0.45\) that can be observed in Figure 3 arises mainly from the intrinsic error rate of the signals.