## 4 Performance Comparison between Protocols
### Simplified Security Analysis
The use of this key rate amounts to the assumption that the sources are perfect and single-photon, while the other components (fibers and detectors) are imperfect. This is not entirely true, but we want to focus on the imperfections of the fiber link. Moreover, to compute (26), we assume that we can model the optical signal loss as a beam splitter (BS) on the channel [20], followed by perfect detectors (detector efficiency is included in the losses). One of the ports of the BS has a vacuum input; the input of the other port is the optical signal. One of the exit ports leaks photons to the environment, while the other is the channel'mode'. If we compute the density operator of this mode and trace out the environment, we can find the yield \(Y_{11}\) (which, in this case, is equal to the gain as the source is perfect, that is, \(p\)(single photon emission) \(=1\)) as the quantity \(1-p\) (no click), with \(p\) meaning the probability. If we include dark counts and take into account that we are computing coincidences, we have \(Y_{11}=[1-(1-Y_{0})(1-\eta)]^{2}\). As said, we considered different scenarios. The first one is the BBM92 protocol with full perturbations. In this case, we need to compute the expected values of Equations (6) and (7) with the PDFs above. We have that  
\[E_{opt}=\frac{\langle p_{err}^{Z}\rangle+\langle p_{err}^{X}\rangle}{2}. \tag{29}\]  
Note that for each value of the channel length \(L\) (with total achievable distance equal to \(2L\)), the variance of the PDFs is different. The results are plotted in Figure 5. Along this error rate, we plot the error rate due only to phase perturbations, that is, the quantity  
\[E_{opt}^{phase}=\frac{1}{2}\langle p_{err}^{X}\rangle, \tag{30}\]  
where, in this case, \(p_{err}^{X}\) is given by Equation (10). Next, we evaluate the secret key rate as a function of distance using Equation (25). For the autocompensating case, we formally have \(\gamma_{\delta}=\gamma_{\theta}=0\). For the non autocompensating case, we have already made estimations in the previous section, that is, \(\gamma_{\theta}\approx 5\times 10^{-5}\) km\({}^{-1}\), but \(\gamma_{\delta}\) has huge values experimentally. Nevertheless, in order to make the critical effect of optical fiber perturbations clear, although they will become quite small, we will assume the following hypothetical value: \(\gamma_{\delta}\approx 3\times 10^{-2}\) km\({}^{-1}\), that is, six orders of magnitude smaller than the real perturbation. Finally, for attenuation, we assume a value of \(0.15\) dB/km. For the detector efficiency, we assume \(\eta_{d}=60\%\). The dark count yield is \(\gamma_{0}=6.02\times 10^{-6}\). Finally, for the error inefficiency, given the values of our errors, we take a pessimistic [47]\(f=1.35\) value. In Figure 5, the variation in the optical error with the distance is shown, which is in turn used to obtain the results shown in Figure 6. We must emphasize that the curves of the secret key rate are critically influenced by the value of fiber attenuation. In fact, if this parameter becomes large enough, then protocols where the light needs to travel longer distances, for example, in BBM92, in order to achieve phase autocompensation exhibit a significant reduction in the secret key rate. On the other hand, without autocompensation, the maximum transmission distances to achieve a secure secret key are relatively short (a few tens of kilometers, as shown by the blue and orange curves in Figure 6), despite having used hypothetical, small values for the phase drift. Therefore, for real optical fibers, such  
Figure 5: Optical error comparison against channel length for the BBM92 protocol for \(\gamma_{\delta}=3\times 10^{-2}\), \(\gamma_{\theta}=5\times 10^{-5}\) (units of km\({}^{-1}\)). The red solid line represents the full error due to a phase drift and coupling perturbation (\(E_{opt}\)), while the dotted blue line represents the error due only to phase drift (\(E_{opt}^{phase}\)). distances would be reduced by up to a few meters.