## 4 Performance Comparison between Protocols
### Simplified Security Analysis  
We now analyze the impact of the perturbation-motivated errors on the key rate. As this paper is focused on optical hardware for QKDs, it will be enough to present an elemental security analysis. Recall that in the first place, we assumed [PERSON] and [PERSON] exchange an infinite number of single-photon signals. We shall follow [17] and, more specifically, the simple case contained in it. We also assume a symmetric channel, where detector efficiencies, attenuation, and dark counts are the same for [PERSON] and [PERSON]. We use a basic version of the key rate, emphasizing the errors due to perturbations and assuming that [PERSON] does not have any information about the basis used for measurement, so the bit error rate and the phase error rates are equal [17; 45]. With these conditions, the secret key rate is given by [45]  
\[R\geq qY_{11}[1-fH(e_{11})-H(e_{11})]. \tag{25}\]where the meanings of the parameters contained in the equation above are the following: \(q\) is the base-sift factor (\(q=1/2\) for BBM92, \(q=1\) for BSEP); \(Y_{1}\) is the single-photon yield, given by  
\[Y_{11}=[1-(1-Y_{0})(1-\eta)]^{2} \tag{26}\]  
where \(Y_{0}\) is the dark-count-associated yield and \(\eta\) is the product of the channel transmissivity, depending on channel length \(L\), attenuation \(\alpha_{att}\) (given by \(10^{-\alpha_{att}L/10}\)), and the detector efficiency \(\eta_{d}\). Therefore, \(\eta=\eta_{d}10^{-\alpha_{att}L/10}\). Moreover, we consider that the attenuation along the Alice-Charlie and Bob-Charlie paths (each of length \(L\)) is the same. The value of \(\eta\) roughly represents the probability of detecting a photon at the end of a lossy fiber at a detector of efficiency \(\eta_{d}\). Furthermore, we assume that every detector has the same dark count probability. \(e_{11}\) is the single-photon error, which can be estimated from the measured error rates on the \(Z\) and \(X\) basis [45] as \(e_{11}=(e_{x}+e_{z})/2\). This error has three contributions: one due to the perturbations, which we indicate as \(E_{opt}\) (optical error); another due to the dark counts; and a third one, a residual error term \(e_{res}\), that accounts for additional sources of errors not included in the perturbations. When we compensate for the perturbations, then \(E_{opt}=0\). Regarding the physical causes of the residual error, they may be due to misalignments in the measuring apparatus. This particular error we cannot eliminate even when we autocompensate the protocol, but we expect it not to be too big if the optical hardware is reliable enough. We use a typical value of \(e_{res}=1.5\%\)[46]. We emphasize that the addition of such a residual term, if small, does not affect our conclusions regarding the benefits of autocompensation, as it is common in every protocol. With all of this in place, the single-photon error becomes  
\[e_{11}=e_{0}\bigg{(}1-\frac{\eta^{2}}{Y_{11}}\bigg{)}+\frac{(E_{opt}+e_{res}) \eta^{2}}{Y_{11}}. \tag{27}\]  
Finally, \(H\) is the well-known binary Shannon entropy, given by the following formal expression  
\[H(x)=-x\log_{2}x-(1-x)\log_{2}(1-x). \tag{28}\]  
We must recall that the error rate is the fraction of erroneous events (detector misfires) in the total events; therefore, the interpretation of Equation (27) is as follows. We have two separate contributions: On the one hand, the optical error contribution (plus the residual term) occurs with fraction \(\eta^{2}/Y_{11}\), that is, the probability \(\eta^{2}\) of coincidences of two photons over the probability that two photons arrive at the detectors, given that [PERSON] produces a two-photon entangled state. On the other hand, dark counts contribute when no photon arrives at the detectors but clicks happen nonetheless; this rate is reduced whenever dark counts and true coincidences happen at the same time (this being a case that happens with so negligible a probability that it is usually disregarded [17]).