## Appendix B Gain and QBER  
In this Appendix, we obtain a mathematical expression for the observed gains \(Q^{i}\) and error rates \(E^{i}\), with \(i\in\{s,d\}\), for the passive QKD transmitter with two intensity settings introduced in Section 2. For that, we employ the typical channel model in the absence of eavesdropping [6, 59]; it just consists of a BS of transmittance \(\eta_{\mathrm{channel}}=10^{-\frac{\mathrm{qd}}{10}}\), where \(\alpha\) denotes the loss coefficient of the channel measured in dB/km and \(d\) is the transmission distance. Moreover, for simplicity, we consider that [PERSON] employs an active BB84 detection setup with two threshold detectors. The action of [PERSON]'s measurement device can be described by two positive operator value measures (POVMs), one for each of the two BB84 polarization bases \(\beta\in\{l,c\}\), with \(l\) denoting a linearpolarization basis and \(c\) a circular polarization basis. Each POVM contains four elements: \(G_{\rm vac}^{\beta}\), \(G_{0}^{\beta}\), \(G_{1}^{\beta}\), and \(G_{\rm dc}^{\beta}\). The first one corresponds to the case of no click in the detectors, the following two POVM operators give precisely one detection click, and the last one, \(G_{\rm dc}^{\beta}\), gives rise to both detectors being triggered. These operators can be written as [42]  
\[G_{\rm vac}^{\beta} = [1-\epsilon_{\rm B}(2-\epsilon_{\rm B})]F_{\rm vac}^{\beta},\] \[G_{0}^{\beta} = (1-\epsilon_{\rm B})\epsilon_{\rm B}F_{\rm vac}^{\beta}+(1- \epsilon_{\rm B})F_{0}^{\beta},\] \[G_{1}^{\beta} = (1-\epsilon_{\rm B})\epsilon_{\rm B}F_{\rm vac}^{\beta}+(1- \epsilon_{\rm B})F_{1}^{\beta},\] \[G_{\rm dc}^{\beta} = I-G_{\rm vac}^{\beta}-G_{0}^{\beta}-G_{1}^{\beta}. \tag{21}\]  
Here we assume that the background rate is, to a good approximation, independent of the signal detection. Moreover, for easiness of notation, we consider only a background contribution coming from the dark count rate \(\epsilon_{\rm B}\) of [PERSON]'s detectors and we neglect other background contributions like, for instance, stray light arising from timing pulses which are not completely filtered out in reception. The operators \(F_{\rm vac}^{\beta}\), \(F_{0}^{\beta}\), \(F_{1}^{\beta}\), and \(F_{\rm dc}^{\beta}\) have the form  
\[F_{\rm vac}^{\beta} = \sum_{n,m=0}^{\infty}(1-\eta_{\rm sys})^{n+m}|n,m\rangle_{\beta} \langle n,m|,\] \[F_{0}^{\beta} = \sum_{n,m=0}^{\infty}[1-(1-\eta_{\rm sys})^{n}](1-\eta_{\rm sys}) ^{m}|n,m\rangle_{\beta}\langle n,m|,\] \[F_{1}^{\beta} = \sum_{n,m=0}^{\infty}[1-(1-\eta_{\rm sys})^{m}](1-\eta_{\rm sys} )^{n}|n,m\rangle_{\beta}\langle n,m|,\] \[F_{\rm dc}^{\beta} = \sum_{n,m=0}^{\infty}[1-(1-\eta_{\rm sys})^{n}][1-(1-\eta_{\rm sys })^{m}]\times|n,m\rangle_{\beta}\langle n,m|, \tag{22}\]  
with \(\beta\in\{l,c\}\). The signals \(|n,m\rangle_{l}\left(|n,m\rangle_{c}\right)\) represent the state which has \(n\) photons in the horizontal (circular left) polarization mode and \(m\) photons in the vertical (circular right) polarization mode. The parameter \(\eta_{\rm sys}\) denotes the overall transmittance of the system. This quantity can be written as \(\eta_{\rm sys}=\eta_{\rm B}\eta_{\rm channel}\), where \(\eta_{\rm B}\) is the overall transmittance of [PERSON]'s detection apparatus, _i.e._, it includes the transmittance of any optical component within [PERSON]'s measurement device together with the efficiency of his detectors. In the scenario considered, it turns out that the gains \(Q^{i}\) are independent of the actual polarization of the signals \(\sigma_{i,\psi}\) given by Equation (9) and the basis \(\beta\) used to measure them. We obtain  
\[Q^{i}=1-{\rm Tr}\big{(}G_{\rm vac}^{\beta}\sigma_{i,\psi}\big{)}=1-\frac{(1- \epsilon_{\rm B})^{2}}{p_{i}\pi}\int_{\theta_{\xi_{i}}}e^{-\eta_{\rm sys}\gamma (\theta)}\;{\rm d}\theta, \tag{23}\]  
where \(p_{s}=\theta_{\Lambda}/\pi\), \(p_{d}=1-p_{s}\), \(\theta_{\xi_{s}}=[0,\theta_{\Lambda}]\), and \(\theta_{\xi_{d}}=[\theta_{\Lambda},\pi]\).