## 2 Passive Decoy-State BB84 Transmitter  
The basic setup is illustrated in Figure 1.  
Let us start considering, for simplicity, the interference of two pure coherent states of frequency \(w_{1}\), both prepared in \(+45^{\circ}\) linear polarization and with arbitrary phase relationship, \(|\sqrt{2\mu}e^{i\theta_{1}}\rangle_{a_{0},+45^{\circ}}\) and \(|\sqrt{2\mu}e^{i\theta_{2}}\rangle_{b_{0},+45^{\circ}}\), at a \(50:50\) BS. The output states in modes \(a_{1}\) and \(b_{1}\) (see Figure 1) are given by  
\[|\sqrt{\mu}(e^{i\theta_{1}}+e^{i\theta_{2}})\rangle_{a_{1},+45^{\circ}}\otimes |\sqrt{\mu}(e^{i\theta_{1}}-e^{i\theta_{2}})\rangle_{b_{1},+45^{\circ}}. \tag{1}\]  
Then, we have that the output states in modes \(c_{1}\) and \(d_{1}\) have the form  
\[|\sqrt{\tfrac{\mu}{2}}(e^{i\theta_{1}}+e^{i\theta_{2}})\rangle_{c_{1},+45^{ \circ}}\otimes|\sqrt{\tfrac{\mu}{2}}(e^{i\theta_{1}}+e^{i\theta_{2}})\rangle_ {d_{1},+45^{\circ}}. \tag{2}\]If these two states are combined with two coherent states of frequency \(w_{2}\), \(|\sqrt{\mu}e^{i\theta_{3}}\rangle_{a_{2},+45^{\circ}}\) and \(|\sqrt{\mu}e^{i\theta_{4}}\rangle_{b_{2},+45^{\circ}}\), in a nonlinear medium using the SFG process, the resulting output states at frequency \(w_{3}=w_{1}+w_{2}\), after the polarization rotation \(R\), can be written as (see Appendix A)  
\[|\tfrac{-\sqrt{\mu}e^{i\theta_{3}}(e^{i\theta_{1}}+e^{i\theta_{2}})}{\sqrt{2} }\rangle_{c_{2},+45^{\circ}}\otimes|\tfrac{-\sqrt{\mu}e^{i\theta_{4}}(e^{i \theta_{1}}+e^{i\theta_{2}})}{\sqrt{2}}\rangle_{d_{2},-45^{\circ}}. \tag{3}\]  
These two beams are now re-combined at a PBS in the \(\pm 45^{\circ}\) linear polarization basis [54]. We obtain that the output state in mode \(a_{3}\) (see Figure 1) is a coherent state of the form  
\[|\sqrt{\zeta(\theta)}e^{i\phi}\rangle_{\psi,a_{3}}=e^{-\zeta(\theta)/2}\sum_{ n=0}^{\infty}\frac{\left(\sqrt{\zeta(\theta)}e^{i\phi}\right)^{n}}{\sqrt{n!}}|n_{ \psi}\rangle, \tag{4}\]  
where \(\zeta(\theta)=2\mu(1+\cos\theta)\), \(\theta=\theta_{2}-\theta_{1}\), \(\phi=\pi+\theta_{1}+\theta_{3}+\arg\left(1+e^{i\theta}\right)\), and the Fock states \(|n_{\psi}\rangle\) are given by  
\[|n_{\psi}\rangle=\frac{[\tfrac{1}{\sqrt{2}}(a_{+45^{\circ}}^{\dagger}+e^{i \psi}a_{-45^{\circ}}^{\dagger})]^{n}}{\sqrt{n!}}|\mathrm{vac}\rangle, \tag{5}\]  
with \(|\mathrm{vac}\rangle\) denoting the vacuum state and \(\psi=\theta_{4}-\theta_{3}\). Finally, [PERSON] sends the quantum state given by Equation (4) through a BS of transmittance \(t\ll 1\). Then, the output states in modes \(c_{3}\) and \(d_{3}\) are given by  
\[|\sqrt{t\zeta(\theta)}e^{i\phi}\rangle_{\psi,c_{3}}\otimes|\sqrt{(1-t)\zeta( \theta)}e^{i\phi}\rangle_{\psi,d_{3}}. \tag{6}\]  
The analysis of the case where the global phase of each input signal \(\rho_{i}\), with \(i\in\{1,\ldots,4\}\), is randomized and inaccessible to the eavesdropper is now straightforward. It can be solved by just  
Figure 1: Basic setup of a passive decoy-state BB84 QKD source with polarization encoding using phase-randomized strong coherent pulses. The mean photon number of the signal states \(\rho_{i}\), with \(i\in\{1,\ldots,4\}\), can be chosen very high; for instance, \(\approx 10^{8}\) photons. BS denotes a beamsplitter, PBS represents a polarizing beamsplitter in the \(\pm 45^{\circ}\) linear polarization basis, such PBS transmits \(-45^{\circ}\) linear polarization and reflects \(+45^{\circ}\) linear polarization [54]. These two orthogonal linear polarizations have creation operators \(a_{\pm 45^{\circ}}^{\dagger}=1/\sqrt{2}(a_{H}^{\dagger}\pm a_{V}^{\dagger})\). F is an optical filter, R denotes a polarization rotator changing \(+45^{\circ}\) linear polarization to \(-45^{\circ}\) linear polarization, \(|\mathrm{vac}\rangle\) represents the vacuum state, and \(t\) denotes the transmittance of a BS; it satisfies \(t\ll 1\). integrating the signals \(|\sqrt{t\zeta(\theta)}e^{i\phi}\rangle_{\psi,c_{3}}\) and \(|\sqrt{(1-t)\zeta(\theta)}e^{i\phi}\rangle_{\psi,d_{3}}\) given by Equation (6) over all angles \(\theta,\phi\), and \(\psi\).