## 2 Passive Decoy-State BB84 Transmitter
In particular, we have that the output state \(\sigma\) in this scenario (see Figure 1) can be written as  
\[\sigma = \frac{1}{(2\pi)^{3}}\iiint_{\phi,\theta,\psi}|\sqrt{t\zeta(\theta )}e^{i\phi}\rangle_{\psi,c_{3}}\langle\sqrt{t\zeta(\theta)}e^{i\phi}|\;\mathrm{ d}\phi\mathrm{d}\theta\mathrm{d}\psi \tag{7}\] \[= \frac{1}{(2\pi)^{2}}\int_{\theta}e^{-\gamma(\theta)}\sum_{n=0}^{ \infty}\frac{\gamma(\theta)^{n}}{n!}\int_{\psi}|n_{\psi}\rangle\langle n_{ \psi}|\;\mathrm{d}\theta\mathrm{d}\psi,\]  
where the intensity \(\gamma(\theta)\) is given by \(\gamma(\theta)=t\zeta(\theta)\). As already mentioned previously, note that the generation of phase-randomized coherent states can be achieved, for example, by using strong current modulation of the laser diode, well above and below threshold, which ensures a true random phase (unknown to the eavesdropper) for each prepared pulse [21,22]. The weak intensity signal \(\sigma\) in mode \(c_{3}\) is suitable for QKD and [PERSON] sends it to [PERSON] through the quantum channel. In addition, she uses the strong intensity signal available in mode \(d_{3}\) to measure both its intensity and polarization. This last measurement can be realized, for example, by means of a passive BB84 detection scheme where the basis choice is performed by a \(50:50\) BS, and on each end there is a PBS and two classical photodetectors. From the different intensities observed in each of these four photodetectors, [PERSON] can determine both the value of the angle \(\psi\) and the total intensity of the signal. Note that, by assumption, we have that the intensity of the input states \(\rho_{i}\) is very high. Intuitively speaking, the working principle of the setup illustrated in Figure 1 is quite simple. In a first step, the setup passively generates in mode \(a_{1}\) coherent states of random amplitude. This is done by combining two coherent states or equal amplitude but random phase at a BS. This part follows the approach proposed in [37,38] to passively prepare decoy states. Then, in a second step, the setup generates signals whose photons are randomly polarized within the X-Z plane of the Bloch sphere. For this, it first prepares two phase-randomized coherent states of the same amplitude in \(+45^{\circ}\) and \(-45^{\circ}\) linear polarization, and then interferes them at a PBS. This approach is inspired by the solution introduced in [42]. Indeed, this is the role of the interferometric part of the scheme, where the SFG process is actually used to imprint a random phase to each state. Finally, in a third step, the setup attenuates the generated signals to the single-photon level before they are sent to [PERSON], and it also determines both the intensity and polarization of the signals prepared. For simplicity, let us assume for the moment that the polarization measurement is perfect, _i.e._, for each incoming signal it provides [PERSON] with a precise value for the measured angle \(\psi\), while the intensity measurement only tells her whether the measured intensity is below or above a certain threshold value \(\Lambda\) that satisfies \(0<\Lambda<4\mu(1-t)\). That is, \(\Lambda\) is between the minimal and maximal possible values of the intensity \((1-t)\zeta(\theta)\) of the optical pulses in mode \(d_{3}\). The first intensity interval, \(\xi_{d}=[0,\Lambda]\), can be associated, for instance, to the generation of a decoy state in output mode \(c_{3}\) (that we shall denote as \(\sigma_{d}\)), while the second intensity interval, \(\xi_{s}=[\Lambda,4\mu(1-t)]\), corresponds to the case of preparing a signal state (\(\sigma_{s}\)). Note, however, that the analysis presented in this section can be straightforwardly adapted to cover as well the case of several intensity intervals \(\xi_{i}\) (_i.e._, the generation of several decoy states). Figure 2 (case A) shows a graphical representation of the intensity \((1-t)\zeta(\theta)\) in mode \(d_{3}\) versus the angle \(\theta\), together with the threshold value \(\Lambda\) and the intensity intervals \(\xi_{d}\) and \(\xi_{s}\).