## 2 Passive Decoy-State BB84 Transmitter
The threshold angle \(\theta_{\Lambda}\) that satisfies \((1-t)\zeta(\theta_{\Lambda})=\Lambda\) is given by  
\[|\sqrt{\mu}(e^{i\theta_{1}}+e^{i\theta_{2}})\rangle_{a_{1},+45^{\circ}}\otimes |\sqrt{\mu}(e^{i\theta_{1}}-e^{i\theta_{2}})\rangle_{b_{1},+45^{\circ}}. \tag{1}\]0  
In this simplified scenario, the conditional quantum states that are sent to [PERSON] can be written as  
\[|\sqrt{\mu}(e^{i\theta_{1}}+e^{i\theta_{2}})\rangle_{a_{1},+45^{\circ}}\otimes |\sqrt{\mu}(e^{i\theta_{1}}-e^{i\theta_{2}})\rangle_{b_{1},+45^{\circ}}. \tag{1}\]1  
where \(i=\{s,d\}\), and the probabilities \(p_{n}^{i}\) are given by  
\[|\sqrt{\mu}(e^{i\theta_{1}}+e^{i\theta_{2}})\rangle_{a_{1},+45^{\circ}}\otimes |\sqrt{\mu}(e^{i\theta_{1}}-e^{i\theta_{2}})\rangle_{b_{1},+45^{\circ}}. \tag{1}\]2 \[|\sqrt{\mu}(e^{i\theta_{1}}+e^{i\theta_{2}})\rangle_{a_{1},+45^{\circ}}\otimes |\sqrt{\mu}(e^{i\theta_{1}}-e^{i\theta_{2}})\rangle_{b_{1},+45^{\circ}}. \tag{1}\]3  
In practice, however, it is not necessary that [PERSON] determines the value of \(\psi\) accurately and restricts herself to only those events where she actually prepares a perfect BB84 polarization state (_i.e._, when the angle \(\psi\) satisfies \(\psi\in\{0,\pi/2,\pi,3\pi/2\}\)) [42]. Note that the probability associated with these ideal events tends to zero. Instead, it is sufficient if the polarization measurement tells her the value of \(\psi\) within a certain interval around the desired ideal values. This situation is illustrated in Figure 2 (case B), where [PERSON] selects some valid regions (marked with gray color in the figure) for the angle \(\psi\)[42]. These regions depend on an acceptance parameter \(\Omega\in[0,\pi/4]\) that we optimize. In particular, whenever the value of \(\psi\) lies within any of the valid regions, [PERSON] considers the pulse emitted by the source as a valid signal. Otherwise, the pulse is discarded afterwards during the post-processing phase of the protocol, and it does not contribute to the key rate. The probability that a pulse is accepted, \(p_{\mathrm{acc}}\), is given by  
\[|\sqrt{\mu}(e^{i\theta_{1}}+e^{i\theta_{2}})\rangle_{a_{1},+45^{\circ}}\otimes |\sqrt{\mu}(e^{i\theta_{1}}-e^{i\theta_{2}})\rangle_{b_{1},+45^{\circ}}. \tag{1}\]4  
\[|\sqrt{\mu}(e^{i\theta_{1}}+e^{i\theta_{2}})\rangle_{a_{1},+45^{\circ}}\otimes |\sqrt{\mu}(e^{i\theta_{1}}-e^{i\theta_{2}})\rangle_{b_{1},+45^{\circ}}. \tag{1}\]5 (Case A) Graphical representation of the intensity \((1-t)\zeta(\theta)\) in mode \(d_{3}\) (see Figure 1) versus the angle \(\theta\). \(\Lambda\) represents the threshold value of the classical intensity measurement, \(\theta_{\Lambda}\) is its associated threshold angle, and \(\xi_{d}\) and \(\xi_{s}\) denote the resulting intensity intervals. (Case B) Graphical representation of the valid regions for the angle \(\psi\). These regions are marked in gray. They depend on an acceptance parameter \(\Omega\in[0,\pi/4]\). There is a trade-off on the acceptance parameter \(\Omega\). A high acceptance probability \(p_{\rm acc}\) favors \(\Omega\approx 0\), but this action also results in an increase of the quantum bit error rate (QBER) of the protocol. A low QBER favors \(\Omega\approx\pi/4\), but then \(p_{\rm acc}\approx 0\). Note that in the limit where \(\Omega\) tends to \(\pi/4\) we recover the standard decoy-state BB84 protocol.