# Free-Space QKD with Modulating Retroreflectors Based on the B92 Protocol
## Appendix A  
For ease of analysis, we transformed the round-trip MRR terminal in Figure 1 into a unidirectional transmission model, as shown in Figure A1. MQW3 was omitted because it does not affect the result. The interrogation beam incident on the MRR terminal is a coherent state with right-handed circular polarization. \[|\psi_{0}\rangle=|\alpha_{R}\rangle_{0} \tag{1}\]  
where \(\alpha\) is the amplitude of the coherent state, and the subscript 0 indicates the transmission path shown in Figure 11.  
Any pure coherent state can be expressed as the direct product state of two coherent states, whose polarizations are orthogonal to each other. \[|\psi_{0}\rangle=D(\alpha_{R})|0\rangle=\left|\frac{1}{\sqrt{2}}\alpha_{H} \right\rangle_{0}\left|\frac{1}{\sqrt{2}}e^{i\frac{\pi}{2}}\alpha_{V}\right\rangle _{0} \tag{2}\]  
The right-handed circular polarization coherent state can be expressed as the direct product of the horizontal and vertical polarization coherent states. Their intensity are equal, which is half of the interrogation beam, and the phase difference is \(\pi/2\). Because the PBS transmits only the horizontal \(|H\rangle\) and reflects the vertical \(|V\rangle\) polarizations, after passing through PBS1, the interrogation beam will evolve into  
\[|\psi_{1}\rangle=\left|\frac{1}{\sqrt{2}}\alpha_{H}\right\rangle_{1}\left| \frac{1}{\sqrt{2}}e^{i\frac{\pi}{2}}\alpha_{V}\right\rangle_{2} \tag{3}\]  
The phase shift generated by PBS has been ignored because it does not influence the conclusion. The beam transmitted along Paths 1 and 2 will pass through a free-space path, which is equivalent to a phase shifter. Then the joint state before incident \(IM_{1}\) and \(IM_{2}\) is:  
\[ \begin{split}|\psi_{2}\rangle&=\hat{P}_{1}\hat{P}_ {2}|\psi_{1}\rangle\\ &=e^{i\hat{p}_{1}\hat{n}_{1}}e^{i\hat{p}_{2}\hat{n}_{2}}\bigg{|} \frac{1}{\sqrt{2}}\alpha_{H}\bigg{\rangle}_{1}\left|\frac{1}{\sqrt{2}}e^{i\frac {\pi}{2}}\alpha_{V}\right\rangle_{2}\\ &=\left|\frac{1}{\sqrt{2}}e^{i\hat{p}_{1}}\alpha_{H}\right\rangle _{1}\left|\frac{1}{\sqrt{2}}e^{i\left(\frac{\pi}{2}+\hat{n}_{2}\right)}\alpha_ {V}\right\rangle_{2}\end{split}  \tag{4}\]  
where \(\hat{P}_{1}\) is the phase shifter operator acting on Path 1, \(\hat{n}_{1}\) is the photon number operator acting on Path 1. Afterward, these two beams are modulated by intensity modulator \(IM_{1}\) and \(IM_{2}\) respectively. The output state is as follows(for details, see Appendix B). \[|\psi_{3}\rangle=\left|\frac{1}{\sqrt{2}}\sqrt{M_{1}}e^{i\hat{p}_{1}}\alpha_{ H}\right\rangle_{1^{\prime}}\left|\frac{1}{\sqrt{2}}\sqrt{M_{2}}e^{i\left( \frac{\pi}{2}+\hat{n}_{2}\right)}\alpha_{V}\right\rangle_{2^{\prime}} \tag{5}\]  
Then the beam passes through the free-space path again, and we have  
\[|\psi_{4}\rangle=\left|\frac{1}{\sqrt{2}}\sqrt{M_{1}}e^{i2\hat{p}_{1}}\alpha_{ H}\right\rangle_{1^{\prime}}\left|\frac{1}{\sqrt{2}}\sqrt{M_{2}}e^{i\left( \frac{\pi}{2}+2\hat{n}_{2}\right)}\alpha_{V}\right\rangle_{2^{\prime}} \tag{6}\]  
Finally, these two beams are combined by the PBS  
\[|\psi_{out}\rangle=\left|\frac{1}{\sqrt{2}}\sqrt{M_{1}}e^{i2\hat{p}_{1}}\alpha _{H}\right\rangle_{\theta^{\prime}}\left|\frac{1}{\sqrt{2}}\sqrt{M_{2}}e^{i \left(\frac{\pi}{2}+2\hat{n}_{2}\right)}\alpha_{V}\right\rangle_{\theta^{ \prime}} \tag{7}\]If \(2\theta_{1}=\pi/2+2\theta_{2}\), \(|\psi_{out}\rangle\) is converted to  
\[|\psi_{out}\rangle=\bigg{|}\frac{1}{\sqrt{2}}\sqrt{M_{1}}\alpha_{H}\bigg{\rangle} \bigg{|}\frac{1}{\sqrt{2}}\sqrt{M_{2}}\alpha_{V}\bigg{\rangle} \tag{10}\]  
It is the direct product state of the horizontal and vertical polarization coherent states, which is linearly polarized. If \(\sqrt{M_{1}}=C\cos\theta\), and \(\sqrt{M_{2}}=C\sin\theta\), the state is  
\[|\psi_{out}\rangle=\big{|}C^{\prime}cos\theta\alpha_{H}\big{\rangle}\big{|}C^{ \prime}sin\theta\alpha_{V}\big{\rangle}=|cos\theta\beta_{H}\rangle|sin\theta \beta_{V}\rangle=|\beta_{\theta}\rangle \tag{11}\]  
where \(\beta=C^{\prime}\alpha\) is the amplitude and \(\theta\) is the polarization direction of the output beam.