## 2 The Q-M Interferometer Scheme and Theoretical Analysis
\begin{array}{ll}U(\gamma,s_{1},s_{2},s_{3})&=\sigma_{0}\cos\gamma+i(s_{1} \sigma_{1}+s_{2}\sigma_{2}+s_{3}\sigma_{3})\sin\gamma\\ &=\left[\begin{array}{cc}\cos\gamma+is_{1}\sin\gamma&is_{2}\sin\gamma-s_{3} \sin\gamma\\ is_{2}\sin\gamma+s_{3}\sin\gamma&\cos\gamma-is_{1}\sin\gamma\end{array}\right] \\ \sigma_{0}=\left[\begin{array}{cc}1&0\\ 0&1\end{array}\right]\sigma_{1}=\left[\begin{array}{cc}1&0\\ 0&-1\end{array}\right]\\ \sigma_{2}=\left[\begin{array}{cc}0&1\\ 1&0\end{array}\right]\sigma_{3}=\left[\begin{array}{cc}0&i\\ -i&0\end{array}\right]\end{array}  \tag{1}\]  
where the angle \(\gamma/2\) corresponds to the birefringence strength of the transmission medium, and Stokes parameters, \(s_{1},s_{2},s_{3}\), originate from the \(X\)-\(Y\) and \(+45^{\circ}\) and \(-45^{\circ}\) components of rectangular birefringence, and from the circular birefringence, respectively. Then, for the Q-M interferometer, the operator  
Figure 1: Unbalanced-arm Michelson interferometer with two quarter-wave plate reflectors (QWPRs) as mirrors, a polarization maintaining coupler (PMC), a phase shifter (PS), and PM optical fibers. Figure 2: The forward light (or incident light) and backward light (or output light) after the reflection by a QWPR, where the angle between the direction of \(X\)-polarization state, and the slow axis of the QWP (\(x\)-direction) is 45 degrees. of PM optical fiber in the long arm (short arm) can be expressed as \(l\) (s) = \(U\)(\(\delta\)/2,1,0,0), where \(\delta\) is the birefringence strength of the PM fiber, and the operator of the QWPR can be expressed as \(QR_{\lambda/4}\) = \(U\)(\(\pi\)/4,0,1,0)\({}^{t}\). \(U\)(\(\pi\)/4,0,1,0) = \(U\)(\(\pi\)/2,0,1,0). Then, the long arm operator \(L\) is:  
\[ \begin{array}{ll}L&=\overset{\leftarrow}{l}\cdot QR_{\lambda/4}\cdot \overset{\rightarrow}{l}\\ &=U(\frac{\delta}{2},1,0,0)^{t}\cdot U(\frac{\delta}{2},0,1,0)\cdot U(\frac{ \delta}{2},1,0,0)\\ &=U(\frac{\delta}{2},1,0,0)\cdot U(\frac{\delta}{2},0,1,0)\cdot U(\frac{ \delta}{2},1,0,0)\\ &=U(\frac{\delta}{2},0,1,0)=QR_{\lambda/4}=i\sigma_{2}\end{array}  \tag{2}\]  
where the arrows \(\leftarrow\) and \(\rightarrow\) indicate the backward and forward propagation, respectively, and the superscript \(t\) designates the transposed matrix. For reciprocal optical element, the backward propagation notation \(\overset{\leftarrow}{l}\) equals the transposed matrix of the notation \(\overset{\rightarrow}{l}\) in the forward coordinate system. The conclusion also applies to the short arm operator \(S\). Now, consider double Q-M interferometers in two distant participants [PERSON] and [PERSON], respectively, connected with each other by commercial optical fiber quantum channel as seen in Figure 3. We only set up a phase shifter in the Q-M interferometer on [PERSON]'s side because the phase shifter is used for compensating the relative phase drifting of the two Q-M interferometers. When [PERSON] sends a quantum pulse to [PERSON], there are two paths for the pulse to interfere on [PERSON]'s side [17]:  
\[\text{Path 1: }L_{a}\rightarrow\text{channel}\to S_{b}\]  
\[\text{Path 2: }S_{a}\rightarrow\text{channel}\to L_{b}\]  
where the subscripts \(a\) and \(b\) represent the operators of the components on [PERSON]'s and [PERSON]'s side, respectively. \(L_{i}\) and \(S_{i}\) (\(i=a\) or \(b\)) are the long and short arm operators of the Q-M interferometer on [PERSON]'s or [PERSON]'s side, respectively. According to Equation (2) and considering the phase resulting from transmission along the long and short arms, the transformation matrices of the two paths can be described respectively by  
\[