## 2 The Q-M Interferometer Scheme and Theoretical Analysis  
According to Ref. [17], for the phase encoding QKD systems with an unbalanced-arm Mach-Zehnder or Michelson interferometer as the decoder, the polarization disturbance in optical fiber quantum channel affects the stability of the systems, and the anti-disturbance condition is \(L^{+}\cdot S=I\) or \(L=S\), where \(L\) and \(S\) represent the operators of the whole long and short arms of the unbalanced-arm interferometer, respectively, and they are unitary. We propose the Q-M interferometer in this paper which satisfies the anti-disturbance condition. The Q-M interferometer is composed of a polarization maintaining coupler (PMC) and two unbalanced arms (the upper and lower arms), as shown in Figure 1. Both the upper and lower arms are comprised of the polarization maintaining (PM) optical fiber, a quarter-wave plate (QWP), and a reflector. The QWP and the reflector can be fabricated into an integral optical component, i.e., a quarter-wave plate reflector (QWPR). We denote that the slow and fast axes of the QWP are along \(x\)- and \(y\)-directions, respectively, and the slow and fast axes of the PM optical fiber are along \(X\)- and \(Y\)-directions, respectively. The angle between the slow axes of the PM optical fiber and the QWP is 45 degrees. In addition, there is a phase shifter (PS, as shown in Figure 1) or a phase modulator in one arm, for example in the upper arm, for compensating phase drifting or implementing phase encoding. Through the analysis below, we find that either arm of the Q-M interferometer corresponds to Pauli Matrix \(\sigma_{2}\) while that of the Faraday Michelson interferometer corresponds to Pauli Matrix \(\sigma_{3}\). The two Pauli matrices will be defined below. It is easy to confirm that the Pauli Matrix \(\sigma_{2}\) naturally satisfies the anti-disturbance condition and makes QKD systems immune to polarization-induced signal fading [17]. In the article, the mathematical notations are the same as Ref. [22]. Physically, a QWPR can turn an \(X\)-direction linear polarization light to \(Y\)-direction, and vice versa, when the angle between the polarization direction of the linear light and the slow axis of the QWPR equals 45 degrees. As shown in Figure 2, a forward \(X(Y)\) polarization incident light along the slow (fast) axis of PM optical fiber can be transformed into a backward \(Y(X)\) polarization output light along the fast (slow) axis of PM optical fiber after reflected by the QWPR. Due to the same phase accumulation during the round-trip transmission, only the exchange of the \(X\)- and \(Y\)-polarization states happens between the input and output light, namely, the operator of the long or short arm can be written as Pauli Matrix \(\sigma_{2}\left[ \begin{array}{cc}0&1\\ 1&0\end{array} \right]\). Therefore, the output polarization state is independent of PM optical fiber in the interferometer arms and can be expressed as the product of the incident polarization state and Pauli Matrix \(\sigma_{2}\). For proving the function of the Q-M interferometer decoder, we give a rigorous theoretical analysis. To describe the change of the polarization state more intuitively, we take the coordinate system as right-handed and the light propagation direction as \(+z\) direction when the light propagates to the QWPR along PM optical fiber, and the coordinate system as left handed after the reflection by the reflector. For the convenience of theoretical analysis, we use the same notations as those in Ref. [22]. According to Ref. [22], the operator \(U\) of any optical component can be described by using Stokes parameters \(s_{1}\), \(s_{2}\), and \(s_{3}\) and Pauli matrices \(\sigma_{1}\), \(\sigma_{2}\), and \(\sigma_{3}\) and 2D unit matrix \(\sigma_{0}\) as follows:  
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