# Asymmetric Protocols for Mode Pairing Quantum Key Distribution with Finite-Key Analysis
## 3 Simulation and Discussion  
Since the number of single-photon pair events \(M_{11}^{Z}\) in the \(Z\)-basis and the phase-error rate associated with the single-photon pair events \(e_{11}^{Z,\text{ph}}\) in the \(Z\)-basis cannot be directly observed, through the decoy state method [26; 27; 28; 29], we can discover the yield of \(Z\)-pair single photon pulse pairs  
\[y_{11}^{Z}=\frac{F^{L}-F^{U}}{a_{1}^{\
u_{a}}a_{1}^{\
u_{a}}\left(b_{1}^{\
u_{ b}}b_{2}^{\
u_{b}}-b_{1}^{\mu_{b}}b_{2}^{\
u_{b}}\right)}, \tag{8}\]  
where  
\[ \begin{split} F^{L}=&\frac{a_{1}^{\mu_{b}}b_{2}^{ \
u_{b}}}{N_{(\
u_{a},\
u_{b})}^{Z}}\underline{u}_{(\
u_{a},\
u_{b})}^{Z}+ \frac{a_{1}^{\
u_{a}}b_{2}^{\
u_{b}}a_{0}^{\mu_{a}}}{N_{(\
u_{a},\mu_{b})}^{Z} }\underline{u}_{(\
u_{a},\mu_{b})}^{Z}+\frac{a_{1}^{\
u_{a}}b_{2}^{\
u_{b}}b_ {0}^{\mu_{b}}}{N_{(\mu_{a},\mu_{b})}^{Z}}\underline{u}_{(\
u_{a},\mu_{b})}^{Z }+\frac{a_{1}^{\mu_{b}}b_{2}^{\
u_{b}}a_{0}^{\
u_{b}}b_{0}^{\
u_{b}}-a_{1}^{ \
u_{b}}b_{2}^{\
u_{b}}a_{0}^{\mu_{b}}b_{0}^{\
u_{b}}}{N_{(\
u_{a},\mu_{b})}^{ Z}}\underline{u}_{(\
u_{a},\mu_{b})}^{Z},\\ F^{U}=&\frac{a_{1}^{\
u_{a}}b_{2}^{\
u_{b}}}{N_{( \mu_{a},\mu_{b})}^{Z}}\overline{u}_{(\mu_{a},\mu_{b})}^{Z}+\frac{a_{1}^{\mu_{ b}}b_{2}^{\mu_{b}}a_{0}^{\
u_{a}}}{N_{(\
u_{a},\mu_{b})}^{Z}}\overline{u}_{(\
u_{a}, \
u_{b})}^{Z}+\frac{a_{1}^{\mu_{b}}b_{2}^{\mu_{b}}b_{0}^{\
u_{b}}}{N_{(\
u_{a},\mu_{b})}^{Z}}\overline{u}_{(\
u_{a},\mu_{b})}^{Z}.\end{split}  \tag{9}\]  
Here, \(a_{m}^{k_{a}}=\frac{(k_{a})^{m}-k_{a}}{m! },b_{m}^{k_{b}}=\frac{(k_{b})^{m}e^{ -k_{b}}}{m! }\), \(\overline{u}_{(k_{a},k_{b})}^{Z}\) and \(\underline{u}_{(k_{a},k_{b})}^{Z}\) are the number of the effective detection for the intensity pairs \((k_{a},k_{b})\) after the statistical fluctuations in the \(Z\)-pairs, \(N_{(k_{a},k_{b})}^{Z}\) is the expected number of pairs with \((k_{a},k_{b})\). For simplicity, here we only use the \(Z\)-basis pairs with intensities \((\mu_{a},\mu_{b})\) for key generation. Thus, \(M_{11}^{Z}=N_{(\mu_{a},\mu_{b})}^{Z}\mu_{a}\mu_{b}e^{-\mu_{a}-\mu_{b}}y_{11}^{ Z}\). The single-photon bit error rate of \(X\)-pair can be expressed as  
\[e_{11}^{X,\text{bit}}=\frac{T^{U}-T^{L}}{a_{1}^{2\
u_{a}}b_{1}^{2\
u_{b}}y_{11 }^{Z}}, \tag{10}\]  
where  
\[ \begin{split} T^{U}&=\frac{1}{N_{(2\
u_{a},2\
u_{ b})}^{X}}\underline{t}_{(2\
u_{a},2\
u_{b})}^{X}+\frac{a_{2}^{2\
u_{b}}b_{0}^{2 \
u_{b}}}{N_{(2\
u_{a},2\
u_{b})}^{X}}\overline{t}_{(2\
u_{a},2\
u_{b})}^{X}, \\ T^{L}&=\frac{a_{0}^{2\
u_{a}}}{N_{(0_{a},2\
u_{b})}^{X}} \underline{t}_{(0_{a},2\
u_{b})}^{X}+\frac{b_{0}^{2\
u_{b}}}{N_{(2\
u_{a},\
u_ {b})}^{X}}\underline{t}_{(2\
u_{a},\
u_{b})}^{X},\end{split}  \tag{11}\]  
and \(\bar{t}_{(k_{a},k_{b})}^{X}\) and \(\underline{t}_{(k_{a},k_{b})}^{X}\) are the error effective detection for the \((k_{a},k_{b})\) after the statistical fluctuations in the \(X\)-pairs. Through a random-sampling theory without replacement [30; 31; 32], \(e_{11}^{Z,\text{ph}}\) can be written as  
\[e_{11}^{Z,\text{ph}}\leq e_{11}^{X,\text{bit}}+\Gamma\left(\xi_{ex},e_{11}^{X, \text{bit}},M_{11}^{X},M_{11}^{Z}\right), \tag{12}\]where  
\[\Gamma(a,b,c,d)=\sqrt{\frac{(c+d)(1-b)b}{cd}\ln\!\left(\frac{c+d}{2\pi cd(1-b)ba^{2 }}\right)}. \tag{13}\]  
\(\xi_{ac}\) is the failure probability of random sampling without replacement. \(M_{11}^{X}\) represent the number of single-photon pair events in the \(X\)-basis. After calculating \(M_{11}^{Z}\) and \(e_{11}^{Z,\text{ph}}\), the secure key length for the asymmetric MP-QKD protocol under finite key analysis can be obtained by applying Equation (6). The SKR is defined as \(R=\mathbf{L}/N\), where \(N\) is the number of total pluses. It should be clearly stated that although some studies have analyzed asymmetric MP-QKD, there remain certain limitations. For instance, [PERSON]. [10] focuses solely on the advantage of asymmetric MP-QKD over TF-QKD, without examining the impact of channel asymmetry on the intensities and probabilities of decoy states. Additionally, [PERSON].