# Asymmetric Protocols for Mode Pairing Quantum Key Distribution with Finite-Key Analysis
## 2 Finite-Key Analysis for MP-QKD with Asymmetric Channels  
Since the key size in practical implementations is not infinite, it is crucial to develop a framework that addresses the finite-key size effect. In previous studies on asymmetric MP-QKD [17, 18], the key size was often assumed to be infinite, leading to overly idealized parameter estimation. In the following analysis, we extend the consideration from the infinite-key regime to the realistic finite-key scenario. At the end of the asymmetric MP-QKD protocol, Alice and Bob each share a key string, denoted as \(\mathbb{S}\) and \(\mathbb{S}^{\prime}\), respectively. According to the universally composable framework [19], if these key strings satisfy both \(\epsilon_{\text{cor}}\)-correct and \(\epsilon_{\text{sec}}\)-secret, they are considered secure keys. \(\epsilon_{\text{cor}}\)-correct refers to \(\Pr(\mathbb{S}\
eq\mathbb{S}^{\prime})\leq\epsilon_{\text{cor}}\). And \(\epsilon_{\text{sec}}\)-secret requires satisfaction of the condition  
\[\frac{1}{2}\|\rho_{AE}-U_{A}\otimes\rho_{E}\|_{1}\leq\epsilon_{\text{sec}}, \tag{1}\]  
where \(\|\cdot\|_{1}\) denotes the trace norm, \(\rho_{AE}\) is the density operator of the system of [PERSON] and [PERSON], \(U_{A}\) is the uniform mixture of all possible values of the bit string \(\mathbb{S}\), and \(\rho_{E}\) is density operator of [PERSON]'s system. In this way, the protocol with finite-key can be regarded as \(\epsilon\)-secure, where \(\epsilon=\epsilon_{\text{cor}}+\epsilon_{\text{sec}}\). In addition, we fix the security bound to \(\epsilon=10^{-10}\). In the following finite-key analysis of the asymmetric MP-QKD protocol, we assume the distances from [PERSON] and [PERSON] to [PERSON] are denoted as \(L_{A}\) and \(L_{B}\), where \(L_{A}\leq L_{B}\) and \(\Delta L=L_{B}-L_{A}\). In the \(i\)-th round, [PERSON] prepares the coherent state \(\left|e^{\text{i}\theta_{a}}\sqrt{k_{a}^{i}}\right\rangle\), where intensity \(k_{a}^{i}\) is randomly selected from \(\{\mu_{a},\
u_{a},\sigma_{a}\}\) with probabilities \(\{p_{\mu_{a}},p_{\
u_{a}},p_{\sigma_{a}}\}\). And modulated phase \(\theta_{a}^{i}\) is randomly chosen from \(\left\{0,\frac{2\pi}{\Delta},\frac{4\pi}{\Delta},...,\frac{2\pi(\Delta-1)}{ \Delta}\right\}\), where \(\Delta\) is typically set to 16. In parallel, [PERSON] randomly selects \(k_{b}^{i}\) and \(\theta_{b}^{i}\) to prepare the coherent state \(\left|e^{\text{i}\theta_{b}^{i}}\sqrt{k_{b}^{i}}\right\rangle\) with probabilities \(\left\{p_{\mu_{b}},p_{\
u_{b}},p_{\sigma_{b}}\right\}\). It is important to note that in asymmetric channels, except for the case where \(\sigma_{a}=\sigma_{b}=0\), the intensity and probability settings of [PERSON] and [PERSON] are unequal. Therefore, full parameter optimization is required to achieve the maximum SKR. After interference measurement, [PERSON] announces the measurement outcomes \(D_{L},D_{R}\in\{0,1\}\) (0 represents \"no click\", while 1 represents \"click\"). \(D_{L}\oplus D_{R}=1\) is considered as an effective detection. After \(N\) rounds, [PERSON] and [PERSON] pair each click with its immediate next neighbor within a maximum pairing interval \(l\) to form a successful pairing. According to the intensities of the paired \(i\)-th and \(j\)-th rounds (\(j\leq i+l\)), [PERSON] ([PERSON]) labels the \"basis\" as illustrated in Table 1.  
Subsequently, [PERSON] and [PERSON] announce the basis, and the sum of the intensity pairs \((k_{a},k_{b})=\left(k_{a}^{i}+k_{a}^{j},k_{b}^{i}+k_{b}^{j}\right)\) for each pairing. Subsequently, they perform the pair assignment based on Table 2. Notably, the 0-pair data are retained specifically for decoy-state parameter estimation. For each \(Z\)-pair on location \(i\), \(j\), [PERSON] ([PERSON]) extracts a bit 0 when \(k_{a}^{i}\
eq k_{a}^{j}=o_{a}\) (\(k_{b}^{j}\
eq k_{b}^{i}=o_{b}\)), and a bit 1 when \(k_{a}^{i}\
eq k_{a}^{i}=o_{a}\) (\(k_{b}^{i}\
eq k_{b}^{j}=o_{b}\)).