# Asymmetric Protocols for Mode Pairing Quantum Key Distribution with Finite-Key Analysis
## 2 Finite-Key Analysis for MP-QKD with Asymmetric Channels
For each \(X\)-pair on location \(i\), \(j\), [PERSON] ([PERSON]) extracts a bit from the relative phase \(\left[\left(\theta_{a(b)}^{i}-\theta_{a(b)}^{j}\right)\right/\pi\right]\) mod 2 and announces the alignment angle \(\theta_{a(b)}=\left(\theta_{a(b)}^{i}-\theta_{a(b)}^{j}\right)\) mod \(\pi\). [PERSON] and [PERSON] only retain the results with \(\left|\theta_{a}-\theta_{b}\right|\leq\Delta\) or \(\left|\theta_{a}-\theta_{b}\right|\geq\pi-\Delta\). In particular, when \(\left|\theta_{a}-\theta_{b}\right|\geq\pi-\Delta\), [PERSON] flips the bit. Thus, [PERSON] ([PERSON]) can then obtain the sifted key strings \(\mathbb{Z}\) (\(\mathbb{Z}^{\prime}\)) and \(\mathbb{X}\) (\(\mathbb{X}^{\prime}\)), derived, respectively, from the \(Z\)-pair and the \(X\)-pair. According to the decoy state method, \(\mathbb{Z}\) (\(\mathbb{Z}^{\prime}\)) is employed for generating secure keys, while \(\mathbb{X}\) (\(\mathbb{X}^{\prime}\)) is used to estimate the phase-error rate. In finite-key scenarios, sifted keys may contain some errors. Here, error correction and privacy amplification are required to ensure both \(\epsilon_{\text{cor}}\)-correct and \(\epsilon_{\text{sec}}\)-secret of the keys. \begin{table}
\begin{tabular}{c c c c} \hline \(k_{a(b)}^{j}\)\(k_{a(b)}^{i}\)\(k_{a(b)}^{i}\)\(\mu_{a(b)}\)\(\mu_{a(b)}\)\(\
u_{a(b)}\)\(\
u_{a(b)}\)\(\
u_{a(b)}\)\(\mu_{a(b)}\)\(\mu_{a(b)}\)\(\mu_{  
[PERSON] sends \(\lambda_{\text{EC}}\) bits to [PERSON] for performing key reconciliation, through which [PERSON] computes an estimate \(\mathbb{Z}^{\prime}\) of \(\mathbb{Z}^{\prime}\). [PERSON] computes a hash of \(\mathbb{Z}\) of length \(\log_{2}\frac{2}{c_{\text{cor}}}\) with a random universal\({}_{2}\) hash function [20], which she sends to [PERSON] together with the hash. If \(\text{hash}(\mathbb{Z}^{\prime})=\text{hash}(\mathbb{Z})\), this guarantees the \(\epsilon_{\text{cor}}\)-correct of the keys; otherwise, the protocol aborts. To ensure the security of final secure keys, [PERSON] and [PERSON] employ the privacy amplification based on the Quantum Leftover Hash Lemma [21; 22], which offers a clear operational interpretation of smooth min-entropy. According to a random universal\({}_{2}\) hash function is used to extract an \(\epsilon_{\text{sec}}\)-secret key of length \(\mathbf{L}\) from \(\mathbb{Z}\), where  
\[\epsilon_{\text{sec}}=2\epsilon+\frac{1}{2}\sqrt{2^{\mathbf{L}-H_{\text{min}} ^{\epsilon}(\mathbb{Z}|\mathbb{E}^{\prime})}}. \tag{2}\]  
Here, \(\mathbb{E}^{\prime}\) summarizes all information [PERSON] learned about \(\mathbb{Z}\) during the protocol. Based on a chain rule inequality [23], the smooth min-entropy \(H_{\text{min}}^{\epsilon}(\mathbb{Z}|\mathbb{E}^{\prime})\) can be represented as  
\[H_{\text{min}}^{\epsilon}\big{(}\mathbb{Z}|\mathbb{E}^{\prime}\big{)}\geq H_{ \text{min}}^{\epsilon}(\mathbb{Z}|\mathbb{E})-\lambda_{EC}-\log_{2}\frac{2}{ \epsilon_{\text{cor}}}, \tag{3}\]  
where \(\mathbb{E}\) is the information before error correction, \(\lambda_{\text{EC}}=fM_{(\mu,\mu)}h\big{(}E_{(\mu,\mu)}\big{)}\), \(f\) is the error correction efficiency, \(M_{(\mu,\mu)}\) represents the number of pairs \((\mu,\mu)\) for \(Z\)-basis, and \(E_{(\mu,\mu)}\) denotes the corresponding bit error rate. Furthermore, the strings \(\mathbb{Z}\) can be partitioned into two subsets: \(\mathbb{Z}_{11}\), which contains bits where both [PERSON] and [PERSON] each send a single photon, and \(\mathbb{Z}_{\text{rest}}\), which comprises all remaining bits. Considering the smooth entropies from the uncertainty relation [22; 24], we find that  
\[H_{\text{min}}^{\epsilon}(\mathbb{Z}|\mathbb{E})\geq H_{\text{min}}^{\epsilon}(\mathbb{Z}_{11}|\mathbb{Z}_{\text{rest}} \mathbb{E})+H_{\text{min}}^{\epsilon^{\prime}}(\mathbb{Z}_{\text{rest}}| \mathbb{E})-2\log_{2}\frac{\sqrt{2}}{\hat{\epsilon}}, \tag{4}\]  
where \(\epsilon=2\tilde{\epsilon}+\epsilon^{\prime}+\hat{\epsilon}\) and \(H_{\text{min}}^{\epsilon^{\prime}}(\mathbb{Z}_{\text{rest}}|\mathbb{E})\geq 0\).