# Asymmetric Measurement-Device-Independent Quantum Key Distribution through Advantage Distillation
## 2 Methods
### Asymmetric MDI-QKD with AD  
Many previous works have demonstrated that the AD method can further improve the performance of QKD [20; 21; 22; 23]. In this section, we improve the secure key rate and transmission distance of the asymmetric seven-intensity MDI-QKD protocol with the AD method. An additional AD method is performed between parameter estimation and post-processing step, and highly correlated bit pairs are discriminated from weakly correlated information. The security of AD method will be analyzed in an entanglement-based scheme. [PERSON] prepares the quantum state \(\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)\) and sends the second particle to [PERSON] through the quantum channel. Since [PERSON] controls the quantum channel by certain value \(\lambda_{i}\)(i = 0, 1, 2, 3), the quantum state shared between [PERSON] and [PERSON] after transmission can be expressed by the following formula:  
\[ \begin{split}\sigma_{AB}=\lambda_{0}|\phi_{0}\rangle\langle\phi_{0}|+ \lambda_{1}|\phi_{1}\rangle\langle\phi_{1}|+\lambda_{2}|\phi_{2}\rangle\langle \phi_{2}|+\lambda_{3}|\phi_{3}\rangle\langle\phi_{3}|,\end{split}  \tag{7}\]\[ \begin{split}|\phi_{0}\rangle&=\frac{1}{\sqrt{2}}(|00 \rangle+|11\rangle),\\ |\phi_{1}\rangle&=\frac{1}{\sqrt{2}}(|00\rangle-|11 \rangle),\\ |\phi_{2}\rangle&=\frac{1}{\sqrt{2}}(|01\rangle+|10 \rangle),\\ |\phi_{3}\rangle&=\frac{1}{\sqrt{2}}(|01\rangle-|10 \rangle),\end{split}  \tag{8}\]  
and \(\lambda_{0}+\lambda_{1}+\lambda_{2}+\lambda_{3}=1\). For the quantum state \(\sigma_{AB}\), the bit error rate of [PERSON] and [PERSON]'s measurements on different bases can be expressed as \(\lambda_{1}+\lambda_{3}=e_{1}^{x}\) (four-state or six-state protocol), \(\lambda_{2}+\lambda_{3}=e_{1}^{z}\) (four-state or six-state protocol), and \(\lambda_{1}+\lambda_{2}=e_{1}^{y}\) (six-state protocol). [PERSON] can steal information and reduce the key rate by choosing the certain value \(\lambda_{i}\) and the secret key rate can be given by [20]:  
\[ \begin{split} R&\geq min_{\lambda_{0},\lambda_{1}, \lambda_{2},\lambda_{3}}[H(X|E)-H(X|Y)]\\ &=min_{\lambda_{0},\lambda_{1},\lambda_{2},\lambda_{3}}[1-( \lambda_{0}+\lambda_{1})h(\frac{\lambda_{0}}{\lambda_{0}+\lambda_{1}})-( \lambda_{2}+\lambda_{3})h(\frac{\lambda_{2}}{\lambda_{2}+\lambda_{3}})-h( \lambda_{0}+\lambda_{1})].\end{split}  \tag{9}\]  
In the AD method, [PERSON] and [PERSON] divide their own raw bits into blocks \((x_{1},\ldots,x_{b})\) and \((y_{1},\ldots,y_{b})\) of size \(b\). Then, choosing a random binary value \(c\), [PERSON] sends \((x_{1}\oplus c,\ldots,x_{1}\oplus c)\) to [PERSON]. [PERSON] compares this bitstring with their bitstring \((y_{1},\ldots,y_{b})\) and accepts the security of information only if the results are either all zeros or all ones in the block. In the two cases accepted, [PERSON] ([PERSON]) saves the first bit \(x_{1}\) (\(y_{1}\)) of the initial string as the raw key. Thus, AD can discern highly correlated bitstring from weakly correlated information as the final raw key. Obviously, the successful probability of the AD method on a certain block of size \(b\) can be calculated by:  
\[P_{succ}=(\lambda_{0}+\lambda_{1})^{b}+(\lambda_{2}+\lambda_{3})^{b}. \tag{10}\]  
After performing the AD step, the practical QBER value \(\lambda_{2}+\lambda_{3}\) in the \(Z\) basis can be replaced by \(\frac{(\lambda_{3}+\lambda_{3})}{P_{succ}}\), and the practical QBER in the \(X\) basis also can be recalculated. The quantum state shared between [PERSON] and [PERSON] can be replaced by:  
\[ \begin{split}\sigma_{AB}=\overline{\lambda}_{0}|\phi_{0}\rangle \langle\phi_{0}|+\overline{\lambda}_{1}|\phi_{1}\rangle\langle\phi_{1}|+ \overline{\lambda}_{2}|\phi_{2}\rangle\langle\phi_{2}|+\overline{\lambda}_{3} |\phi_{3}\rangle\langle\phi_{3}|,\end{split}  \tag{11}\]  
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