## 2 Information Reconciliation in QKD Post-Processing
### Rate-Adaptive LDPC Codes and Efficiency Metric of Information Reconciliation
Specifically, \(N\) corresponds to the size of the sifted key in the primary round of each post-processing cycle, and \(N_{sb}\) pertains to the size of the subblock in the additional round, employed only after a failure of the confirmation step. Each of these two irregular LDPC codes includes a set of mother code rates, as follows:  
\[\mathcal{R}=\Big{\{}R_{\mathbb{C}_{1}}^{0},\ R_{\mathbb{C}_{2}}^{0},\ \ldots\,\ R_{\mathbb{C}_{n}}^{0}\Big{\}}. \tag{8}\]  
These rates are fine-tuned using puncturing and shortening techniques to select the appropriate code rate for the rate-adaptive information reconciliation step. **(2) Syndrome-based QBER estimation:** This step utilizes the properties of a maximum likelihood estimator based on the syndrome information of LDPC codes to estimate QBER over the possible ranges of errors in a QKD system. The process consists of the following subsequent steps:  
**(2.1) Syndrome encoding:**_[PERSON]_ and _[PERSON]_ generate their syndrome information, denoted as \(S_{A}\) and \(S_{B}\), by encoding their sifted keys \(K_{\textit{sifted}}^{A}\) and \(K_{\textit{sifted}}^{B}\) respectively. Both parties employ the syndrome encoding formula \(S_{(A\ /\ B)}=K_{\textit{sifted}}^{(A\ /\ B)}\cdot\textbf{H}_{R_{\mathbb{C}}^{(mn)}}^{T}\), where \(\textbf{H}_{R_{\mathbb{C}}^{(mn)}}^{T}\) is the transpose of the parity-check matrix associated with the maximum code rate \(R_{\mathbb{C}}^{(max)}\) with block length \(N\). Then, _[PERSON]_ transmits \(S_{A}\) to _[PERSON]_ over the authenticated classical channel. **(2.2) Calculation of syndrome discrepancy**: On _[PERSON]_'s side, the syndrome discrepancy, denoted as \(S_{\textit{dis}}\), is determined by calculating the difference between the syndrome information \(S_{A}\) and \(S_{B}\). Specifically, \(S_{\textit{dis}}=S_{A}\oplus S_{B}\), where \(\oplus\) signifies the bitwise XOR operation. **(2.3) QBER estimation**: _[PERSON]_ computes the initial estimate of QBER (\(q_{\textit{est}}\)) using a maximum likelihood estimator (MLE) based on \(S_{\textit{dis}}\)[21; 22; 23]. The estimation is determined as follows:  
\[q_{\textit{est}}=\underset{q\in[0,\ q_{\textit{threshold}}]}{\operatorname{ argmax}}\ L(q|S_{\textit{dis}}), \tag{9}\]  
where \(L(q\ |\ S_{\textit{dis}})\) is the likelihood function for estimating the value of QBER \(q\) based on \(S_{\textit{dis}}\), and \(q_{\textit{threshold}}\) is the maximum QBER threshold that ensures the security of the QKD system. Within this context, each \(i^{\text{th}}\) syndrome bit is probabilistically determined by a Bernoulli distribution. The explicit expression for \(L(q\ |\ S_{\textit{dis}})\) is given as follows:  
\[L(q|S_{\textit{dis}})=\prod_{i,j\ =\ 1}^{m} \begin{cases}p(q,\ d_{c}^{(i)})&\text{if }S_{ \textit{dis}}[i]=1\\ 1-p(q,\ d_{c}^{(j)})&\text{if }S_{\textit{dis}}[j]=0\end{cases}  \tag{10}\]  
where \(\Pr(S_{\text{dis}}[i]=1)=p(q,\ d_{c}^{(i)})\), \(\Pr(S_{\text{dis}}[j]=0)=1-p(q,\ d_{c}^{(j)})\), and \(d_{c}^{(i)}\) denotes the number of 1s in the \(i^{\text{th}}\) row of the parity-check matrix **H** for irregular LDPC codes. Specifically, when the syndrome bits originate from a mother code with its maximum rate \(R_{\mathbb{C}}^{(max)}\), the likelihood function \(L(q\ |\ S_{\textit{dis}})\) can be reformulated as follows:  
\[L(q|S_{\textit{dis}})=\prod_{i=1}^{m}\Big{(}1-S_{\textit{dis}}[i]+(2\cdot S_{ \textit{dis}}[i]-1)\cdot p(q,\ \ d_{c}^{(i)})\Big{)}, \tag{11}\]where the function \(p\left(q,\ d_{c}^{(i)}\right)\), which signifies the probability that \(S_{dis}[i]=1\), is defined as:  
\[p(q,\ d_{c}^{(i)})=\sum_{ \begin{array}{c}k\ =\ 1\\ k\ \mbox{mod}\ 2\ =\ 1\end{array} }^{\frac{d_{c}^{(i)}}{2}}\left( \begin{array}{c}d_{c }^{(i)}\\ k\end{array} \right)q^{k}(1-q)^{d_{c}^{(i)}-k}. \tag{12}\]  
Furthermore, the syndrome bits are generated from the parity-check matrix, which is affected by the puncturing and shortening positions, denoted as \(p\) and \(s\), respectively.