## 2 Information Reconciliation in QKD Post-Processing
### Rate-Adaptive LDPC Codes and Efficiency Metric of Information Reconciliation
For accurate QBER estimation, it is essential to use only syndrome bits that reflect QBER influences, excluding those adjusted by \(p\) and \(s\) at the position \(S_{dis}[i]\). In this context, \(\omega_{i}\) denotes the set of positions with entries of \(1\) in the \(i\)th row of the parity-check matrix \(\mathbf{H}\), and essentially identifies the position \(S_{dis}\ [\ i\ ]\). To mitigate the effects of punctured and shortening, it is necessary to ensure that \(\omega_{i}\cap p=\emptyset\). The count of bit positions in the \(i\)th row of matrix \(\mathbf{H}\) corresponding to \(\omega_{i}\) is then defined as \(d_{C}^{(i)}=d_{C}-\left|\omega_{i}\cap s\right|\)[22]. Consequently, the likelihood function \(L(q\ |\ S_{dis})\) in Equation (11) can be reformulated as follows:  
\[L(q|S_{dis})=\prod_{ \begin{array}{c}i=1\\ \omega_{i}\cap\ p\ =\ \varnothing\end{array} }^{m}\Big{(}1-S_{dis}[i]+(2 \cdot S_{dis}[i]-1)\cdot p(q,\ d_{c}^{(i)})\Big{)}. \tag{13}\]  
In the syndrome-based QBER estimation, the source of the syndrome information determines the selected likelihood function. If the syndrome information is derived from the original matrix \(\mathbf{H}\) of the mother code \(R_{C}^{\ 0}\), the likelihood function \(L(q\ |\ S_{dis})\), as given in Equation (11), is employed. Conversely, when the syndrome information is generated based on the matrix \(\mathbf{H}\) that incorporates both puncturing and shortening, the likelihood function \(L(q\ |\ S_{dis})\) specified in Equation (13) is considered as the appropriate approach for syndrome-based QBER estimation. After this step, \(Bob\) obtains the value of the estimated QBER (\(q_{est}\)). This value is then communicated to _[PERSON]_ and is subsequently used to determine the optimal coding rate \(R_{C}^{\ (opt)}\) in the information reconciliation step. If \(q_{est}\) exceeds the maximum tolerable QBER of \(11\%\)[32], the system is required to abort, thereby preventing the use of these sifted keys in subsequent steps to ensure security. **(3) Rate-adaptive information reconciliation**: In this step, _[PERSON]_ and \([PERSON]\) employ the value of the estimated QBER (\(q_{est}\)) to optimize the initial coding rate (\(R_{C}^{\ (opt)}\)). This optimization requires setting up the baseline efficiency metric for information reconciliation (\(\eta_{IR}^{(base)}\)), which is used to calculate the number of puncturing bits (\(n_{p}\)) and shortening bits (\(n_{s}\)) based on \(q_{est}\), as defined in Equation (6). This metric ensures that _[PERSON]_ can generate sufficient syndrome information, allowing [PERSON][PERSON]\) to decode and correct errors within his sifted key. It is essential to note that \(\eta_{IR}^{(base)}\) is obtained from the performance evaluation when deploying the specific irregular LDPC codes in the experimental settings. This process comprises the following subsequent steps:  
**(3.1) Code rate optimization**: _[PERSON]_ and \([PERSON]\) collaboratively select a set of mother code rates \(\mathcal{R}\), associated with the block length \(N\). They also agree on the baseline efficiency metric for information reconciliation (\(\eta_{\mathrm{IR}}^{(base)}\)), which is employed to calculate the optimal coding rate (\(R_{C}^{\ (opt)}\)) based on the entropy function of the estimated QBER (\(H(q_{est})\)). In this scenario, the mother code rate (\(R_{C}^{\ 0}\)) is selected from \(\mathcal{R}\) such that its value is closest to the calculated \(R_{C}^{(opt)}\). According to Equation (6), \(R_{C}^{(opt)}\) is derived using:  
\[R_{C}^{(opt)}=1-[\eta_{IR}\cdot H(q_{est})]. \tag{14}\]  
To identify the desired \(R_{C}^{\ 0}\) from the set \(\mathcal{R}\), the selection criteria is determined by  
\[R_{C}^{0}=\operatorname*{argmin}_{R\in\mathcal{R}}\ \Big{|}R-R_{C}^{(opt)} \Big{|}, \tag{15}\]  
where \(R_{C}^{\ 0}=\frac{N-M}{N}\). After selection, \(R_{C}^{\ 0}\) is then adapted to attain the value of \(R_{C}^{(opt)}\) by adjusting the parameters of \(n_{p}\) and \(n_{s}\), given as Equation (4).