## 2 Information Reconciliation in QKD Post-Processing
### Rate-Adaptive LDPC Codes and Efficiency Metric of Information Reconciliation  
Low-density parity-check (LDPC) codes [29] are a class of linear block code characterized by their sparse parity-check matrix. Typically, LDPC codes utilize iterative algorithms such as the bit-flipping algorithm, belief propagation algorithm, and min-sum algorithm for the decoding process. According to their structures, LDPC codes can be broadly categorized into regular LDPC codes and irregular LDPC codes. For the regular LDPC codes, each variable node connects to a fixed number of check nodes, which ensures a consistent and fixed number of non-zero entries in each row and column of the parity-check matrix, leading to a uniform degree distribution. Conversely, for the irregular LDPC codes with a non-uniform degree distribution, specific variable nodes can connect to varying numbers of check nodes. The degree distribution of these irregular LDPC codes is designed for specific applications such as wireless communications, digital television broadcasting, and satellite communications. These codes exhibit superior error-correcting capabilities when compared to conventional regular LDPC codes. For information reconciliation with irregular LDPC codes, a prior error estimation is essential to optimize the channel coding rates, which significantly enhances the efficiency of rate-adaptive information reconciliation. Consider an irregular LDPC code with a parity-check matrix \(\mathbf{H}\) of size \(M\times N\). The mother code rate, denoted as \(R_{C}^{0}\), is defined by \(R_{C}^{0}=\frac{N-M}{N}\). To fine-tune the LDPC's mother code rate, puncturing and shortening techniques [30; 31] are employed to modulate the optimal coding rate, represented as \(R_{C}^{(opt)}\). In this context, \(n_{p}\) and \(n_{s}\) represent the number of punctured and shortened bits within block length \(N\), respectively. The equation for \(R_{C}^{(opt)}\) is given by:  
\[R_{C}^{(opt)}=\frac{N-M-n_{s}}{N-n_{p}-n_{s}}. \tag{4}\]  
The theoretically secret key rate after QKD post-processing, denoted as \(r_{th}\), can be expressed by \(r_{th}=H(X|Z)\ H(X|Y)\). Within the QKD context, \(X\) and \(Y\) correspond to the sifted key of _[PERSON]_ and _[PERSON]_, respectively, and \(Z\) represents the information that _[PERSON]_ extracts from the quantum channel. The conditional entropy \(H(X|Z)\) measures the uncertainty associated with _[PERSON]_'s knowledge about _[PERSON]_'s sifted key. \(H(X|Y)\) represents the theoretical information leakage during the information reconciliation phase, which depends on the minimal number of syndrome bits \(\min(\,|S_{X}|)\), as expressed in Equation (1). However, the number of syndrome bits, which corresponds to the Slepian-Wolf compression rate \(R_{S}\), must exceed the quantity of theoretical information leakage \(H(X|Y)\). Therefore, the evaluation of information reconciliation is determined by the ratio of practical information leakage to the theoretical limit, as expressed in the following:  
\[\eta_{IR}=\frac{R_{S}}{H(X|Y)}=\frac{M}{N\cdot H(q)}\geq 1, \tag{5}\]  
where \(\eta_{IR}\) denotes the efficiency metric of information reconciliation, and \(H(q)\) represents the binary entropy function of QBER, which can be calculated as \(H(q)=q\log_{2}q\ (1\ q)\log_{2}(1\ q)\). Considering the lower bound of the Slepian-Wolf compression rate, represented as \(R_{S}\) = \(H(X|Y)\), it is noted that \(\eta_{IR}\) must be equal to one to achieve \(min(\,|S_{A}|)\). Notably, the maximum tolerable QBER to ensure information-theoretic security is 11% [32]. For the rate-adaptive reconciliation, the optimal coding rate for LDPC codes \(R_{C}^{(opt)}\) is modulated using puncturing and shortening techniques, as derived from Equation (4).