## 2 Information Reconciliation in QKD Post-Processing
### Information Reconciliation Based on Channel Coding Theorem  
In the field of communication over classical channels, the channel coding theorem [27] aims to optimize the transmission rate while ensuring reliable communication in the presence of channel noise.  
In this section, the problem of information reconciliation is addressed by the concept of the channel coding theorem, as illustrated in Figure 1. Let \(C\) represent a linear code that contains a parity-check matrix \(\mathbf{H}\) of size \(M\times N\), defined over GF(2) (GF refers to a Galois field). The inputs of the two legitimate parties (_Alice_ and _Bob_) are random binary key strings \(X=\{x_{1},\,\ldots,\,x_{n}\}\) and \(Y=\{y_{1},\,\ldots,\,y_{n}\}\), respectively. These inputs, \(X\) and \(Y\), are distributed according to a joint probability distribution \(P_{XY}\left(x,y\right)\), where \(x\in X\) and \(y\in Y\). In the context of a QKD system, \(P_{XY}\left(x,y\right)\) can be utilized to determine the discrepancy between \(X\) and \(Y\), which is known as the quantum bit error rate (QBER) value. The simple information reconciliation protocol, which is based on channel coding, enables one-way communication from _[PERSON]_ to _[PERSON]_ over a classical channel. The protocol can be summarized in the following two main steps:  
**(1) Encoding:**_[PERSON]_ performs the encoding function of the linear code \(C\) to generate the syndrome \(S_{A}\), where \(S_{A}=X\cdot\mathbf{H}^{T}\). The syndrome \(S_{A}\) is then transmitted over the classical channel to _[PERSON]_.
**(2) Decoding:** Upon receiving the syndrome \(S_{A}\), _[PERSON]_ computes \(\hat{e}\) using the decoding function of \(C\), denoted by \(\mathrm{dec}_{\mathbf{H}}\), where \(\hat{e}=\mathrm{dec}_{\mathbf{H}}(Y,S_{A})\). The value of \(\hat{e}\) indicates the error position in \(Y\), and _[PERSON]_ calculates the output value \(X^{\prime}\) by performing \(X^{\prime}=Y\oplus\hat{e}\).  
The effectiveness of the information reconciliation protocol based on channel coding relies on the capability of the syndrome decoding process to generate an output syndrome \(S_{B}=X^{\prime}\cdot\mathbf{H}^{T}\) that matches the received syndrome \(S_{A}\) from _Alice_ (\(S_{A}=S_{B}\)). If the decoding process fails, the system must be aborted or reprocessed until successful decoding is achieved.  
In this system, the cardinality of \(S_{A}\), denoted as \(\mid S_{A}\mid\), represents the information leakage, which depends on the correlation of channel capacity in the channel coding scheme. The minimum theoretical information leakage of information reconciliation based on channel coding, represented as \(min(\mid S_{A}\mid)\), can be calculated as  
\[\frac{\min\left(\mid S_{A}\mid\right)}{N}=H(X|Y), \tag{1}\]  
where \(N\) is the size of input \(X\), and \(H(X|\,Y)\) denotes the conditional entropy of \(X\) given \(Y\). However, _Eve_ can obtain information leakage when communication occurs over a public channel. Importantly, \(\mid S_{A}\mid\) is a fundamental parameter for evaluating the efficiency of the information reconciliation protocol based on channel coding.