## 2 Information Reconciliation in QKD Post-Processing
### Application of Slepian-Wolf Coding to Information Reconciliation Based on Channel Coding  
The Slepian-Wolf theorem [28] is a fundamental principle in information theory that addresses the task of conducting effective lossless compression of two correlated data  
Figure 1: Information reconciliation protocol based on channel coding scheme.  
sources. This theorem has significant applications for resolving issues of information reconciliation through channel coding scenarios. In the context of QKD post-processing, the sifted keys of _Alice_ and _Bob_ are not perfectly correlated; they are represented as binary random variables \(X\) and \(Y\), respectively.  
In general, a binary symmetric channel (BSC) is a simplified model that is commonly used to characterize transmission errors in the context of discrete-variable quantum key distribution (DV-QKD) protocols. Therefore, the framework of the information reconciliation protocol based on the channel coding scheme, as illustrated in Figure 1, can be employed within the system of Slepian-Wolf coding. The objective of this framework is to transform the sifted keys \(Y\) and \(X\) into a pair of identical keys, such that the reconciled key \(X^{\prime}\) of _Bob_ is identical to \(X\) with probability equal to one, denoted as \(P[X^{\prime}=X]=1\). To achieve this goal, _Bob_ requires a certain minimum quantity of syndrome \(\mid S_{X}\mid\) from _Alice_. This requirement is defined by the Slepian-Wolf lower bound on the compression rate \(R_{S}\), which must be at least equal to the conditional entropy \(H(X\!\mid\!Y)\), denoted as \(R_{S}\geq H(X\!\mid\!Y)\).  
In Figure 2, a simple Tanner graph is shown; it corresponds to a binary linear block code \(C\) with a parity-check matrix \(\mathbf{H}\) of size \(M\times N\). Generally, the syndrome \(S\) can be calculated by compressing the main information \(X\), where \(S=X\!\cdot\!\mathbf{H}^{T}\). Correspondingly in Slepian-Wolf coding, the compression rate of the syndrome is represented as \(R_{S}=\frac{M}{N}\). This rate is equivalent to the channel coding rate of the linear code \(C\), expressed as \(R_{C}=\frac{N-M}{N}\). Therefore, the relationship between Slepian-Wolf compression rate \(R_{S}\) and channel coding rate \(R_{C}\) can be expressed as:  
\[R_{S}=1-R_{C} \tag{2}\]  
To achieve efficient information reconciliation, the channel coding rate \(R_{C}\) must be optimized to satisfy the Slepian-Wolf lower bound, which is \(R_{S}\geq H(X\!\mid\!Y)\). Then, it can be rewritten as:  
\[1-R_{C}\geq H(X\!\mid\!Y)\text{ and }H(X\!\mid\!Y)=H(q)\,, \tag{3}\]  
where \(q\) is the cross-over probability distribution between \(X\) and \(Y\) over BSC. In the context of a QKD system, \(q\) corresponds to the quantum bit error rate (QBER), which quantifies the joint probability distribution among the correlated information from _[PERSON]_, _[PERSON]_, and _[PERSON]_.  
Figure 2: Tanner graph representation of a binary linear block code \(C\) with a parity-check matrix \(\mathbf{H}\) of dimension \(M\times N\). This graph illustrates the relationship between channel coding and Slepian–Wolf coding: **(a)** Structure of the tanner graph for channel coding, where \(N\) is the number of variable nodes, \(M\) is the number of check nodes, and the channel coding rate of linear code (\(R_{C}\)) is \(\frac{N-M}{N}\). **(b)** Structure of the Tanner graph for Slepian–Wolf coding, where \(N\) is the number of variable nodes, \(M\) is the number of syndrome nodes, and the compression rate of syndrome (\(R_{S}\)) is \(\frac{M}{N}\).