## Appendix A  
As mentioned earlier, the readback signals from the desired track and two adjacent tracks convey some information about the \(k\)-th bit of the \(l\)-th track (or the desired track). Therefore, the \(k\)-th log-likelihood ratio (LLR) sample produced from the \((l-1)\)-th track, \(\left\{\lambda_{k,l-1}\right\}\), and the \((l+1)\)-th track, \(\left\{\lambda_{k,l+1}\right\}\), can be utilized to improve the quality and reliability of the \(k\)-th LLR sample from the \(l\)-th track, \(\lambda_{k,l}\), according to Equation (7) for the bit-summation detection (BSD) technique, which can be proven as follows.  
Consider the LLRs of a BPMR channel model in Figure 4. An improved version of the LLR of the \(k\)-th bit from the \(l\)-th track can be written as  
\[\lambda_{k,l}^{\text{new}}=\text{log}\Bigg{(}\frac{\Pr\big{[}\lambda_{k,l}=1 \big{|}\{\lambda\}\big{]}}{\Pr\big{[}\lambda_{k,l}=-1\big{|}\{\lambda\}\big{]} }\Bigg{)}, \tag{11}\]  
where log(x) is a natural logarithm of x, and \(\lambda_{k,l}^{\text{new}}\) is the mutual information of \(\left\{\lambda_{k,l-1}\right\}\), \(\left\{\lambda_{k,l}\right\}\), and \(\left\{\lambda_{k,l+1}\right\}\). We have applied the conditional independence relationship in variable form according to [37]  
\[\begin{array}{l}\Pr[A=x|E]=\Pr[A=x|B\cap C\cap D]\\ =\Pr[A=x|B]\cdot\\ \Pr[A=x|C]\cdot\Pr(A=x|D),\end{array} \tag{12}\]  
where A = \(\in\){\(\pm 1\)}, E = x \(\mid\) B\(\cap\)C\(\cap\)D are random variables, and the events B, C, and D are mutually independent. Then, (11) can be expanded to  
\[\lambda_{k,l}^{\text{new}}=\text{log}\Bigg{(}\frac{\Pr\big{[}u_{k,l}=1\big{|} \big{\{}\lambda_{k,l-1}\big{\}}\big{]}\cdot\Pr\big{[}u_{k,l}=1\big{|}\big{\{} \lambda_{k,l}\big{\}}\big{]}\cdot\Pr\big{[}u_{k,l}=1\big{|}\big{\{}\lambda_{k, l+1}\big{\}}\big{]}}{\Pr\big{[}u_{k,l}=-1\big{|}\big{\{}\lambda_{k,l-1}\big{\}} \big{]}\cdot\Pr\big{[}u_{k,l}=-1\big{|}\big{\{}\lambda_{k,l}\big{\}}\big{]} \cdot\Pr\big{[}u_{k,l}=-1\big{|}\big{\{}\lambda_{k,l+1}\big{\}}\big{]}}\Bigg{)}, \tag{13}\]  
where \(\left\{l-1,\,l,\,l+1\right\}\) are the upper track, desired track, and lower track, respectively. Therefore, it can be shown that  
\[\Pr\big{[}u_{k,l}=1\big{|}\big{\{}\lambda_{k,l}\big{\}}\big{]}=\frac{e^{ \lambda_{k,l}/2}}{e^{\lambda_{k,l}/2}+e^{-\lambda_{k,l}/2}}, \tag{14}\]  
and  
\[\Pr\big{[}u_{k,l}=-1\big{|}\big{\{}\lambda_{k,l}\big{\}}\big{]}=\frac{e^{- \lambda_{k,l}/2}}{e^{\lambda_{k,l}/2}+e^{-\lambda_{k,l}/2}}. \tag{15}\]  
By substituting (14) and (15) into (13), one obtains  
\[\lambda_{k,l}^{\text{new}}=\text{log}\Bigg{(}\frac{\Big{(}\frac{e^{\lambda_{k,l}-1/2}}{e^{\lambda_{k,l}-1/2}+e^{-\lambda_{k,l}-1/2}}\Big{)}\cdot\Big{(}\frac{e ^{\lambda_{k,l}/2}}{e^{\lambda_{k,l}/2}+e^{-\lambda_{k,l}/2}}\Big{)}\cdot\Big{(} \frac{e^{\lambda_{k,l+1}/2}}{e^{\lambda_{k,l+1}/2}+e^{-\lambda_{k,l+1}/2}} \Big{)}}{\Big{(}\frac{e^{-\lambda_{k,l}-1/2}}{e^{\lambda_{k,l-1}/2}+e^{- \lambda_{k,l}-1/2}}\Big{)}\cdot\Big{(}\frac{e^{-\lambda_{k,l}/2}}{e^{\lambda_{ k,l}/2}+e^{-\lambda_{k,l}/2}}\Big{)}\cdot\Big{(}\frac{e^{-\lambda_{k,l+1}/2}}{e^{ \lambda_{k,l+1}/2}+e^{-\lambda_{k,l+1}/2}}\Big{)}}\Bigg{)}, \tag{16}\]  
which can be rewritten as  
\[\begin{array}{l}\lambda_{k,l}^{\text{new}}=\text{log}\Big{(}\frac{e^{\lambda_{ k,l-1}/2},e^{\lambda_{k,l}/2},e^{\lambda_{k,l+1}/2}}{e^{-\lambda_{k,l-1}/2}e^{- \lambda_{k,l}/2}e^{-\lambda_{k,l+1}/2}}\Big{)},\\ =\text{log}\big{(}e^{\lambda_{k,l-1}}\big{)}+\text{log}\big{(}e^{\lambda_{k,l} }\big{)}+\text{log}\big{(}e^{\lambda_{k,l+1}}\big{)},\\ =\lambda_{k,l-1}+\lambda_{k,l}+\lambda_{k,l+1}.\end{array} \tag{17}\]  
Clearly, (17) is the same as (7).