# Loop-Back Quantum Key Distribution (QKD) for Secure and Scalable Multi-Node Quantum Networks
## 2 Loop-Back QKD in a Linear Topology
### Multi-Pulse Loop-Back QKD
Classical reconciliation: [PERSON] and [PERSON] communicate over a classical channel to confirm which events were accepted, without revealing their bases or individual state values. 5. Key assignment: The secret key bits are assigned based on [PERSON]'s polarization choices: -- If [PERSON] used the \(Z\) basis, the accepted pair represents bit 0. -- If [PERSON] used the \(X\) basis, the accepted pair represents bit 1. Table 5 presents the probability distribution for the Loop-Back QKD protocol when [PERSON] sends the state pair (\(0_{Z}\), \(0_{Z}\)). The table details the basis selection by [PERSON], the possible polarization outcomes, and the results obtained when [PERSON] performs a \(Z\)-basis measurement. Only cases where [PERSON]'s measurement results in (\(1_{Z}\), \(1_{Z}\)) are considered successful, as these satisfy the orthogonality condition required for key generation. The probability of success is calculated as \(4\times 0.03125=0.125\), indicating that the acceptance rate in this scheme is lower than in the single-pulse protocol, but with an improved robustness against errors. Entries where at least one state does not meet the orthogonality condition are marked as ambiguous and discarded, reducing the probability of an erroneous bit being included in the final key. As a result, doubling the number of pulses per round reduces the effective QBER to \(e^{2}\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**[PERSON]** & **[PERSON]’s** & **[PERSON]’s Polarization** & **Alice** & **Measurement** & **Prob.** \\
**Sends** & **Basis Choices** & **Results** & **Z-Measurement** & **Result** & **Prob.** \\ \hline \multirow{8}{*}{\((0_{Z},0_{Z})\)} & \multirow{8}{*}{\((X,X)_{0.5}\)} & \multirow{8}{*}{\((0_{Z},1_{Z})_{0.25}\)} & \((0_{Z},1_{Z})_{0.25}\) & - & 0.03125 \\ \cline{3-6}  & & & \((0_{Z},0_{Z})_{0.25}\) & - & 0.03125 \\ \cline{3-6}  & & & \((1_{Z},1_{Z})_{0.25}\) & - & 0.03125 \\ \cline{3-6}  & & & \((0_{Z},1_{Z})_{0.25}\) & - & 0.03125 \\ \cline{3-6}  & & & \((0_{Z},0_{Z})_{0.25}\) & - & 0.03125 \\ \cline{3-6}  & & & \((1_{Z},1_{Z})_{0.25}\) & - & 0.03125 \\ \cline{3-6}  & & & \((1_{Z},0_{Z})_{0.25}\) & - & 0.03125 \\ \cline{3-6}  & & & \((0_{Z},1_{Z})_{0.25}\) & - & 0.03125 \\ \cline{3-6}  & & & \((0_{Z},1_{Z})_{0.25}\) & - & 0.03125 \\ \cline{3-6}  & & & \((1_{Z},0_{Z})_{0.25}\) & - & 0.03125 \\ \cline{3-6}  & & & \((0_{Z},1_{Z})_{0.25}\) & - & 0.03125 \\ \cline{3-6}  & & & \((1_{Z},1_{Z})_{0.25}\) & - & 0.03125 \\ \cline{3-6}  & & & \((1_{Z},1_{Z})_{0.25}\) & - & 0.03125 \\ \cline{3-6}  & & & \((1_{Z},1_{Z})_{0.25}\) & - & 0.03125 \\ \cline{3-6}  & & & \((1_{Z},1_{Z})_{0.25}\) & - & 0.03125 \\ \cline{3-6}  & & & \((1_{Z},1_{Z})_{0.25}\) & - & 0.03125 \\ \cline{3-6}  & & & \((1_{Z},0_{Z})_{0.25}\) & - & 0.03125 \\ \hline \((Z,Z)_{0.5}\) & \((0_{Z},0_{Z})_{1.0}\) & \((0_{Z},0_{Z})_{1.0}\) & - & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Probability distribution for the Loop-Back QKD protocol with double pulses. The success probability, corresponding to the cases where [PERSON] measures (\(1_{Z}\), \(1_{Z}\)), is given by \(4\times 0.03125=0.125\). All cases marked with (-) are discarded due to ambiguity. Compared to the single-pulse variant, the double-pulse approach offers the following advantages:  
* The effective error rate is reduced from \(e\) to \(e^{2}\), significantly minimizing the impact of quantum bit errors on the final key. * The requirement for both states to be orthogonal on [PERSON]'s side enhances resistance to MitM attacks. * [PERSON] can more reliably determine [PERSON]'s basis choices, reducing bit loss due to basis ambiguity.