# Loop-Back Quantum Key Distribution (QKD) for Secure and Scalable Multi-Node Quantum Networks
## 2 Loop-Back QKD in a Linear Topology
### Single-Pulse Loop-Back QKD
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**[PERSON]** & **[PERSON]** & **[PERSON]** & **Result** & **Prob.** \\ \hline \multirow{4}{*}{\(0_{X}\)} & \(X_{0.5},0_{X}\) & \(X,0_{X}\) & - & 0.5 \\ \cline{2-5}  & \multirow{2}{*}{\(Z_{0.5},0_{Z0.5}\)} & \(X,0_{X0.5}\) & - & 0.125 \\ \cline{2-5}  & & \(X,1_{X0.5}\) & \(\checkmark\) & 0.125 \\ \cline{2-5}  & \multirow{2}{*}{\(Z_{0.5},1_{Z0.5}\)} & \(X,0_{X0.5}\) & - & 0.125 \\ \cline{2-5}  & & \(X,1_{X0.5}\) & \(\checkmark\) & 0.125 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The table represents conditional probabilities using subscripts. Each row’s total probability is computed by multiplying the probabilities in the subscripts (if no probability appears in the subscript, it is assumed to be 1). These probabilities originate either from the choice of measurement basis or from the outcome of a measurement. The notation used for the symbols in this table is detailed in Table 2. \begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Description** \\ \hline \(B_{p}\) & [PERSON]’s measurement basis, where \(B\in\{X,Z\}\) and \(p\) is the probability of choosing that basis. \\ \hline \(b_{B_{p}}\) & [PERSON]’s measurement outcome, where \(b\in\{0,1\}\), conditioned on [PERSON]’s basis choice with probability \(p\). \\ \hline \(P(b_{B_{p}})\) & Total probability of an event, computed as \(P(b_{B_{p}})=p\cdot P(b\mid B)\). If no subscript appears, the probability is assumed to be 1. \\ \hline \(\checkmark\) & [PERSON] correctly identifies [PERSON]’s basis (successful event). \\ \hline \(-\) & Inconclusive case where [PERSON] cannot determine [PERSON]’s basis; these cases are discarded. \\ \hline \(\mathbf{\kappa}\) & [PERSON] assumes success but misidentifies [PERSON]’s basis. \\ \hline \(\delta\) & Successful event for [PERSON] and [PERSON], but [PERSON] misidentifies the result, giving [PERSON] and [PERSON] an advantage. \\ \hline \(Th_{ab}\) & The throughput of the protocol, representing the total number of events used for key distillation. \\ \hline \(\psi\) & Quantum state which is written as \(b_{B_{p}}\). \\ \hline \(\
eg\psi\) & Orthogonal quantum state. \\ \hline \(e\) & Quantum Bit Error Rate (QBER), representing the probability of a bit error in the quantum channel. \\ \hline \(n\) & Number of pulses encoding a single bit in the multi-pulse-based protocol. \\ \hline \(\eta\) & The efficiency of the sifting process. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Notation used in the Loop-Back-based protocols. We have used the color green (\(\checkmark\)) to represent successful events and red (\(\mathbf{\varphi}\)) to denote erroneous events. This color-coding has been employed to provide an immediate and intuitive visual distinction between successful and failed outcomes throughout the document. [PERSON] measures the returned state using the same basis in which she originally prepared it. 4. If the state remains unchanged, [PERSON] cannot determine [PERSON]'s basis choice. However, if the measurement result is orthogonal to the originally prepared state (\(\
eg\psi\)), this indicates that [PERSON] used a basis different from [PERSON]'s preparation basis, allowing her to infer [PERSON]'s choice. 5. [PERSON] and [PERSON] use the basis identification process to generate a shared secret key.