# 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  
This enhanced version of the Loop-Back QKD protocol introduces a double-pulse scheme, which increases security and error tolerance by mitigating quantum bit errors. The general case of \(n\) pulses will be addressed in Section Efficiency Trade-Off. However, this improvement comes at the cost of lower efficiency compared to the single-pulse version. Unlike BB84, where key generation relies on basis matching, in this protocol, the key is determined based on [PERSON]'s polarization choices. A key motivation for using double pulses is the reduction in the effective QBER. In a single-pulse scheme, the probability of an error occurring is given by \(e\), where \(e\) represents the intrinsic error rate of the quantum channel. In the double-pulse scheme, a valid key bit is generated only when both received states satisfy the orthogonality condition. If an error occurs in only one of the pulses, the event is classified as ambiguous and discarded. The basic principle is that [PERSON] transmits two quantum states per round, ensuring that both states are prepared in the same basis, either \(X\) or \(Z\). The transmitted pairs are selected from the following sets:  
* In the \(Z\) basis: \((0_{Z},0_{Z}),(0_{Z},1_{Z}),(1_{Z},0_{Z}),(1_{Z},1_{Z})\)
* In the \(X\) basis: \((0_{X},0_{X}),(0_{X},1_{X}),(1_{X},0_{X}),(1_{X},1_{X})\)  
[PERSON] is the only one who knows the basis on which each pair was prepared. Upon reception, [PERSON] independently selects a random basis (\(X\) or \(Z\)) for the pair, polarizes the states accordingly, and returns them to [PERSON]. An attacker cannot distinguish whether a transmitted pair belongs to the set of parallel states \(\{(0_{Z},0_{Z}),(1_{Z},1_{Z}),(0_{X},0_{X}),(1_{X},1_{X})\}\) or to the set of orthogonal states \(\{(0_{Z},1_{Z}),(1_{Z},0_{Z}),(0_{X},1_{X}),(1_{X},0_{X})\}\). Moreover, even if partial information about one of the states were extracted, it would not reveal additional information about the other state in the pair. This fundamental quantum property ensures that no meaningful knowledge about the transmitted pairs can be obtained by an eavesdropper. [PERSON] then measures the received states in the same basis as the original preparation (\(X\) or \(Z\)). Thus, she only accepts events where the measured pair is orthogonal to the originally sent pair. The specific steps of the Multi-Pulse Loop-Back QKD protocol are detailed in the following procedure. 1. State preparation: [PERSON] randomly selects a basis \(B_{A}\in\{X,Z\}\) and prepares a pair of quantum states \((\psi_{1B_{A}},\psi_{2B_{A}})\) and she sends these states to [PERSON]. 2. [PERSON]'s polarization: Upon receiving the states, [PERSON] selects a quantum basis \(B_{B}\in\{X,Z\}\) to polarize both states:  
* If \(B_{B}=B_{A}\), the state remains unchanged. * If \(B_{B}\
eq B_{A}\), the state is polarized onto the new basis with equal probability (50%). [PERSON] then returns the polarized states to [PERSON]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Protocol** & \(\delta\) & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) & \(Th_{ab}\) \\ \hline BB84 & - & 0.5 & - & 0.5 \\ BB84 with MitM & 0.25 & 0.125 & 0.125 & 0.5 \\ Loop-Back QKD & - & 0.25 & - & 0.25 \\ Loop-Back with MitM & 0.125 & 0.125 & 0.125 & 0.375 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of protocol performance under ideal quantum channel conditions. The throughput (\(Th_{ab}\)) is the sum of all sifting events, \(\delta\) and \(\checkmark\), as well as \(\checkmark\). The values for each protocol indicate the corresponding probabilities for each type of event. [PERSON]'s measurement and validation: [PERSON] measures the received states in the original basis \(B_{A}\). She accepts only those cases where the measured pair satisfies the orthogonality condition: \[(\psi_{1B_{A}},\psi_{2B_{A}})\rightarrow(\
eg\psi_{1B_{A}},\
eg\psi_{2B_{A}}).\] Any other outcome is considered ambiguous and discarded. 4.