# Loop-Back Quantum Key Distribution (QKD) for Secure and Scalable Multi-Node Quantum Networks
## 4 Discussion
### Efficiency Trade-Off  
Taking Table 5 as a reference, we calculate the success probability \(\eta\) when [PERSON] sends \(n\) pulses to [PERSON] as described in Section 2.3. Table 7 illustrates the probability of a successful measurement outcome in a scenario where [PERSON] sends \(n\) pulses and [PERSON] performs measurements in randomly chosen bases. Since only one of [PERSON]'s basis choices is orthogonal to [PERSON]'s states, its probability is \(1/2\). Each polarization result that [PERSON] obtains, also occurring with probability \(1/2^{n}\), propagates through [PERSON]'s measurement process, where each measurement outcome maintains the same probability. Given that there are \(2^{n}\) possible states in [PERSON]'s system and that each step contributes a factor of \(1/2^{n}\), the overall success probability, denoted as \(\eta\), follows as indicated in Equation (1).  
\[\eta=\frac{1}{2^{n+1}} \tag{1}\]  
Beyond the degradation of the success probability, the occurrence of errors in the interpretation of multiple detection events follows a rate of \(e^{n}\), as each detection error occurs independently with probability \(e\), leading to a cumulative probability. By generalizing the secret key rate expression \(\frac{1}{2}(1-2H(e))\), originally derived for the BB84 protocol to evaluate individual attacks, we obtain the generalized form given in Equation (2).  
\[R=\frac{1}{2^{n+1}}(1-2H(e^{n})) \tag{2}\]  
where \(H\) denotes the Shannon entropy, defined as \(H(x)=-x\log_{2}x-(1-x)\log_{2}(1-x)\).  
The analysis of Figure 5 highlights a trade-off between security and efficiency as \(n\) increases. The zero-crossing points mark the threshold beyond which the protocol becomes  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Alice’s** & **Bob’s Basis** & **Bob’s** & **Alice’s** & \\
**Number of** & **Choices Prob.** & **Polarization** & **Measurement** & **Success Probability**\(\eta\) \\
**Pulses** & **Choices Prob.** & **Prob.** & **Prob.** & \\ \hline \hline \(n\) & \(\frac{1}{2}\) & \(\frac{1}{2^{n}}\) & \(\frac{1}{2^{n}}\) & \(2^{n}\left(\frac{1}{2}\right)\left(\frac{1}{2^{n}}\right)\left(\frac{1}{2^{n}}\right)\) \\ \hline \hline \end{tabular}
\end{table}
Table 7: Only one combination of [PERSON]’s bases is orthogonal to [PERSON]’s states, with a probability of \(\frac{1}{2}\). Each of [PERSON]’s polarization outcomes is \(2^{n}\) in total, containing a single successful outcome with probability \(\frac{1}{2^{n}}\). Moreover, each of [PERSON]’s polarization outcomes propagates to \(2^{n}\) measurement results for [PERSON], where each retains a probability of success of \(\frac{1}{2^{n}}\). Therefore, the total probability of success is computed as \(\frac{1}{2^{n+1}}\).  
insecure, as [PERSON] may acquire more information than [PERSON] and [PERSON]. For BB84, this threshold is approximately \(e\approx 0.11\). In contrast, for \(n=1,2,3\), and \(4\), it shifts to \(e\approx 0.11,0.33,0.48\), and \(0.58\), respectively, indicating that higher \(n\) allows for greater quantum bit error rates (QBER) before security is compromised. Moreover, as \(n\) increases, the quantum channel error tolerance asymptotically approaches unity, since \(R=0\) when \(H(x)=0.5\) for \(x=0.1\). Provided \(x=e^{n}\), then, \(e^{n}=0.1\) holds while \(n\) increases, so \(e\) must vary accordingly. We obtain \(n\ln e=\ln 0.1\), which leads to the derivation of Equation (3).  
\[e=10^{-1/n} \tag{3}\]  
This results in Equation (4).  
\[\lim_{n\to\infty}e=\lim_{n\to\infty}10^{-1/n}=1 \tag{4}\]  
Thus, as \(n\) grows, \(e\) asymptotically approaches \(1\). However, this advantage comes at the cost of a reduced initial secret key rate, which declines from \(\frac{1}{4}\) to \(\frac{1}{2^{n+1}}\). Therefore, while the protocol becomes more resilient to errors, it experiences a lower key generation rate.  
The previous results contrast with the reported limits for BB84 (11%), SARG04 (9.68%), and Six-State (12.6%) as presented in [28]. Future experimental studies may provide further validation of these findings.