## 4 Analysis of Results
### Example Optimization  
In this section, an optimization of a fielded decoy state enabled QKD system is demonstrated. As one of the most well documented decoy state protocol implementations and a major milestone in the world's largest QKD network, [PERSON] and coworkers' work lends itself well to detailed analysis [55]. The protocol's configuration is provided in Table 6.  
Assuming \(N_{\mu}=100,000\) detections per quantum exchange and an arbitrarily small vacuum state occurrence percentage \(S_{Y_{0}}=0.005\), the decoy state protocol occurrence percentages can be optimized to \(S_{\mu}=0.99435\), \(S_{\
u}=0.00065\) using the approach described in Equations (9)-(16). This optimized configuration is particularly advantageous as it results in a >30% increase in key rate (i.e., a signal state occurrence percentage 99.435% instead of 75%) and the ability to detect PNS attack with 99.9% confidence at no additional cost. This optimization accounts for expected real-world variations in the source but does not account for significant disturbances in the quantum channel which would quickly eliminate the ability to reliably perform QKD regardless of the decoy state configuration. Figure 6 presents detailed results of the optimized protocol while operating under normal conditions and when subject to PNS attacks. Shown on the left, during normal operations the signal and decoy state efficiencies (blue and red) overlap as expected. Shown in the middle, PNS attacks cause \begin{table}
\begin{tabular}{c c} \hline \hline
**Protocol Configuration** & **Operational Results** \\ \hline \(S_{\mu}=0.75\) & \(\eta=0.00985\) \\ \(S_{\
u}=0.125\) & \(Q_{\mu}=6.36\ \times 10^{-3}\) \\ \(S_{Y_{0}}=0.125\) & \(Q_{\
u}=8.61\times 10^{-4}\) \\ \(\mu=0.65\) & \(Y_{0}=1.0\ \times 10^{-4}\) \\ \(\
u=0.08\) & — \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Example Decoy State Protocol Configuration. the signal and decoy state efficiencies (green and purple) to become non-overlapping. In particular, since the protocol is configured to operate with a minimum number of decoy state detections, the PNS attack reduces the decoy state from a small number of detections to zero during nearly every round of quantum exchange. This results in a reported decoy state mean efficiency of 0.000 with relatively little variation (see Figure 7 for further details). Consequently, the optimized decoy state protocol configuration serves to emphasize the negative impact of the PNS attack by forcing the decoy state's efficiency below the measured dark count rate (shown in brown with a detailed inlay) because so few decoy state detections are expected per round of quantum exchange. Figure 6: Simulation results show the optimized decoy state protocol for detecting PNS attacks based on the fielded QKD system [55]. The inlay details the expected dark count rate complete with outliers to illustrate the modeled noise over the QKD communication channel. Figure 7: Simulation results detailing the number of decoy state detections per round of quantum exchange for the optimized decoy state protocol based on the fielded QKD system parameters shown in Table 6.  
Figure 7 displays the number of decoy state detections per round of quantum exchange during normal operations (shown in green) and when subject to PNS attacks (shown in red). During normal operations, the optimized configuration results, shown in green, demonstrate at least one decoy state detection per 100,000 detections and a mean of nine detections. Conversely, as shown in red, very few decoy state detections are expected during PNS attacks. Detections occur in only 134 out of the 1000 rounds of quantum exchange, which constitute statistical outliers. In terms of efficiency, the mean decoy state efficiency is 0.0096 during normal operations (shown in red in Figure 6) and drops to 0.0013 (shown in purple in Figure 6) during PNS attacks.