## 4 Analysis of Results
### Studying Detection Results  
Figure 4 illustrates the normal operating conditions for 20 configurations over an operational distance of 50 km (the 20 km results are not shown because they are very similar). The results are grouped with respect to signal and decoy MPNs with each treatment labeled across the bottom of the graph by signal-decoy-vacuum occurrence percentages (e.g., 60-30-10 means 60% signal states, 30% decoy states, and 10% vacuum states). The overlapping box plots imply \(\eta^{signal}=\eta^{decoy}\); thus, the system is operating in a secure state. Of note, variation in the signal state remains relatively fixed, while variation in the decoy state increases as the occurrence percentage lessens from 30% to 0.5%. Likewise, the lower MPN (i.e., 0.1 compared to 0.2) results in slightly more variation in each configuration. This occurs because less decoy states are sent by [PERSON], and therefore, detected by [PERSON], causing more variation. In all 40 configurations studied without PNS attacks at both 20 and 50 km, the signal and decoy state efficiencies are overlapping with no statistically significant differentiation. Figure 5 illustrates results over the 50 km operational distance from 20 configurations when subject to PNS attacks (the 20 km results are not shown because they are very similar). For each configuration studied, there is a clear separation between the decoy state efficiencies and the signal state efficiencies. This is because [PERSON] inadvertently blocks most of the decoy state pulses since the majority of them contain only a single photon due to its lower MPN. Conversely, relatively few signal state pulses are  
Figure 3: Variation in weak coherent pulse Mean Photon Number (MPN) emitted from [PERSON]’s modeled laser due to expected fluctuations and performance limitations in the modeled decoy state generator (i.e., the variable optical attenuator). The upper and lower Prediction Intervals (PI) are shown to bound the expected performance. blocked since the higher MPN generates more multi-photon pulses. Thus, Eve significantly reduces the decoy state efficiency. This behavior is precisely why the decoy state protocol requires two different MPNs in otherwise indistinguishable states (i.e., [PERSON] is unaware of the pulse type she is acting upon, since any of the pulses (signal, decoy, or vacuum) _could_ consist of 0, 1, or \(\geq\)2 photons). Additionally, as can be seen in the downward trending efficiencies, these responses are tempered by the protocol's occurrence percentages and [PERSON]'s gain matching. Figure 4: Simulation results are shown for the for the 50 km decoy state protocol configurations examined when operating under normal conditions. In each configuration studied, the signal and decoy state efficiencies are the same \(\eta^{signal}=\eta^{decoy}\pm\Delta\) (within expected variation tolerances). Figure 5: Simulation results are shown for the 50 km decoy state protocol configurations examined when subject to PNS attacks. In each configuration studied, the signal and decoy state efficiencies are statistically different \(\eta^{signal}\
eq\eta^{decoy}\pm\Delta\) (outside expected variation tolerances). In the 40 configurations considered at both 20 and 50 km distances, the PNS attack was successfully detected in all 40,000 trials (i.e., 1000 trials in each of the 40 PNS attack configurations simulated). For example, in the worst case scenario, when the signal and decoy state MPNs are closest (0.5 and 0.2) with the least amount of decoy states (99% signal, 0.5% decoy, and 0.5% vacuum) and the most loss (10 dB loss over the 50 km channel), there is very strong statistical evidence that the PNS attack will be detected because \(\eta^{signal}\
eq\eta^{decoy}\pm\Delta\). More specifically, based on 1000 simulations in the worst case configuration, the decoy state enabled QKD system has less than one in a thousand chance of not detecting the attack with a low probability of \(p<0.001\).