## 2 Quantum Key Distribution
### Detecting PNS Attacks  
The decoy state protocol is designed to detect PNS attacks by comparing the signal and decoy states during quantum exchange, and specifically, the photon number dependent yields of the signal state \(\gamma_{n}^{signal}\), the decoy state \(Y_{n}^{decoy}\), and the expected yield \(\gamma_{n}^{expected}\) are compared in the security condition [24].  
\[Y_{n}^{signal}=Y_{n}^{decoy}=Y_{n}^{expected}, \tag{3}\]  
where \(Y_{n}\) represents the conditional probability that [PERSON] detects a pulse given [PERSON] sent an \(n\)-photon pulse. Ideally, \(Y_{n}^{signal}\) and \(Y_{n}^{decoy}\) are measured, while \(\gamma_{n}^{expected}\) is calculated (or estimated) from a known quantum channel efficiency \(\eta\).  
\[Y_{n}^{expected}=Y_{0}+\eta_{n}-Y_{0}\eta_{n}\approx Y_{0}+\eta_{n} \tag{4}\]  
where \(Y_{0}\) is the measured dark count rate and \(\eta_{n}=1-(1-\eta)^{n}\) is the photon number specific efficiency based on the number of photons, \(n\), in each pulse and the measured quantum efficiency \(\eta\). Note, the joint probability \(Y_{0}\eta_{n}\) is typically disregarded because it is insignificant compared to \(Y_{0}\) and \(\eta_{n}\).  
As indicated by Equation (3), under normal operational conditions (i.e., when no PNS attacks are occurring), the signal and decoy state yields should be the same as the expected photon number dependent yield; for example, \(Y_{1}^{signal}=Y_{1}^{decoy}=Y_{1}^{expected}\). This security condition should always be true for a decoy state enabled QKD architecture because the signal and decoy state yields are primarily  
\begin{table}
\begin{tabular}{c p{142.3 pt}} \hline \hline
**Parameter** & **Description** \\ \hline \multirow{3}{*}{\(q\)} & The protocol efficiency represents the overall efficiency of the QKD protocol (e.g., \(q<1\)). \\  & For example, in the classical BB84 protocol shown in Table 1, 50\% of the detections will be sifted out because of [PERSON]’s random choice of basis measurement. \\ \hline \multirow{3}{*}{\(Q_{1}\)} & The estimated gain of pulses emitted by [PERSON] with one photon (i.e., the single photons prepared by [PERSON] and successfully measured by [PERSON]). This value is typically calculated (or bounded) by several operational parameters such as, \(\
u\), \(Q_{\mu}\), and \(Y_{0}\). \\ \hline \multirow{3}{*}{\(e_{1}\)} & The estimated error rate associated with pulses emitted by [PERSON] with a single photon. This value is typically calculated (or bounded) by several operational parameters such as, \(\
u\), \(E_{\mu}\), \(Q_{\mu}\) and \(Y_{0}\). \\ \hline \multirow{3}{*}{\(Q_{\mu}\)} & The gain of the signal state is calculated from system measurements, where \(Q_{\mu}=\frac{Number\cdot\text{of signal state detections}}{Total number\cdot\text{of sent signal state pulses}}\) \\ \hline \multirow{3}{*}{\(E_{\mu}\)} & The QBER of the signal state is calculated from system measurements, where \(\frac{Number\cdot\text{of signal state bit rates}}{Total number\cdot\text{of matched signal state detections}}\) \\ \hline \multirow{3}{*}{\(f\left(E_{\mu}\right)\)} & The error reconciliation efficiency is dependent upon the signal state QBER \(E_{\mu}\) with typical values of \(\leq 1.15\) for QBERs \(\leq 5\%\). \\ \hline \(H_{2}\left(\left\{e_{1},E_{\mu}\right\}\right)\) & Uncertainty in the error rate \(e_{1}\) or \(E_{\mu}\) is calculated using [PERSON]’s binary entropy limit [15]. \\ \hline \hline \end{tabular}
\end{table}
Table 3: Secure Key Rate Parameters [7].  
based on the fixed quantum efficiency \(\eta\) and not the state type. If ever \(Y_{1}^{signal}\
eq Y_{1}^{decoy}\
eq Y_{1}^{expected}\), an eavesdropper is assumed to be actively listening on the key distribution channel and the secret key is considered compromised. While [24] also proposes an error-based condition \(e_{n}^{signal}=e_{n}^{decoy}=e_{n}^{expected}\), it is not considered in this work since the PNS attack does not introduce errors [19].