## 2 Quantum Key Distribution
### Unconditionally Secure Key Generation  
While the decoy state protocol was introduced to detect PNS attacks, to date it has been primarily used to increase unconditionally secure key rates. More specifically, decoy state research has focused on understanding and bounding [PERSON]'s single photon generation rate, \(Q_{1}\), as it pertains to the secret key generation rate \(R\)[25].  
\[R\geq~{}q\{Q_{1}[1~{}-~{}H_{2}(e_{1})]-~{}Q_{\mu}~{}f(E_{\mu})H_{2}(E_{\mu})\}, \tag{2}\]  
where \(q\) is the protocol efficiency (e.g., \(<\)1), \(Q_{1}\) is the estimated single photon contribution, \(e_{1}\) is the estimated error rate of single photon detections, \(Q_{\mu}\) is the signal state gain, \(E_{\mu}\) is the signal state QBER, \(f\big{(}E_{\mu}\big{)}\) is the error reconciliation efficiency, and \(H_{2}(\{E_{\mu},e_{1}\})\) is [PERSON]'s binary information function [15]. In a general sense, the positive contribution of Equation (2) accounts for pulses emitted by Alice containing a single photon (i.e., \(Q_{1}\))--those which can contribute to the QKD-generated \"unconditionally secure\" key rate \(R\), while the negative contribution accounts for insecure multi-photon pulses and errors which are also consider insecure contributions [25]. More specifically, the parameters of Equation (2) are described in Table 3. For a more detailed, yet readily accessible discussion of these parameters, please see [7]. For more comprehensive treatments, please see the references listed below.  
\begin{table}
\begin{tabular}{c l c c} \hline \hline
**State** & **Purpose** & **MPN** & **Occurrence Percentage** \\ \hline \multirow{4}{*}{Signal \(\mu\)} & The signal state is used to generate secret key and facilitates & & \\  & improved performance by using a higher MPN (i.e., 0.5 is & \\  & greater than the value 0.1 typically employed in non-decoy & \\  & state protocol QKD systems). & & \\ \hline \multirow{2}{*}{Decoy \(\
u\)} & The decoy state is used to increase the likelihood of detecting & & \\  & unauthorized eavesdropping on the quantum channel & & \\  & through statistical differential analysis with the signal state. & & \\ \hline \multirow{2}{*}{Vacuum \(Y_{0}\)} & The vacuum state is used to determine the noise on the & & \\  & quantum channel known as the “dark count” (i.e., detections & 0.0 & 10\% \\ \cline{1-1}  & when no photons are sent). & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Example Decoy State Protocol Configuration.  
In their 2005 work, [PERSON] et al. optimized the single photon generate rate \(Q_{1}\) while simultaneously bounding the error rates \(e_{1}\) and \(E_{\mu}\) to maximize the secure key rate \(R\)[25]. More specifically to the decoy state protocol configuration, this optimization results in recommend signal and decoy state MPNs of \(\mu\cong 0.5\) and \(\
u\cong 0.1\) because of implementation limitations in classical laser sources (as described in Section 2.2) [25]. Following this seminal work, many others have studied this optimization problem to more fully understand and bound the single photon estimate \(Q_{1}\), the negative impact of associated error rates \(e_{1}\) and \(E_{\mu}\), and expected fluctuations in commercially available laser sources [31; 32; 33; 34; 35; 36; 37; 38; 39; 40]. Additionally, considerations for finite key size statistics (i.e., limitations due to the number of detections) have been carefully investigated by others [41; 28; 42; 43]. Here, we also note a recent work that benchmarks secret key rates over ideal quantum communication channels to include the decoy state protocol [44]. In addition to these works, several practically-oriented experimental demonstrations have been accomplished as detailed in Table 4 (discussed in Section 3).