## 4 The Photon Number Splitting Attack
As mentioned before, the eavesdropper blocks the one-photon states and she performs a channel substitution to adjust the transmittance of the channel, \(T_{AB}\). However, this activity produces error gains in the single- and double-detection events that are verifiable by [PERSON]. The QPEG after [PERSON] blocks the one-photon states can be written as \(\Delta Q=Q_{1}\), where \(Q_{1}\) is the gain of the one-photon states, and it must be computed for the single- and the double-detection events. The error gain is \(\Delta Q_{(+,+)}=Q_{1_{(+)}}^{2}=Q_{1}^{2}\) and \(\Delta Q_{(\pm,\mp)}=Q_{1}\cdot Q_{(-)}\) for double-detection events and single-detection events, respectively, where \(Q_{1_{(+)}}=(Y_{0}+\eta-Y_{0}\eta)\mu e^{-\mu}\), \(Q_{(-)}=e^{-\mu\eta}-Y_{0}\), \(\eta\) is the transmittance of the channel and the detectors at [PERSON]'s side of the one-photon states and \(Y_{0}\) is the background noise according to [8]. To remain hidden in the channel, the eavesdropper must adjust the transmittance, \(T_{AB}\), to achieve the two reference photonic gains, \(Q_{(+,+)}\) and \(Q_{(\pm,\mp)}\), for the double detection events and single detection events, respectively. Since \(Q_{(+,+)}\
eq Q_{(\pm,\mp)}\), [PERSON] can adjust \(T_{AB}\) to \(Q_{1}^{2}\) or \(Q_{1}\cdot Q_{(-)}\), but not both simultaneously. In other words, she cannot fulfill the conditions \(\Delta Q_{(+,+)}=0\) and \(\Delta Q_{(\pm,\mp)}=0\); thus, the attack becomes detectable. If the eavesdropper adjusts \(T_{AB}\), so that it produces a photonic deviation in one or in both gains, she will introduce a detectable QBER to the system. As a consequence, [PERSON] knows that she must not change \(T_{AB}\), otherwise she will be detected. Now, the QBER that [PERSON] produces is \(\frac{0.5Q_{0}+0.5^{2}Q_{1}+0.5^{3}Q_{2}+...}{Q_{0}+Q_{1}+Q_{2}+...}\), because the QBER of single-detection events is \(0.5^{2}\), as in the BB84. In contrast, in absence of the attack, the QBER of the system is given by \(\frac{0.5Q_{0}+e_{d}(Q_{1}+Q_{2}+Q_{3}+...)}{Q_{0}+Q_{1}+Q_{2}+...}\), where \(e_{d}\) is the detection error according to [8]. Since the probability of getting a (compatible) matching measurement from the non-orthogonal double detection events is \(0.5^{2}\), we derived the error rate of the non-orthogonal double detection events as \(\frac{0.5(Q_{0}+Q_{1})+0.5^{2}Q_{2}+0.5^{3}Q_{3}+...}{Q_{0}+Q_{1}+Q_{2}+...}\). The QBER from the multi-photonic non-orthogonal states decrease by half for each copy of quantum states in [PERSON]'s memory. By contrast, the multi-photonic parallel states do not contribute to increasing the QBER, because [PERSON] reveals the basis measurements he used. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Quantum Flow** & **[PERSON]** & **Bob** \\ \hline \(Q_{(-,-)}\) & \(e^{-2\mu}\) & \((e^{-\mu\eta_{BT}}-Y_{0})^{2}\) \\ \(Q_{(+,+)}\) & \((1-e^{-\mu})^{2}\) & \((Y_{0}+1-e^{-\mu\eta_{BT}})^{2}\) \\ \(Q_{(\pm,\mp)}\) & \(2e^{-\mu}(1-e^{-\mu})\) & \(2(e^{-\mu\eta_{BT}}-Y_{0})(Y_{0}+1-e^{-\mu\eta_{BT}})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The reference photonic gain of the double \((+,+)\) detection events is written as \(Q_{(+,+)}\), and the gain of the single detection events \((\pm,\mp)\) is represented as \(Q_{(\pm,\mp)}\), where \(Y_{0}\) is the background noise, \(\mu\) is the mean value of the photonic source and \(\eta_{BT}\) is the overall efficiency at Bob’s side.