# One-Photon-Interference Quantum Secure Direct Communication
## Appendix B Details of Security Analysis
We use \(|\Psi^{\pm}\rangle,|\Phi^{\pm}\rangle\) to mark Bell states, where  
\[|\Psi^{-}_{AB}\rangle =(|-+\rangle_{AB}-|+-\rangle_{AB})/\sqrt{2}\] (A2) \[=(|01\rangle_{AB}-|10\rangle_{AB})/\sqrt{2},\] \[|\Psi^{+}_{AB}\rangle =(|++\rangle_{AB}-|--\rangle_{AB})/\sqrt{2}\] \[=(|01\rangle_{AB}+|10\rangle_{AB})/\sqrt{2},\] \[|\Phi^{-}_{AB}\rangle =(|+\rangle_{AB}+|-+\rangle_{AB})/\sqrt{2}\] \[=(|00\rangle_{AB}-|11\rangle_{AB})/\sqrt{2},\] \[|\Phi^{+}_{AB}\rangle =(|++\rangle_{AB}+|--\rangle_{AB})/\sqrt{2}\] \[=(|00\rangle_{AB}+|11\rangle_{AB})/\sqrt{2}.\]  
We define \(|\pm\rangle_{A(B)}\equiv(|0\rangle_{A(B)}\pm|1\rangle_{A(B)})/\sqrt{2}\), where \(|0\rangle_{A(B)}\) represents the vacuum state, and \(|1\rangle_{A(B)}\) represents the single-photon state for optical pulse A (B). A coherent state \(|\alpha\rangle\) is defined as  
\[|\alpha\rangle\equiv e^{-\frac{|\alpha|^{2}}{2}}\sum_{n=0}^{\infty}\frac{( \alpha)^{n}}{\sqrt{n!}}|n\rangle.\] (A3)And we use the following transformation matrix corresponding to a 50:50 beam splitter:  
\[\left( \begin{array}{c}c^{\dagger}\\ d^{\dagger}\end{array} \right)=\frac{1}{\sqrt{2}}\bigg{(} \begin{array}{cc}1&1 \\ 1&-1\end{array} \bigg{)}\bigg{(} \begin{array}{c}a^{\dagger}\\ b^{\dagger}\end{array} \bigg{)}, \tag{10}\]  
where \(a^{\dagger}\), \(b^{\dagger}\) represents the input optical pulse of a, b, respectively, and \(c^{\dagger}\), \(d^{\dagger}\) represents the output optical pulse of C, D, respectively. Next, we analyze the eavesdropping process and calculate \(I(A:E)\). We first focus on the joint state \(\rho_{AB}\) shared by [PERSON] and [PERSON] in the **Step 3** after [PERSON]'s measurement. If \(\rho_{AB}\) is a pure state, it can be considered that there is no information leakage. Without the loss of generality, we can assume that [PERSON] and [PERSON] currently share the entangled state \(|\Psi^{+}_{AB}\rangle\). If eavesdropping occurs, \(\rho_{AB}\) will become a mixed state. To facilitate the calculation, we can assume that [PERSON] and [PERSON] use some of the same unitary operations to change \(\rho_{AB}\) into a diagonal matrix [16], then  
\[ \begin{split}\rho_{AB}=&\lambda_{1}|\Psi^{-}_{AB}\rangle \langle\Psi^{-}_{AB}|+\lambda_{2}|\Psi^{+}_{AB}\rangle\langle\Psi^{+}_{AB}|\\ &+\lambda_{3}|\[PERSON]^{-}_{AB}\rangle\langle\[PERSON]^{-}_{AB}|+\lambda_{4}| \[PERSON]^{+}_{AB}\rangle\langle\[PERSON]^{+}_{AB}|.\end{split}  \tag{11}\]  
The parameters satisfies \(\sum_{i=1}^{4}\lambda_{i}=1\). We assume that [PERSON] uses an auxiliary system \(|E\rangle\) for eavesdropping and performs a collective attack, which can be extended to coherent attacks through the quantum de [PERSON] theorem [56]. Then, a purification state \(|\phi_{ABE}\rangle\) of \(\rho_{AB}\) and \(|E\rangle\) can be written as  
\[|\phi_{ABE}\rangle=\sqrt{\lambda_{1}}|\Psi^{-}_{AB}\rangle|E_{1}\rangle+\sqrt{ \lambda_{2}}|\Psi^{+}_{AB}\rangle|E_{2}\rangle+\sqrt{\lambda_{3}}|\Phi^{-}_{ AB}\rangle|E_{3}\rangle+\sqrt{\lambda_{4}}|\Phi^{+}_{AB}\rangle|E_{4}\rangle, \tag{12}\]  
where \(|E_{1}\rangle\), \(|E_{2}\rangle\), \(|E_{3}\rangle\) and \(|E_{4}\rangle\) are the orthogonal states of [PERSON]'s auxiliary system \(|E\rangle\). If [PERSON] and [PERSON] do not share a Bell state \(|\Psi^{+}_{AB}\rangle\), but instead share a Bell state \(|\Phi^{+}_{AB}\rangle\) or \(|\Phi^{-}_{AB}\rangle\), and they perform measurements in the Z basis, which will lead to a phase-error rate. Thus, the phase-error rate \(E^{Z}_{u}\) is given by  
\[E^{Z}_{u}=\lambda_{3}+\lambda_{4}. \tag{13}\]  
[PERSON] can intercepts the qubits from [PERSON] and acquires system AE:  
\[\rho_{AE}=\mathit{Tr}_{B}(|\phi_{ABE}\rangle\langle\phi_{ABE}|)=\frac{1}{2}(| \varphi_{1}\rangle\langle\varphi_{1}|+|\varphi_{2}\rangle\langle\varphi_{2}|), \tag{14}\]  
where  
\[