# One-Photon-Interference Quantum Secure Direct Communication
## Appendix C Derivation Details of Parameterized OPI-QSDC Protocol  
To determine the performance of OPI-QSDC protocol, we use the channel model in [42], which transfer \(a^{\dagger}\), \(b^{\dagger}\) according to  
\[ \begin{split} a^{\dagger}\rightarrow&\sqrt{\eta}a^{ \dagger}+\sqrt{1-\eta}a^{\dagger}_{\perp},\\ b^{\dagger}\rightarrow&\sqrt{\eta}b^{\dagger}+ \sqrt{1-\eta}b^{\dagger}_{\perp},\end{split}  \tag{114}\]  
where \(\eta=\eta_{d}\sqrt{\eta_{c}}\) is the transmittance, and \(\eta_{d}\), \(\eta_{c}\) are the detection efficiencies and the channel losses function, respectively. The channel losses function is \(\eta_{c}=10^{\wedge}(-\zeta d/10)\), where \(\zeta\) represents the attenuation coefficient, and \(d\) represents the transmission distance. The transformation matrix of the beam splitter is the same as Equation (114), which leads to the the transformations of the coherent state as  
\[|\sqrt{u}e^{i\phi_{a}}\rangle_{a}|\sqrt{u}e^{i\phi_{b}}\rangle_{b}\rightarrow| \sqrt{\frac{\eta u}{2}}(e^{i\phi_{a}}+e^{i\phi_{b}})\rangle_{C}|\sqrt{\frac{ \eta u}{2}}(e^{i\phi_{a}}-e^{i\phi_{b}})\rangle_{D}. \tag{115}\]Here, \(\phi_{a},\phi_{b}\in\{0,\pi\}\). We consider the case where only detector \(\mathrm{D}_{0}\) click, i.e., \(\phi_{a}=\phi_{b}\). To model the phase mismatch between [PERSON] and [PERSON]'s signals arriving at [PERSON], we introduce a parameter \(\delta\equiv\phi_{b}-\phi_{a}\). We first consider the case when there is only one photon in the channel \((a^{\mathrm{t}}+e^{i\delta}b^{\mathrm{t}})|00\rangle_{ab}\). Hence the detection probabilities of [PERSON] are  
\[ \begin{split}& q_{1}^{0}=1-\eta,\\ & q_{1}^{\mathrm{C}}=\eta\cos^{2}(\frac{\delta}{2}),\\ & q_{1}^{\mathrm{D}}=\eta\sin^{2}(\frac{\delta}{2}),\\ & q_{1}^{\mathrm{C,D}}=0,\end{split}  \tag{108}\]  
where \(q_{1}^{0}\), \(q_{1}^{\mathrm{C}}\), \(q_{1}^{\mathrm{D}}\) and \(q_{1}^{\mathrm{C,D}}\) are the probabilities for no click, \(\mathrm{D}_{0}\) click, \(\mathrm{D}_{1}\) click an double click, respectively. Then we consider the case when there is \(n\) photons in the channel \((a^{\mathrm{t}}+e^{i\delta}b^{\mathrm{t}})^{n}|00\rangle_{ab}\), for which we can regard as \(n\) identical and independent click events of the one-photon case [42]. We take into the effects of the detector dark counts \(p_{d}\); thus, the detection probabilities of the detector \(\mathrm{D}_{0}\) and \(\mathrm{D}_{1}\) are  
\[ \begin{split} q_{n}^{\mathrm{C}}=&\ p_{d}(1-p_{d})( 1-\eta)^{n}+(1-p_{d})\{[1-\eta+\eta\cos^{2}(\frac{\delta}{2})]^{n}-(1-\eta)^{n }\},\\ & q_{n}^{\mathrm{D}}=&\ p_{d}(1-p_{d})(1-\eta)^{n}+( 1-p_{d})\{[1-\eta+\eta\sin^{2}(\frac{\delta}{2})]^{n}-(1-\eta)^{n}\}.\end{split}  \tag{109}\]  
Based on the above equation, the yield \(\gamma_{n}^{\mathrm{C}}\) of the \(n\)-photon state is given by  
\[ \begin{split}\gamma_{n}^{\mathrm{C}}=&\ q_{n}^{ \mathrm{C}}+q_{n}^{\mathrm{D}}\\ =&\ 1-(1-2p_{d})(1-\eta)^{n}.\end{split}  \tag{110}\]  
Here, we take the first order of a small \(\delta\). Finally, we consider the case of both [PERSON] and [PERSON] to send the coherent state  
\[ \begin{split}&|\sqrt{\text{\it i}}e^{i\phi_{b}}\rangle_{a}|\sqrt{ \text{\it i}}e^{i\phi_{b}}\rangle_{b}\\ =&\ e^{-\text{\it i}}e^{\sqrt{\text{\it i}}e^{\text{ \it i}}(a^{\mathrm{t}}+e^{i\delta}b^{\mathrm{t}})}|00\rangle_{ab}\\ =&\ e^{-\text{\it i}}\sum_{n}\frac{(\sqrt{\text{\it i }}e^{i\phi_{b}})^{n}}{\sqrt{n!}}(a^{\mathrm{t}}+e^{i\delta}b^{\mathrm{t}})^{n} |00\rangle_{ab}.\end{split}