# Improving Eight-State Continuous Variable Quantum Key Distribution by Applying Photon Subtraction
## 2 Eight-State CVQKD with Photon Subtraction Operation
### Eight-State CVQKD Protocol  
In the eight-state protocol, the eight coherent displaced states \(\mid\alpha_{k}^{8}\rangle\) prepared by [PERSON] are given by  
\[\mid\alpha_{k}^{8}\rangle=\mid\alpha e^{\frac{8\pi}{4}}\rangle\,, \tag{1}\]  
where \(k\) is the number of coherent displacement states (ranging from 0 to 7), and \(\alpha\) is a positive real number obtained from the modulation variance of the coherent state. After that, [PERSON] sends one state to [PERSON], the state being randomly obtained from eight states. When this state reaches [PERSON]'s station by the quantum channel, [PERSON] measures it using a homodyne or heterodyne detector. [PERSON]'s received noise variance can be expressed as \((1+\tau\epsilon)\,N_{0}\). This noise is mainly generated by the interaction of the excess noise \(\epsilon\) and the transmission efficiency \(\tau\), where \(N_{0}\) is expressed as the shot-noise limit. Therefore, the total noise received at the [PERSON]'s station can be given by \(X_{line}=1/\tau+\epsilon-1\)[18]. Finally, they achieve the final secret key by taking advantage of privacy amplification and error correction. In the EB protocol, [PERSON] needs to prepare a pure two-mode quantum state. The received matrix \(\rho_{8}\) can be expressed as  
\[ \begin{split}\rho_{8}&=\frac{1}{8}\sum_{k=0}^{7} \mid\alpha_{k}^{8}\rangle\,\langle\alpha_{k}^{8}\mid\\ &=\sum_{k=0}^{7}\lambda_{k}\mid\phi_{k}\rangle\,\langle\phi_{k} \mid\,,\end{split}  \tag{2}\]  
where  
\[ \begin{split}\lambda_{0(4)}&=\frac{1}{4}e^{-\alpha ^{2}}\left[cosh(\alpha^{2})+cos(\alpha^{2})\pm 2 cos(\frac{\alpha^{2}}{ \sqrt{2}})cosh(\frac{\alpha^{2}}{\sqrt{2}})\right],\\ \lambda_{1(5)}&=\frac{1}{4}e^{-\alpha^{2}}\left[ sinh(\alpha^{2})+sin(\alpha^{2})\pm\sqrt{2}cos(\frac{\alpha^{2}}{\sqrt{2}}) sinh(\frac{\alpha^{2}}{\sqrt{2}})\pm\sqrt{2}sin(\frac{\alpha^{2}}{\sqrt{2}}) cosh(\frac{\alpha^{2}}{\sqrt{2}})\right],\\ \lambda_{2(6)}&=\frac{1}{4}e^{-\alpha^{2}}\left[ sinh(\alpha^{2})-sin(\alpha^{2})\pm 2 sin(\frac{\alpha^{2}}{\sqrt{2}})sinh(\frac{ \alpha^{2}}{\sqrt{2}})\right],\\ \lambda_{3(7)}&=\frac{1}{4}e^{-\alpha^{2}}\left[ sinh(\alpha^{2})-sin(\alpha^{2})\mp\sqrt{2}cos(\frac{\alpha^{2}}{\sqrt{2}}) sinh(\frac{\alpha^{2}}{\sqrt{2}})\pm\sqrt{2}sin(\frac{\alpha^{2}}{\sqrt{2}}) cosh(\frac{\alpha^{2}}{\sqrt{2}})\right],\end{split}  \tag{3}\]  
and  
\[\mid\phi_{k}\rangle=\frac{e^{-\alpha^{2}/2}}{\sqrt{\lambda_{k}}}\sum_{n=0}^{ \infty}e\frac{\alpha^{8n+k}}{\sqrt{(8n+k)!}}\mid sn+k\rangle\,,k\in\{0,1,2...,7\}\,. \tag{4}\]When we apply the annihilation operator \(\hat{\alpha}\) to \(\mid\phi_{k}\rangle\), we obtain \(\hat{\alpha}\mid\phi_{0}\rangle=-\alpha\sqrt{\lambda_{7}}/\sqrt{\lambda_{0}} \mid\phi_{7}\rangle\), and \(\hat{\alpha}\mid\phi_{k}\rangle=\alpha\sqrt{\lambda_{k-1}}/\sqrt{\lambda_{k}} \mid\phi_{k-1}\rangle\,,k\in\{0,1,2...,7\}\). Therefore, the state \(\rho_{8}\) can be further processed as  
\[ \begin{split}\mid\Psi_{8}\rangle&=\sum_{k=0}^{7} \sqrt{\lambda_{k}}\mid\phi_{k}\rangle\mid\phi_{k}\rangle\\ &=\frac{1}{4}\sum_{k=0}^{7}\mid\psi_{k}\rangle\mid a_{k}^{8} \rangle\,,\end{split}  \tag{5}\]  
where the states \(\mid\psi_{k}\rangle\) are the orthogonal non-Gaussian states  
\[\mid\psi_{k}\rangle=\frac{1}{2}\sum_{m=0}^{7}e^{\frac{i(4k+1)m\pi}{4}}\mid\phi _{m}\rangle\,,k\in\{0,1,2...,7\}\,. \tag{6}\]  
Among these, [PERSON] prepares the quantum entangled states \(\mid\psi_{k}\rangle\,\langle\psi_{k}\mid(k\in\{0,1,2...,7\})\) and performs projection measurements, one of which enters the first half of \(\mid\Psi_{8}\rangle\), while the other half is projected to the eight coherent states \(\mid\phi_{k}\rangle\,(k\in\{0,1,2...,7\})\) with equal probability. Now, we analyze the covariance matrix \(\gamma_{8}\) of state \(\mid\Psi_{8}\rangle\), which is given by  
\[\gamma_{8}= \begin{bmatrix}XL_{2}&Z_{8}\sigma_{z}\\ Z_{8}\sigma_{z}&YL_{2}\end{bmatrix} \,, \tag{7}\]  
where the \(L_{2}\) represents a diagonal matrix of \([1,1]\), \(\sigma_{z}\) represents a diagonal matrix of \([1,-1]\), and  
\[