## Appendix A Derivation of the Asymptotic Secret Key Rate  
In this section, we consider the characteristics of the transmitted states in the PS-based PP DM-CVQKD scheme and derive the asymptotic secret key rate of it. Here, we take the asymptotic secret key rate of the four-state PP DM-CVQKD protocol into account without loss of generality. We assume that [PERSON] and [PERSON] perform heterodyne detection and homodyne detection, respectively. In the EB scheme of original DM-CVQKD, [PERSON] prepares a two-mode entangled state \(|\psi\rangle_{A_{0}B}\) given by  
\[ \begin{split}|\psi\rangle_{A_{0}B}&=\sum_{k=0}^{d-1 }\sqrt{\lambda_{k}}|\phi_{k}\rangle|\phi_{k}\rangle\\ &=\frac{2}{d}\sum_{k=0}^{d-1}|\psi_{k}\rangle|a_{k}^{d}\rangle, \end{split}  \tag{10}\]  
where \(d=4\) for four-state protocol and \(d=8\) for eight-state protocol with  
\[|\psi_{k}\rangle=\frac{1}{2}\sum_{r=0}^{d-1}e^{i(\frac{d}{2}k+1)r\pi/8}|\phi_{ r}\rangle, \tag{11}\]  
and  
\[|\phi_{r}\rangle=\frac{e^{-\alpha^{2}/2}}{\lambda_{r}}\sum_{n=0}^{\infty}(-1) ^{n}\frac{\alpha^{dn+r}}{\sqrt{(dn+r)!}}|dn+r\rangle,r=0,1,\ldots,d-1. \tag{12}\]  
In four-state protocol, \(\lambda_{r}\) can be calculated as  
\[ \begin{split}\lambda_{0,2}&=\frac{1}{2}e^{-\alpha ^{2}}\left[\cosh(\alpha^{2})\pm\cos(\alpha^{2})\right],\\ \lambda_{1,3}&=\frac{1}{2}e^{-\alpha^{2}}\left[ \sinh(\alpha^{2})\pm\sin(\alpha^{2})\right],\end{split}  \tag{13}\]  
whereas, in eight-state protocol, it can be expressed as  
\[ \begin{split}\lambda_{0,4}&=\frac{1}{4}e^{-\alpha ^{2}}\left[\cosh(\alpha^{2})+\cos(\alpha^{2})\pm 2\cos(\frac{\alpha^{2}}{ \sqrt{2}})\cosh(\frac{\alpha^{2}}{\sqrt{2}})\right],\\ \lambda_{1,5}&=\frac{1}{4}e^{-\alpha^{2}}\left[ \sinh(\alpha^{2})+\sin(\alpha^{2})\pm\sqrt{2}\cos(\frac{\alpha^{2}}{\sqrt{2}}) \sinh(\frac{\alpha^{2}}{\sqrt{2}})\right.\\ &\left.\pm\sqrt{2}\sin(\frac{\alpha^{2}}{\sqrt{2}})\cosh(\frac{ \alpha^{2}}{\sqrt{2}})\right],\\ \lambda_{2,6}&=\frac{1}{4}e^{-\alpha^{2}}\left[ \sinh(\alpha^{2})-\sin(\alpha^{2})\pm 2\sin(\frac{\alpha^{2}}{\sqrt{2}})\sinh( \frac{\alpha^{2}}{\sqrt{2}})\right],\\ \lambda_{3,7}&=\frac{1}{4}e^{-\alpha^{2}}\left[ \sinh(\alpha^{2})-\sin(\alpha^{2})\mp\sqrt{2}\cos(\frac{\alpha^{2}}{\sqrt{2}}) \sinh(\frac{\alpha^{2}}{\sqrt{2}})\right.\\ &\left.\pm\sqrt{2}\sin(\frac{\alpha^{2}}{\sqrt{2}})\cosh(\frac{ \alpha^{2}}{\sqrt{2}})\right].\end{split}  \tag{14}\]  
The covariance matrix \(\Gamma_{A_{0}B}\) of state \(|\psi\rangle_{A_{0}B}\) can be expressed as  
\[\Gamma_{A_{0}B}= \begin{pmatrix}X\mathbb{I}&Z_{d}\sigma_{z}\\ Z_{d}\sigma_{z}&Y\mathbb{I}\end{pmatrix} , \tag{15}\]where \(\mathbb{I}=\text{diag}(1,1)\), \(\sigma_{z}=\text{diag}(1,-1)\) and  
\[ \begin{split} X&=\langle\psi_{d}|2a^{\dagger}a+1| \psi_{d}\rangle=1+2\alpha^{2},\\ Y&=\langle\psi_{d}|2b^{\dagger}b+1|\psi_{d}\rangle=1+2 \alpha^{2},\\ Z_{d}&=\langle\psi_{d}|ab+a^{\dagger}b^{\dagger}| \psi_{d}\rangle=2\alpha^{2}\sum_{l=0}^{d-1}\frac{\lambda_{l-1}^{3/2}}{\lambda_ {l}^{1/2}}.\end{split}  \tag{10}\]  
Here, \(a,a^{\dagger}\), and \(b,b^{\dagger}\) are the annihilation and creation operators, respectively, on [PERSON]'s and [PERSON]'s mode. The success probability of subtracting \(m\) photons \(P_{(m)}^{\Pi_{1}}\) can be calculated as  
\[ \begin{split} P_{(m)}^{\Pi_{1}}&=\text{tr}_{A_{1}BCF }(\hat{\Pi}_{1}\rho_{A_{1}BCF})\\ &=(1-\xi^{2})\sum_{n=m}^{\infty}C_{n}^{m}\xi^{2n}(1-\mu)^{m}\mu^{ n-m}\\ &=(1-\xi^{2})\bigg{(}\frac{1-\mu}{\mu}\bigg{)}^{m}\sum_{n=m}^{ \infty}C_{n}^{m}(\xi^{2}\mu)^{n}\\ &=(1-\xi^{2})\xi^{2m}\frac{(1-\mu)^{m}}{(1-\xi^{2}\mu)^{m+1}}, \end{split}  \tag{11}\]  
where \(C_{n}^{m}\) is a combination number (\(n\geq m\)) and \(\xi=\sqrt{\frac{V-1}{V+1}}\). After the \(m\)-photon subtraction operation, the covariance matrix \(\Gamma_{A_{1}BF}\) is given in Equation (4), and the parameters are given by  
\[ \begin{split} X^{\prime}&=\frac{\mu\xi^{2}+2m+1}{1 -\mu\xi^{2}},\\ Y^{\prime}&=\frac{\mu\xi^{2}(2m+1)+1}{1-\mu\xi^{2}},\\ Z_{d}^{\prime}&=\frac{\sqrt{\mu}\xi(m+1)}{1-\mu\xi^{2 }}.\end{split}