## 2 Photon-Subtracted Plug-and-Play DM-CVQKD
### Photon-Subtracted Plug-and-Play DM-CVQKD  
In what follows, we suggest a PS-based PP DM-CVQKD scheme. As shown in Figure 2b, different from the above scheme, after the Step 1 in PP DM-CVQKD scheme, the mode \(A_{0}\) would intersect with a vacuum state \(C_{0}\) at beamsplitter with transmittance \(\mu\). Afterwards, the state is split into two modes \(C\) and \(A_{1}\). The yielded tripartite state \(\rho_{A_{1}BCF}\) can be expressed as  
\[\rho_{A_{1}BCF}=U_{BS}[|\psi\rangle_{FBA_{0}}\langle\psi|_{FBA_{0}}\otimes|0 \rangle\langle 0|]U_{BS}^{\dagger}. \tag{2}\]  
The positive operator-valued measurement (POVM) \(\{\hat{\Pi}_{0},\hat{\Pi}_{1}\}\) is applied to the photon number resolving detector (PNRD) so that the PNRD can be used to measure mode \(C\)[49]. The subtracted photon number \(m\) depends on \(\hat{\Pi}_{1}=|m\rangle\langle m|\) and the operation succeeds only when the POVM element \(\hat{\Pi}_{1}\) clicks. After the PS the state \(\rho_{A_{1}BCF}^{\hat{\Pi}_{1}}\) reads  
\[\rho_{A_{1}BCF}^{\hat{\Pi}_{1}}=\frac{\text{tr}_{C}(\hat{\Pi}_{1}\rho_{A_{1}BCF })}{p_{(m)}^{\hat{\Pi}_{1}}}, \tag{3}\]  
where \(\text{tr}_{X}(\cdot)\) represents the partial trace of the multi-mode state and \(P_{(m)}^{\hat{\Pi}_{1}}\) is the success probability of subtracting \(m\) photons, as shown in Appendix A. By applying the photon subtraction operation, the covariance matrix of state \(\rho_{A_{1}BF}\) reads  
\[\Gamma_{A_{1}BF}=\begin{pmatrix}X^{\prime}\mathbb{I}&Z_{d}^{\prime}\sigma_{z} \\ Z_{d}^{\prime}\sigma_{z}&Y^{\prime}\mathbb{I}\end{pmatrix} \tag{4}\]  
where \(X^{\prime}\), \(Y^{\prime}\), and \(Z_{d}^{\prime}\) are calculated in Appendix A. The subsequent quantum state transfer, detection, and post-processing procedure is the same as the PP DM-CVQKD scheme. After passing through the unsafe channel, the covariance matrix of state \(\rho_{A_{2}BF}\) can be represented as  
\[\begin{split}\Gamma_{A_{2}BF}&=\begin{pmatrix}a\mathbb{I} &c\sigma_{z}\\ c\sigma_{z}&b\mathbb{I}\end{pmatrix}\\ &=\begin{pmatrix}X^{\prime}\mathbb{I}&\sqrt{\mathbb{I}}Z_{d}^{\prime}\sigma_{z }\\ \sqrt{\mathbb{I}}Z_{d}^{\prime}\sigma_{z}&T(Y^{\prime}+\chi_{line}+\zeta) \mathbb{I}\end{pmatrix},\end{split} \tag{5}\]where \(\chi_{line}=1/T-1+\varepsilon\), and \(\varepsilon\) represents the excess noise of channel.  
We can derive from [54] that, after photon subtraction operation with a given \(V_{M}\), the \(Z^{\prime}_{d}\) of the two-mode squeezed vacuum (TMSV) state is the same for four-state and eight-state modulation. Namely, the four-state and eight-state modulation protocols demonstrate the similar performance after photon subtraction, so that we take the four-state modulation protocol into account without loss of generality while performing the PS-based PP scheme in the DM-CVQKD. Notice that, after the PS operation, the entanglement degree of the coherent state is strengthened [13].  
Finally, the asymptotic secret key rate of reverse reconciliation against collective attacks can be calculated as  
\[K_{asym}=P_{(m)}^{\mathbb{f}_{1}}[\beta(I(A:B)-S(E:A)], \tag{6}\]  
where \(\beta\) is the reconciliation efficiency, \(S(E:A)\) is the Holevo bound of the mutual information between [PERSON] and [PERSON], and \(I(A:B)\) is the Shannon mutual information between [PERSON] and [PERSON] following the photon subtraction.