# Improving Eight-State Continuous Variable Quantum Key Distribution by Applying Photon Subtraction
## 2 Eight-State CVQKD with Photon Subtraction Operation
### Eight-State CVQKD with Photon Subtraction Operation  
In what follows, as shown in Figure 1, we suggest an improved eight-state CVQKD scheme by using the photon-subtraction operation. Here, [PERSON] takes advantage of a beam splitter (BS) to split modes \(B\) and \(C_{0}\)(vacuum state) into mode \(B_{1}\) and mode \(C\), where the states \(A\),\(C\), and \(B_{1}\) can be expressed as  
\[\rho_{ACB_{1}}=U_{BS}\left[\mid\Psi_{8}\rangle\,\langle\Psi_{8}\mid\otimes \mid 0\rangle\,\langle 0\mid\right]U_{BS}^{\dagger}. \tag{10}\]  
For the measurement of mode \(C\), [PERSON] gets \(\left\{\widehat{\Pi}_{0},\widehat{\Pi}_{1}\right\}\) by utilizing positive operator-valued measurement (POVM) in the photon-number-resolving detector (PNRD). The number of photon reductions is related to \(\widehat{\Pi}_{1}=\left|\ k\right\rangle\left\langle k\right.\). The modes \(A\) and \(B_{1}\) are kept by [PERSON] and [PERSON] when the POVM element \(\widehat{\Pi}_{1}\) clicks. The photon-subtracted state \(\rho_{AB_{1}}^{\widehat{\Pi}_{1}}\) is expressed by  
\[\rho_{AB_{1}}^{\widehat{\Pi}_{1}}=\frac{tr_{C}(\widehat{\Pi}_{1}\rho_{ACB_{1}} )}{tr_{ACB_{1}}(\widehat{\Pi}_{1}\rho_{ACB_{1}})}, \tag{11}\]  
where \(tr_{X}(\cdot)\) is a multimode quantum state part. Our success probability of subtracting \(k\) photons is denoted by \(P_{(k)}^{\widehat{\Pi}_{1}}\), which is given by  
\[\begin{split} P_{(k)}^{\widehat{\Pi}_{1}}&=tr_{ACB_ {1}}(\widehat{\Pi}_{1}\rho_{ACB_{1}})\\ &=(1-\lambda^{2})\sum_{n=k}^{\infty}\lambda^{2n}C_{n}^{k}\mu^{n-k }(1-\mu)^{k}\\ &=\frac{1-\lambda^{2}}{1-\mu\lambda^{2}}\left[\frac{\lambda^{2} (1-\mu)}{1-\mu\lambda^{2}}\right]^{k},\end{split} \tag{12}\]  
where \(\lambda=\sqrt{(V-1)/(V+1)}\). It is worth noting that the state \(\rho_{AB_{1}}^{\widehat{\Pi}_{1}}\) is no longer a Gaussian distribution after the photon-subtraction operation. Here, [PERSON] performs heterodyne detection. Note that it can be converted into an electrical signal without distortion by selecting a suitable photodetector. When the circuit completes the inspection task, the photodetector ultimately detects the required information in the form of an output electrical signal, which helps to generate a key sequence when interacting with [PERSON]. [PERSON] can perform heterodyne or homodyne detection. In this experiment, we focus our attention on the homodyne measurement. Suppose \(\gamma_{8}^{(k)}\) represents the covariance matrix of \(\rho_{AB_{2}}^{\widehat{\Pi}_{1}}\), which is expressed as ([20]):  
\[\gamma_{8}^{(k)}=\begin{bmatrix}X^{{}^{\prime}}L_{2}&Z_{8}^{{}^{\prime}} \sigma_{2}\\ Z_{8}^{{}^{\prime}}\sigma_{2}&Y^{{}^{\prime}}L_{2}\end{bmatrix}, \tag{13}\]  
where  
\[\begin{split} X^{{}^{\prime}}&=\frac{\mu\lambda^{2}+2k+1 }{1-\mu\lambda^{2}},\\ Y^{{}^{\prime}}&=\frac{\mu\lambda^{2}(2k+1)+1}{1-\mu \lambda^{2}},\\ Z_{8}^{{}^{\prime}}&=\frac{2\sqrt{\mu}\lambda(k+1)}{1-\mu \lambda^{2}}.\end{split} \tag{14}\]  
Figure 1: Prototype diagram of an improved eight-state discrete-modulated continuous variable quantum key distribution (CVQKD). [PERSON] prepares one of the eight-state discrete modulation states. The rest of the state is passed through the photon subtraction module and then passed through the untrusted quantum channel to [PERSON]. [PERSON] will use homodyne or heterodyne detection to derive the key sequence shared by [PERSON].