\mu\mathbf{I} & \sqrt{\tau}\mathbf{G} & -\sqrt{1-\tau}\mathbf{G} & \mathbf{0}
\\
\sqrt{\tau}\mathbf{G} & \nu_{B}\mathbf{I} & \gamma\mathbf{I} & \delta
\mathbf{Z} \\
-\sqrt{1-\tau}\mathbf{G} & \gamma\mathbf{I} & \nu_{E}\mathbf{I} & \kappa\mathbf{Z} \\
\mathbf{0} & \delta\mathbf{Z} & \kappa\mathbf{Z} & \omega\mathbf{I}\end{array}
\right) ,$$ where $\mathbf{0}$ is the $2\times2$ zero matrix, and$$\begin{aligned}
\nu_{E} & :=\tau\omega+(1-\tau)\mu,~\gamma:=\sqrt{\tau(1-\tau)}(\omega -\mu),
\\
\delta & :=\sqrt{1-\tau}\sqrt{\omega^{2}-1},~\kappa:=\sqrt{\tau(\omega
^{2}-1)}.\end{aligned}$$

From this global CM, we extract Eve’s reduced CM $\mathbf{V}_{EE^{\prime}}:=\mathbf{V}_{\mathbf{E}}$ describing the two output modes $\mathbf{E}=EE^{\prime}$ of the entangling cloner. This reduced CM has symplectic spectrum [@RMP2]$$\nu_{\mathbf{E}}^{\pm}=\frac{\sqrt{\alpha^{2}+4\beta}\pm\alpha}{2},$$ where $\alpha:=(1-\tau)(\mu-\omega)$ and $\beta:=\tau+(1-\tau)\mu\omega$. The von Neumann entropy of Eve’s average state is then given by $S(\mathbf{E})=h(\nu_{\mathbf{E}}^{+})+h(\nu_{\mathbf{E}}^{-})$, where$$h(x):=\frac{x+1}{2}\log_{2}\frac{x+1}{2}-\frac{x-1}{2}\log_{2}\frac{x-1}{2}.$$

By transforming the global CM under heterodyne detection [@RMP2], we compute Eve’s conditional CMs. First, we derive Eve’s CM conditioned to Bob’s detection $$\mathbf{V}_{\mathbf{E}|Y}=\mathbf{V}_{\mathbf{E}}-\frac{1}{\nu_{B}+1}\left(
\begin{array}{cc}
\gamma^{2}\mathbf{I} & \gamma\delta\mathbf{Z} \\
\gamma\delta\mathbf{Z} & \delta^{2}\mathbf{I}\end{array}
\right) ,$$ which has symplectic spectrum$$\nu_{\mathbf{E}|Y}^{-}=1,~\nu_{\mathbf{E}|Y}^{+}=\frac{\mu+\beta}{1+\mu
\tau+(1-\tau)\omega}.$$ Then, Eve’s CM conditioned to Alice’s detection is $$\mathbf{V}_{\mathbf{E}|X}=\mathbf{V}_{\mathbf{E}}-\frac{(1-\tau)g^{2}}{\mu +1}\left(
\begin{array}{cc}
\mathbf{I} & \mathbf{0} \\
\mathbf{0} & \mathbf{0}\end{array}
\right) ,$$ and has symplectic spectrum$$\nu_{\mathbf{E}|X}^{\pm}=\frac{\sqrt{\theta^{2}+4(\mu+1)\phi}\pm\theta}{2(\mu+1)},$$ where$$\theta:=(1-\tau)g^{2}-(\mu+1)\alpha,~\phi:=(\mu+1)\beta-(1-\tau)\omega
g^{2}.$$

From the previous conditional spectra, we compute Eve’s conditional entropies$$S(\mathbf{E}|X)=h(\nu_{\mathbf{E}|X}^{+})+h(\nu_{\mathbf{E}|X}^{-}),~S(\mathbf{E}|Y)=h(\nu_{\mathbf{E}|Y}^{+}),$$ and, therefore, we can derive the two Holevo quantities $I(\mathbf{E},X)=S(\mathbf{E})-S(\mathbf{E}|X)$ and $I(\mathbf{E},Y)=S(\mathbf{E})-S(\mathbf{E}|Y)$. By subtracting these from Alice and Bob’s mutual information $I(X,Y)$, we finally get the two key rates 