Alice and Bob heterodyne their local modes $A$ and $B$. (Note that other protocols involving homodyne detection for one of the parties or even two homodynes may be considered as well.) One can easily check that Alice remotely prepares thermal states on mode $a$. In fact, by heterodyning mode $A$, the other mode $a$ is collapsed in a Gaussian state $\rho_{a|X}$ with CM $\mathbf{V}_{a|X}=(1+\varepsilon)\mathbf{I}$, where$$\varepsilon:=\mu-1-\frac{g^{2}}{\mu+1}\geq0$$ quantifies the thermalization above the coherent state. This conditional thermal state is randomly displaced in the phase space according to a bivariate Gaussian distribution with variance $\mu-1-\varepsilon$ (so that the average input state on mode $a$ is thermal with the correct CM $\mu\mathbf{I}$).

At the output of the channel, Bob’s average state is thermal with CM $\nu
_{B}\mathbf{I}$, where $\nu_{B}:=\tau\mu+(1-\tau)\omega$. By propagating the conditional thermal state $\rho_{a|X}$, we also get Bob’s conditional state $\rho_{B|X}$, which is randomly displaced and has CM $\nu_{B|X}\mathbf{I}$, where$$\nu_{B|X}:=\tau(1+\varepsilon)+(1-\tau)\omega=\nu_{B}-\frac{\tau g^{2}}{\mu
+1}.$$ Therefore, we can easily compute Alice and Bob’s mutual information, which is equal to $$I(X,Y)=\log_{2}\frac{\nu_{B}+1}{\nu_{B|X}+1}.$$

The next step is the calculation of Eve’s Holevo information on Alice’s and Bob’s variables. We derive the global state of Alice, Bob and Eve, which is pure Gaussian with zero mean and CM$$\mathbf{V}_{ABEE^{\prime}}=\left(
\begin{array}{cccc}
