the secret-key rate expression for Gaussian inputs i.e., $\rvu=\rvx \sim \cN(0,  P_0)$ when $\rvs_\mrm{r} = 0$ and $\rvu=\rvx \sim \cN(0, P_1)$ when $\rvs_\mrm{r} = 1$. Further to satisfy the average power constraint we have that $P_0 +P_1 \le 2P$. An achievable rate from Theorem \[thm:symCSI\] $$\begin{aligned}
R &= I(\rvx;\rvyr|{\mathsf{s}_{r}}) - I(\rvx;\tilde{\rvye}| {\mathsf{s}_{r}}) + H({\mathsf{s}_{r}}| \tilde{\rvye}) \\
&=I(\rvx;\rvyr|{\mathsf{s}_{r}}) - I(\rvx;\rvye,{\mathsf{s}_{e}}| {\mathsf{s}_{r}}) + H({\mathsf{s}_{r}}| {\mathsf{s}_{e}}, \rvye)\\
&= \frac{1}{8}\log(1 + P_1) + \frac{1}{2}E_{\rvye}[H(p(y_e),1-p(y_e))] + \frac{1}{2},  \label{eq:numerical1}\end{aligned}$$ where we have introduced $$p(y_e) = \frac{\cN_{y_e}(0,P_0+1)}{\cN_{y_e}(0,P_0+1) +
\cN_{y_e}(0,P_1+1)}$$ the aposterior distribution $\Pr({\mathsf{s}_{r}}=0|y_e)$ and the notation $\cN_{\rvye}(0,\sigma^2)$ denotes the zero mean Gaussian distribution with variance $\sigma^2$ evaluated at $\rvye$ and where  follows through a straightforward computation.

&= ([\_[r]{}]{}= 1, [\_[e]{}]{}= 0) I(; |[\_[r]{}]{}= 1) +\
&([\_[e]{}]{}= 1)H([\_[r]{}]{}|[\_[e]{}]{}= 1, ) + ([\_[e]{}]{}= 0)H([\_[r]{}]{})\
&= (1 + P\_1) + H([\_[r]{}]{}|[\_[e]{}]{}= 1, ) +\
&= (1 + P\_1) + E\_\[H([\_[r]{}]{}|[\_[e]{}]{}= 1, =y\_e)\] +\

In Fig. \[fig:trade\] we numerically evaluate this rate for $\mrm{SNR} = 17$ dB. For comparison we also plot the corresponding rate with public discussion [@khisti:09] $$R_\mrm{disc}= \frac{1}{8}\log(1 + 2P_1) + \frac{1}{2}E_{\rvye}[H(p(y_e),1-p(y_e))] + \frac{1}{2}.\label{eq:numerical2}$$

In Fig. \[fig:trade\] the solid curves show the secret key rate with and without public discussion, while the dashed curve is the entropy $H({\mathsf{s}_{r}}|{\mathsf{s}_{e}}=1,\rvye)$ 