+ 2\rho\sqrt{PQ})}{P+Q+1+2\rho\sqrt{PQ}}\right).\end{aligned}$$Rearranging, $$\begin{aligned}
P(1-\rho^2) &\ge 1-\frac{1}{P+Q+1 + 2\rho\sqrt{PQ}} \ge 1-\frac{1}{P+Q+1}\end{aligned}$$ as required.

It is worth comparing the choice of the auxiliary variable $\rvu = \rvx + \rvs$ in the present problem with the choice of optimal $\rvu$ in the dirty paper coding problem [@costa:83]. While the input $\rvx$ is independent of $\rvs$ in [@costa:83], as illustrated in Fig. \[fig:dpc\] the optimal $\rvx$ in the secret-key problem has a component along $\rvs$. This is because scaling the interference sequence increases the secret-key rate. Secondly recall that in [@costa:83] we find the auxiliary codeword $\rvu^n$ that is closest to $\al \rvs^n$ where $\al = \frac{P}{P+N}$. In contrast this MMSE scaling is not performed in the secret-key problem.

Proof of Prop. \[prop:ubGaussNoDisc\]
-------------------------------------

We evaluate the upper bound in Theorem \[thm:ubNoDisc\] for the choice $\rvze = \rvzr + \rvz_\delta$, where $\rvz_\delta \sim
\cN(0,\Delta)$ is independent of $\rvzr$.

$$\begin{aligned}
&I(\rvs,\rvx;\rvyr|\rvye) = h(\rvyr|\rvye)-
h(\rvyr|\rvye,\rvx,\rvs)\\
&= h(\rvyr|\rvye)-
h(\rvzr|\rvze)\\
&\le \frac{1}{2}\log\left(P+Q+1 + 2\sqrt{PQ}-
\frac{(P+Q+1 + 2\sqrt{PQ})^2}{P+Q+1+\Delta + 2\sqrt{PQ}}\right)-
\\&\quad-\frac{1}{2}\log\left(1- \frac{1}{1+\Delta}\right)\end{aligned}$$

where we have used the fact that the conditional entropy $h(\rvyr|\rvye)$ is maximized by a Gaussian distribution [@khistiWornell:MISOME]. The above expression gives .

Symmetric CSI {#symmetric-csi}
=============

We establish the secret-key capacity for the case of symmetric 