lower bound on the secret-key agreement capacity is capacity is given by, $$\begin{gathered}
R^- =  \frac{1}{2}\log\left(1 + \frac{\Delta(P+Q+2\rho \sqrt{PQ})}{P+Q+1+\Delta + 2\rho \sqrt{PQ}}\right),
\label{eq:lbGaussNoDisc}\end{gathered}$$ where $|\rho| < 1$ and $$P(1-\rho^2) =1 - \frac{1}{P+Q+1}.\label{eq:rhoCons}$$ \[prop:lbGaussNoDisc\]

An upper bound on the secret-key capacity is given by, $$R^+ = \frac{1}{2}\log\left(1 +
\frac{\Delta(P+Q + 2\sqrt{PQ})}{P+Q+1+\Delta + 2\sqrt{PQ}}\right)\label{eq:ubGaussNoDisc}$$\[prop:ubGaussNoDisc\]

It can be readily verified that the upper and lower bounds are close in several interesting regimes. In Fig. \[fig:GaussNumerical\] we numerically plot these bounds and state some properties below. We omit the proof due to space constraints.

The upper and lower bounds on secret-capacity satisfy the following $$\begin{aligned}
&R_+ - R_- \le \frac{1}{2}~~\mrm{bit/symbol} \label{eq:univGap} \\
&\lim_{P\rightarrow\infty} R_+ - R_- = 0 \label{eq:highINR}\\
&\lim_{Q\rightarrow\infty} R_+ - R_- = 0 \label{eq:lowINR}\end{aligned}$$

Symmetric CSI {#subsec:CSI}
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Consider the special case where the state sequence $\rvs$ is also revealed to the legitimate receiver. In this case we have a complete characterization of the secret-key capacity.

The secret-key capacity for the channel model in section \[subsec:Model\] when the state sequence $\rvs^n$ is also revealed to the decoder is given by $$C_\mrm{sym} = \max_{p_{\rvu|\rvs(\cdot)}p_{\rvx|\rvu,\rvs(\cdot)}} I(\rvu;\rvyr|\rvs) - I(\rvu;\rvye|\rvs) + H(\rvs|\rvye),\label{eq:symCap}$$ where the maximization is over all auxilary random variables $\rvu$ that obey the Markov 