using a common framework (the Poisson functional representation), which is noteworthy since the roles of an encoder and a decoder in an operational setting are very different, and their constructions usually have little in common in conventional approaches.

Notation {#notation .unnumbered}
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Throughout this paper, we assume that $\log$ is to base 2 and the entropy $H$ is in bits. We write $\mathrm{exp}_{a}(b)$ for $a^{b}$.

The set of positive integers is denoted as $\mathbb{N}=\{1,2,\ldots\}$. We use the notation: $X_{a}^{b}:=(X_{a},\ldots,X_{b})$, $X^{n}:=X_{1}^{n}$ and $[a:b]:=[a,b]\cap\mathbb{Z}$. The conditional information density is denoted as $$\iota_{X;Y|Z}(x;y|z):=\log\frac{dP_{XY|Z=z}}{d(P_{X|Z=z}\times P_{Y|Z=z})}(x,y).$$ We consider $\iota_{X;Y|Z}(x;y|z)$ to be defined only if $P_{XY|Z=z}\ll P_{X|Z=z}\times P_{Y|Z=z}$.

For discrete $X$, we write the probability mass function as $p_{X}$. For continuous $X$, we write the probability density function as $f_{X}$. For a general random variable $X$ in a measurable space, we write its distribution as $P_{X}$. The uniform distribution over a finite set $S$ is denoted as $\mathrm{Unif}(S)$. The joint distribution of $X_{1},\ldots,X_{n}\stackrel{iid}{\sim}P_{X}$ is written as $P_{X}^{\otimes n}$. The degenerate distribution $\mathbf{P}\{X=a\}=1$ is denoted as $\delta_{a}$. The conditional independence of $X$ and $Z$ given $Y$ is denoted as $X\leftrightarrow Y\leftrightarrow Z$.

The Q-function and its inverse are denoted as $\mathcal{Q}(x)$ and $\mathcal{Q}^{-1}(\epsilon)$ respectively. For $V\in\mathbb{R}^{n\times n}$ 