positive semidefinite, define $\mathcal{Q}^{-1}(V,\epsilon)=\{x\in\mathbb{R}^{n}:\,\mathbf{P}\{X\le x\}\ge1-\epsilon\}$ where $X\sim N(0,V)$ and $X\le x$ denotes entrywise comparison.

We assume that every random variable mentioned in this paper lies in a Polish space with its Borel $\sigma$-algebra, and all functions mentioned (e.g. distortion measures, the function $x(u,s)$ in Theorem \[thm:channel\_state\]) are measurable. The Lebesgue measure over $\mathbb{R}$ is denoted as $\lambda$. The Lebesgue measure restricted to the set $S\subseteq\mathbb{R}$ is denoted as $\lambda_{S}$. For two measures $\mu,\nu$ over $\mathcal{X}$ (a Polish space with its Borel $\sigma$-algebra) such that $\nu$ is absolutely continuous with respect to $\mu$ (denoted as $\nu\ll\mu$), the Radon-Nikodym derivative is written as $$\frac{d\nu}{d\mu}:\,\mathcal{X}\to[0,\infty).$$ If $\nu_{1},\nu_{2}\ll\mu$ (but $\nu_{1}\ll\nu_{2}$ may not hold), we write $$\frac{d\nu_{1}}{d\nu_{2}}(x)=\frac{d\nu_{1}}{d\mu}(x)\left(\frac{d\nu_{2}}{d\mu}(x)\right)^{-1}\in[0,\infty],\label{eq:rnderiv}$$ which is $0$ if $(d\nu_{1}/d\mu)(x)=0$, and is $\infty$ if $(d\nu_{1}/d\mu)(x)>0$ and $(d\nu_{2}/d\mu)(x)=0$.

The total variation distance between two distributions $P,Q$ over $\mathcal{X}$ is denoted as $\Vert P-Q\Vert_{\mathrm{TV}}=\sup_{A\subseteq\mathcal{X}\,\mathrm{measurable}}|P(A)-Q(A)|$.

Poisson Matching Lemma
======================

We first state the definition of Poisson functional representation in [@sfrl_trans], with a different notation that allows the proofs to be written in a simpler and more intuitive manner.

\[Poisson functional representation\]Let $\{\bar{U}_{i},T_{i}\}_{i\in\mathbb{N}}$ be the points of a Poisson process with intensity measure $\mu\times\lambda_{\mathbb{R}_{\ge0}}$ on $\mathcal{U}\times\mathbb{R}_{\ge0}$ (where $\mathcal{U}$ is a Polish space with its Borel $\sigma$-algebra, and 