the conditional Poisson matching lemma on $(X,\,(U,M),\,(M,Y),\,P_{U|Y}\times\delta_{M})$ (note that $P_{U,M|X}=P_{U|X}\times P_{M}$). Therefore there exists a fixed $\{(\bar{u}_{i},\bar{m}_{i}),t_{i}\}_{i\in\mathbb{N}}$ attaining the desired bound.

This reduces to lossy source coding (without side information) when $Y=\emptyset$. Note that the encoder is designed in the same way with or without side information. An encoder for lossy source coding is sufficient to achieve the bound in Theorem \[thm:lscsi\] even when side information is present. Binning is not required at the encoder.

Similar to the case in Section \[sec:channel\_state\], it can be checked that our bound is stronger than that in Theorem 2 in [@verdu2012nonasymp]. Compared to Corollary 9 in [@watanabe2015nonasymp]: $$\begin{aligned}
P_{e} & \le\mathbf{P}\{\iota_{U;X}(U;X)>\gamma_{\mathrm{c}}\;\mathrm{or}\;\iota_{U;Y}(U;Y)<\gamma_{\mathrm{p}}\;\mathrm{or}\;\mathsf{d}(X,Z)>\mathsf{D}\}+\frac{\mathsf{J}}{2^{\gamma_{\mathrm{p}}}\mathsf{L}}+\frac{1}{2}\sqrt{\frac{2^{\gamma_{\mathrm{c}}}}{\mathsf{J}}}\label{eq:lscsi_watanabe}\end{aligned}$$ for any $\gamma_{\mathrm{p}},\gamma_{\mathrm{c}}>0$, $\mathsf{J}\in\mathbb{N}$, our result is stronger since $$\begin{aligned}
 & \mathbf{E}\left[1-\mathbf{1}\{\mathsf{d}(X,Z)\le\mathsf{D}\}(1+\mathsf{L}^{-1}2^{\iota_{U;X}(U;X)-\iota_{U;Y}(U;Y)})^{-1}\right]\\
 & \le\mathbf{P}\{\iota_{U;X}(U;X)>\gamma_{\mathrm{c}}\;\mathrm{or}\;\iota_{U;Y}(U;Y)<\gamma_{\mathrm{p}}\;\mathrm{or}\;\mathsf{d}(X,Z)>\mathsf{D}\}+\mathsf{L}^{-1}2^{\gamma_{\mathrm{c}}-\gamma_{\mathrm{p}}}\\
 & \le\mathbf{P}\{\iota_{U;X}(U;X)>\gamma_{\mathrm{c}}\;\mathrm{or}\;\iota_{U;Y}(U;Y)<\gamma_{\mathrm{p}}\;\mathrm{or}\;\mathsf{d}(X,Z)>\mathsf{D}\}+\frac{\mathsf{J}}{2^{\gamma_{\mathrm{p}}}\mathsf{L}}+\frac{1}{2}\sqrt{\frac{2^{\gamma_{\mathrm{c}}}}{\mathsf{J}}},\end{aligned}$$ where the last inequality is due to $$a+b\ge(a+b)^{3}=27\left(\frac{a+2(b/2)}{3}\right)^{3}\ge27a(b/2)^{2}\ge4ab^{2}$$ by the AM-GM inequality for $a,b\ge0$, $a+b\le1$ (since the right hand side of $\le1$ for it to be meaningful). We bound $\iota_{U;X}(U;X)-\iota_{U;Y}(U;Y)$ as a single quantity, instead of bounding the two terms separately, resulting in a sharper bound.

One-shot Joint Source-Channel Coding
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The one-shot joint source-channel coding setting is described as follows. Upon observing the source symbol $W\sim P_{W}$, the encoder produces $X\in\mathcal{X}$, which is sent through 