Unified Framework for One-shot Achievability via the Poisson Matching Lemma'
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Introduction
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The Poisson functional representation was introduced by Li and El Gamal [@sfrl_trans] to prove the strong functional representation lemma: for any pair of random variables $(X,Y)$, there exists a random variable $Z$ independent of $X$ such that $Y$ is a function of $(X,Z)$, and $H(Y|Z)\le I(X;Y)+\log(I(X;Y)+1)+4$. The lemma is applied to show various one-shot variable-length lossy source coding results, and a simple proof of the asymptotic achievability in the Gelfand-Pinsker theorem [@gelfand1980coding].

In this paper, we introduce the Poisson matching lemma, which gives a bound on the probability of mismatch between the Poisson functional representations applied on different distributions, and use it to prove one-shot achievability results for various settings, namely channels with state information at the encoder, lossy source coding with side information at the decoder, joint source-channel coding, broadcast channels, distributed lossy source coding, multiple access channels, channel resolvability and wiretap channels. The Poisson matching lemma can replace both the packing and covering lemmas (and generalizations such as the mutual covering lemma) in asymptotic typicality-based proofs. The one-shot bounds in this paper subsume the corresponding asymptotic achievability results by straightforward applications of the law of 