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We introduce the Poisson matching lemma and apply it to prove one-shot achievability results for channels with state information at the encoder, lossy source coding with side information at the decoder, joint source-channel coding, broadcast channels, and distributed lossy source coding. Our one-shot bounds improve upon the best known bounds in the aforementioned settings, with shorter proofs in some settings even when compared to the conventional asymptotic typicality approach. The Poisson matching lemma replaces both the packing and covering lemmas. This paper extends the work of Li and El Gamal on Poisson functional representation for variable-length source coding settings, showing that the Poisson functional representation is a viable alternative to typicality for most problems in network information theory.

A Unified Framework for One-shot Achievability via the Poisson Matching Lemma