  We introduce a fundamental lemma called the Poisson matching lemma, and apply
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. Our one-shot bounds improve upon the best
known one-shot bounds in most of the aforementioned settings (except multiple
access channels, channel resolvability and wiretap channels, where we recover
bounds comparable to the best known bounds), with shorter proofs in some
settings even when compared to the conventional asymptotic approach using
typicality. The Poisson matching lemma replaces both the packing and covering
lemmas, greatly simplifying the error analysis. This paper extends the work of
Li and El Gamal on Poisson functional representation, which mainly considered
variable-length source coding settings, whereas this paper studies fixed-length
settings, and is not limited to source coding, showing that the Poisson
functional representation is a viable alternative to typicality for most
problems in network information theory.
