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

## TL;DR

This paper introduces the Poisson matching lemma, a new fundamental tool that simplifies and improves one-shot achievability bounds across various information theory problems, extending previous work on Poisson functional representation.

## Contribution

The paper presents the Poisson matching lemma and demonstrates its broad applicability, providing improved one-shot bounds and simpler proofs in network information theory.

## Key findings

- Improved one-shot bounds in most settings compared to previous results.
- Simplified proofs by replacing packing and covering lemmas.
- Extension of Poisson functional representation to fixed-length settings.

## Abstract

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.