# Duality of channels and codes

## TL;DR

This paper explores the duality between channels and codes, revealing how uncertainty relations and duality principles connect channel performance, code performance, and tasks like privacy amplification, with implications for capacity and EXIT functions.

## Contribution

It introduces a new framework linking channel duality with code duality, leading to novel insights into polarization, privacy, and capacity-achieving properties.

## Key findings

- Channel duality relates the performance of a code on a channel to its dual code on the dual channel.
- Polar code polarization rates to ideal and useless channels are identical due to duality.
- EXIT functions exhibit sharp transitions at capacity, providing a new approach to capacity proofs.

## Abstract

For any given channel $W$ with classical inputs and possibly quantum outputs, a dual classical-input channel $W^\perp$ can be defined by embedding the original into a channel $\mathcal N$ with quantum inputs and outputs. Here we give new uncertainty relations for a general class of entropies that lead to very close relationships between the original channel and its dual. Moreover, we show that channel duality can be combined with duality of linear codes, whereupon the uncertainty relations imply that the performance of a given code over a given channel is entirely characterized by the performance of the dual code on the dual channel. This has several applications. In the context of polar codes, it implies that the rates of polarization to ideal and useless channels must be identical. Duality also relates the tasks of channel coding and privacy amplification, implying that the finite blocklength performance of extractors and codes is precisely linked, and that optimal rate extractors can be transformed into capacity-achieving codes, and vice versa. Finally, duality also extends to the EXIT function of any channel and code. Here it implies that for any channel family, if the EXIT function for a fixed code has a sharp transition, then it must be such that the rate of the code equals the capacity at the transition. This may give a different route to proving a code family achieves capacity by establishing sharp EXIT function transitions.