# Moderate deviation analysis for classical communication over quantum channels

## TL;DR

This paper extends moderate deviation analysis to quantum channels, characterizing the tradeoff between error probability, rate, and code length, and introduces new techniques for quantum hypothesis testing.

## Contribution

It generalizes moderate deviation analysis to quantum channels and derives fundamental tradeoffs involving capacity and dispersion, with new methods for quantum hypothesis testing.

## Key findings

- Derived tradeoff between error, rate, and code length for quantum channels.
- Established the role of channel dispersion in finite block transmission.
- Developed techniques for asymmetric binary quantum hypothesis testing.

## Abstract

We analyse families of codes for classical data transmission over quantum channels that have both a vanishing probability of error and a code rate approaching capacity as the code length increases. To characterise the fundamental tradeoff between decoding error, code rate and code length for such codes we introduce a quantum generalisation of the moderate deviation analysis proposed by Altug and Wagner as well as Polyanskiy and Verdu. We derive such a tradeoff for classical-quantum (as well as image-additive) channels in terms of the channel capacity and the channel dispersion, giving further evidence that the latter quantity characterises the necessary backoff from capacity when transmitting finite blocks of classical data. To derive these results we also study asymmetric binary quantum hypothesis testing in the moderate deviations regime. Due to the central importance of the latter task, we expect that our techniques will find further applications in the analysis of other quantum information processing tasks.