Finite blocklength converse bounds for quantum channels

TL;DR

This paper establishes general upper bounds on the classical communication rate over quantum channels at finite blocklengths, unifying classical and quantum converse bounds through quantum hypothesis testing.

Contribution

It introduces a channel-agnostic framework for finite blocklength converse bounds, extending classical and quantum results with a semidefinite programming approach.

Findings

Bounds are applicable to both entanglement-assisted and unassisted codes.

The bounds unify classical and quantum converse results.

For memoryless channels, the bounds recover known capacity formulas.

Abstract

We derive upper bounds on the rate of transmission of classical information over quantum channels by block codes with a given blocklength and error probability, for both entanglement-assisted and unassisted codes, in terms of a unifying framework of quantum hypothesis testing with restricted measurements. Our bounds do not depend on any special property of the channel (such as memorylessness) and generalise both a classical converse of Polyanskiy, Poor, and Verd\'{u} as well as a quantum converse of Renner and Wang, and have a number of desirable properties. In particular our bound on entanglement-assisted codes is a semidefinite program and for memoryless channels its large blocklength limit is the well known formula for entanglement-assisted capacity due to Bennett, Shor, Smolin and Thapliyal.