Strong converse exponent for classical-quantum channel coding

TL;DR

This paper derives the exact strong converse exponent for classical-quantum channel coding above the Holevo capacity, using a specific Renyi capacity framework that supports the natural interpretation of quantum divergences.

Contribution

It provides the first exact formula for the strong converse exponent in the quantum setting, extending classical results with a novel use of Renyi capacities based on recent divergence definitions.

Findings

Exact strong converse exponent derived for all rates above Holevo capacity

Exponent expressed as a transform of Renyi capacities with alpha > 1

Supports the naturalness of a specific Renyi divergence in quantum information theory

Abstract

We determine the exact strong converse exponent of classical-quantum channel coding, for every rate above the Holevo capacity. Our form of the exponent is an exact analogue of Arimoto's, given as a transform of the Renyi capacities with parameters alpha>1. It is important to note that, unlike in the classical case, there are many inequivalent ways to define the Renyi divergence of states, and hence the R\'enyi capacities of channels. Our exponent is in terms of the Renyi capacities corresponding to a version of the Renyi divergences that has been introduced recently in [M\"uller-Lennert, Dupuis, Szehr, Fehr and Tomamichel, J. Math. Phys. 54, 122203, (2013)], and [Wilde, Winter, Yang, Commun. Math. Phys. 331, (2014)]. Our result adds to the growing body of evidence that this new version is the natural definition for the purposes of strong converse problems.