Strong Converse Exponent for Classical-Quantum Channel Coding

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 Rényi capacities with parameters α>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha > 1}$$\end{document}. It is important to note that, unlike in the classical case, there are many inequivalent ways to define the Rényi divergence of states, and hence the Rényi capacities of channels. Our exponent is in terms of the Rényi capacities corresponding to a version of the Rényi divergences that has been introduced recently in Müller-Lennert et al. (J Math Phys 54(12):122203, 2013. arXiv:1306.3142), and Wilde et al. (Commun Math Phys 331(2):593–622, 2014. arXiv:1306.1586). Our result adds to the growing body of evidence that this new version is the natural definition for the purposes of strong converse problems.