Geometric R\'{e}nyi Divergence and its Applications in Quantum Channel Capacities

TL;DR

This paper studies the geometric Rényi divergence in quantum information theory, revealing its structural properties, proving a chain rule, and applying it to improve bounds on quantum channel capacities with efficient computability.

Contribution

It introduces new structural insights and chain rule inequalities for the geometric Rényi divergence, leading to improved, computable bounds on quantum channel capacities.

Findings

Proves a chain rule inequality for geometric Rényi divergence.

Addresses the open question of amortization collapse in quantum channel discrimination.

Constructs new channel information measures with sharper bounds.

Abstract

We present a systematic study of the geometric R\'enyi divergence (GRD), also known as the maximal R\'enyi divergence, from the point of view of quantum information theory. We show that this divergence, together with its extension to channels, has many appealing structural properties. For example we prove a chain rule inequality that immediately implies the "amortization collapse" for the geometric R\'enyi divergence, addressing an open question by Berta et al. [arXiv:1808.01498, Equation (55)] in the area of quantum channel discrimination. As applications, we explore various channel capacity problems and construct new channel information measures based on the geometric R\'enyi divergence, sharpening the previously best-known bounds based on the max-relative entropy while still keeping the new bounds single-letter efficiently computable. A plethora of examples are investigated and the improvements are evident for almost all cases.