Non-additivity of Renyi entropy and Dvoretzky's Theorem

TL;DR

This paper connects the analysis of quantum channel entropy to Dvoretzky's Theorem, simplifying the proof that the minimal output p-Renyi entropy is non-additive for p>1.

Contribution

It demonstrates that the non-additivity of quantum entropy can be understood through high-dimensional convex geometry, providing a conceptual simplification of previous proofs.

Findings

Shows the link between quantum entropy and Dvoretzky's Theorem

Simplifies the proof of non-additivity of minimal output p-Renyi entropy

Provides a geometric perspective on quantum information theory

Abstract

The goal of this note is to show that the analysis of the minimum output p-Renyi entropy of a typical quantum channel essentially amounts to applying Milman's version of Dvoretzky's Theorem about almost Euclidean sections of high-dimensional convex bodies. This conceptually simplifies the (nonconstructive) argument by Hayden-Winter disproving the additivity conjecture for the minimal output p-Renyi entropy (for p>1).