 
Summary: NONADDITIVITY OF R´ENYI ENTROPY AND DVORETZKY'S THEOREM
GUILLAUME AUBRUN, STANISLAW SZAREK, AND ELISABETH WERNER
Abstract. The goal of this note is to show that the analysis of the minimum output pR´enyi
entropy of a typical quantum channel essentially amounts to applying Milman's version of Dvoret
zky's Theorem about almost Euclidean sections of highdimensional convex bodies. This con
ceptually simplifies the counterexample by HaydenWinter to the additivity conjecture for the
minimal output pR´enyi entropy (for p > 1).
1. Introduction. Many major questions in quantum information theory can be formulated
as additivity problems. These questions have received considerable attention in recent years,
culminating in Hastings' work showing that the minimal output von Neumann entropy of a
quantum channel is not additive. He used a random construction inspired by previous examples
due to Hayden and Winter, who proved nonadditivity of the minimal output pR´enyi entropy for
any p > 1. In this short note, we show that the HaydenWinter analysis can be simplified (at least
conceptually) by appealing to Dvoretzky's theorem. Dvoretzky's theorem is a fundamental result
of asymptotic geometric analysis, which studies the behaviour of geometric parameters associated
to norms in Rn (or equivalently, to convex bodies) when n becomes large. Such connections
between quantum information theory and highdimensional convex geometry promise to be very
fruitful.
2. Notation. If H is a Hilbert space, we will denote by B(H) the space of bounded linear
operators on H, and by D(H) the set of density matrices on H, i.e., positive semidefinite trace
