Home

About

Advanced Search

Browse by Discipline

Scientific Societies

E-print Alerts

Add E-prints

E-print Network
FAQHELPSITE MAPCONTACT US


  Advanced Search  

 
NON-ADDITIVITY OF RENYI ENTROPY AND DVORETZKY'S THEOREM GUILLAUME AUBRUN, STANISLAW SZAREK, AND ELISABETH WERNER
 

Summary: NON-ADDITIVITY 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 p-R´enyi
entropy of a typical quantum channel essentially amounts to applying Milman's version of Dvoret-
zky's Theorem about almost Euclidean sections of high-dimensional convex bodies. This con-
ceptually simplifies the counterexample by Hayden­Winter to the additivity conjecture for the
minimal output p-R´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 non-additivity of the minimal output p-R´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 high-dimensional 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 semi-definite trace

  

Source: Aubrun, Guillaume - Institut Camille Jordan, Université Claude Bernard Lyon-I

 

Collections: Mathematics