NON ADDITIVITY OF RENYI ENTROPY AND DVORETZKY'S THEOREM

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Niveau: Supérieur, Doctorat, Bac+8
NON-ADDITIVITY OF RENYI ENTROPY AND DVORETZKY'S THEOREM GUILLAUME AUBRUN, STANIS LAW SZAREK, AND ELISABETH WERNER 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 Dvoret- zky's Theorem about almost Euclidean sections of high-dimensional convex bodies. This con- ceptually simplifies the counterexample by Hayden–Winter to the additivity conjecture for the minimal output p-Renyi 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-Renyi entropy for any p > 1. In this short note, we show that the Hayden–Winter 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.

  • any high-dimensional

  • valued quantum

  • quantum channel

  • euclidean norm

  • theory can

  • ?x?q ≤

  • convex bodies


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NON-ADDITIVITY
´ OF RENYI ENTROPY AND DVORETZKY’S THEOREM
GUILLAUMEAUBRUN,STANISLAWSZAREK,ANDELISABETHWERNER
Abstract.The goal of this note is to show that the analysis of the minimum outputpR-yne´i 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 Hayden–Winter to the additivity conjecture for the minimal outputpientropy-R´enyrof(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 outputpR-e´ienyrontfopyr anyp >1. In this short note, we show that the Hayden–Winter 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 n to norms inR(or equivalently, to convex bodies) whennbecomes large. Such connections between quantum information theory and high-dimensional convex geometry promise to be very fruitful.
2. Notation.IfHis a Hilbert space, we will denote byB(H) the space of bounded linear operators onH, and byD(H) the set ofdensity matricesonH, i.e., positive semi-definite trace one operators onH(orstatesonH, or – more properly – states onB(Hoften we will)). Most n n haveH=Cfor somenN, and we will then writeMnforB(C). Forp1, thep-opyentrenyiR´of a stateρis defined as 1 p Sp(ρ) = trρ . 1p (Forp= 1, this should be understood as a limit and coincides with the von Neumann entropy.) A linear map Φ :MmMdis called aquantum channelif it is completely positive and trace-preserving. Theminimal outputpR-yne´tneiyporof Φ is then defined as min S pmin(Φ) = Sp(Φ(ρ)). m ρD(C)
The research of the first named author was partially supported by theAgence Nationale de la Recherchegrant ANR-08-BLAN-0311-03. The research of the second and third named authors was partially supported by their respective grants from theNational Science Foundation(U.S.A.) and from theU.S.-Israel Binational Science Foundationsecond named author thanks the organizers and fellow participants (particularly F. Brandao and. The C. King) of the Workshop on Operator Structures in Quantum Information (Fields Institute, July 2009), which served as a catalyst for this project.
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