Joint convexity of the quantum relative entropyTheorem Lieb For evey Hermitianmatrix Hthe map A TrceHthogaOn positivematrices

is concave

A random matrix ETrceHthosA e TrceHthosEA

q ehthosa a eh

PiGE IRI Krob distr on 9h2 in 3

of prior distr

DcpHq FE i logPq Cgi pi p actual distrILL 1 Tofu prob

O X X xn O

Xi g iidnudhypothes

Stein's Lemma Given X K Nn p or X xn n

Crusher g 4 probMin prob you answer g when 1 as n sX Xn p

opt pterror en Pll8 tolu

1 pkgh

pilogPq Cgi pi24

Joint convexity pig DcpHq convex

For all p pz q qz E IRI prob distus and t C G D

D lpC HC1D lDCH87G DCp

t TP Q

Xi yxnn tp.tl pz X Xn5tIw prob t w pro I t

Nn Pi vsyn up us Xiu shik

Limo E lose O

D P s i 0 logo o

p Ifpilogpi Pi negative entropy

zo for pi ooomf

yep lostPilp.toiDCpHqj

oCp oCq7 L v0Cg p g

su esi g CaATtCb a is convex

p gca.isFo 2gcabi

fIa aYYtr 20 X the 20det o SX.kz o 1,1220

i aot AA

w

FEA jA A o

qIga

fn i Ei I

Tr A ClogA logB CA B

pi bogPig pilogpi logfillin

Alog fB AB

S AUB Tr A dogA log B CA B A B to

T Go al Jointly convex

H EBf A t B jointly concave

fop.com i

Lemmai x B XBtX Atu

HB.lyjoinHywnmxonMCELxHft tfCAItG t1flB2

x g Xyy is convex

Claim fury o

AIeBnIcoIaxi.B7zoesAIGIiu_es

axo.xt If III ETI El

T.EE e1Lemma

Cx.B7tsXBtXfisjointtyconnexon MH.ie dpomt up

ome

f.laBitzBejfxitzxzzxe3ixatxBixi

i I tTu Tu

AµBixxn

Bto

Lemma_Cx B7tXBtx yjayt.lyane

f is jointly convex on MnCE7xHIt

AB7jointyTovexrfCx.BXB X

Liebigtonality thanI

HEA B A tB janty concave

HEAiBJ A ACAtB

A 4 AA

TAIB TICA AtD jointconcue

1 AttBtY A CABA AirsI ATB Atty

I ATB A ATB 1BA ACATB A A ATB B

ACAtB5 B tB9ATB B A A tB

ATB BIA t BI A

ATB Bt A I

AP cpf P tItA5 dt1 1

A Att B YA A 4A ACATBIA1 1

It A B A I A Atty A 2

I It A BA 25

ItX5 I CItx

I L

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