ENTROPIC CHARACTERISTICS OF QUANTUM CHANNELS AND THE ADDITIVITY PROBLEM

A. S. Holevo

Steklov Mathematical Institute, Moscow

• Introduction: quantum information theory

• The classical capacity of quantum channel

• Hierarchy of additivity conjectures

• Global equivalence 

• Partial results 

INTRODUCTION

A brief history

of quantum information theory

Information Theory

• Born: middle of XX century, 1940-1950s

(Shannon,…) • Concepts: random source, entropy, typicality,

code, channel, capacity: • Tools: probability theory, discrete math,

group theory,…• Impact: digital data processing, data compression,

error correction,…

,...ShannonCC

Quantum Information Theory

• Born: second half of XX century

Physics of quantum communication, 1950-60s

(Gabor, Gordon, Helstrom,…):

FUNDAMENTAL QUANTUM LIMITATIONS

ON INFORMATIOM TRANSMISSION

? • Mathematical framework: 1970-80s

Quantum Information Theory

The early age (1970-1980s) Understanding quantum limits• Concepts: random source, entropy, channel,

capacity, coding theorem, …, entanglement• Tools: noncommutative probability, operator

algebra, random matrices (large deviations)…• Implications: …, the upper bound for classical

capacity of quantum channel: χ-capacity

C ≤ Cχ An overview in the book “Statistical structure of quantum theory” (Springer, 2001)

Quantum Information Theory(“Quantum Shannon theory”)

The new age (1990-2000s)

From quantum limitations to quantum advantages

• Q. data compression (Schumacher-Josza,…)• The quantum coding theorem for c-q channels:

C = C χ (Holevo; Schumacher-Westmoreland)• Variety of quantum channel capacities/coding

theorems (Shor, Devetak, Winter, Hayden,…)

Summarized in recent book by Hayashi (Springer, 2006)

Additivity of channel capacity

1

2112

CC

CCC

n n

CLASSICALINFORMATION

CLASSICALINFORMATION

1

2

01001011 11011010

?

?

MEMORYLESS

encoding decoding

The χ-CAPACITY and the CLASSICAL CAPACITY

of QUANTUM CHANNEL

Finite quantum systemHdim

pointspace phase state pure Classical

xstate Classical

ext

projection dim-1state pure pointExtreme

setconvex compact

space State

matrix density state Quantum

)](diag[

;

}:)({)(

}1Tr,0:{)(

2

H

H H

H

pure! are

subsystems of states Partial

state Pure

product Tensor

12121,2

12

2121

21

NOT

entangled

HextHextHHext

HH

1,2

21

Tr

)()()(

Composite quantum systems –entanglement

Quantum channel

1,2,...n forpositive nId

Completely positive (CP) map, Σ(H)→ Σ(H’):

ρ ρ’

nId

12 '12

Product of channels

)Id()Id( 212121

1

2Id

12 '12

2

1Id

The minimal output entropy

ADDITIVITY

ntentangleme no :yClassicall

HHH

state) ( )( on attained

HH

)( on continuousconcave

-)H(Entropy

)2()1()12(

)A()()()(

ext

))((min)(

logTr

2121

purepurepure

pureH

H

?

The χ-capacity

x

xxx

xxp

xx

HppHCxx

))(())(()(

)(

,

max

x

ensemble average conditional output entropy

output entropy

The Additivity Conjecture

)(C)C(

)(C1

lim)C(

,);(C)(C

,

)(A)(C)(C)(C

21

χ2121

n

CAPACITY CLASSICALthe

nn

channels ALL for

nn

n

?

Separate encodings/separate decodings

)(cc aI

CLASSICAL

INFORMATION

CLASSICAL

INFORMATION

.

. n

.

separate q. separate q. encodings decodings

ACCESSIBLE (SHANNON) INFORMATION

Separate encodings/entangled decodings

)()( acc IC

CLASSICAL

INFORMATION

CLASSICAL

INFORMATION

.

. n

.

separate q. entangled encodings decodings

HSW-theorem: χ - CAPACITY!

!

)(C

Entangled encodings/ entangled decodings

)(Cn/)(Clim)(C nn

CLASSICAL

INFORMATION

CLASSICAL

INFORMATION

.

. n

.

entangled entangled encodings decodings

The full CLASSICAL CAPACITY

?

?

HIERARCHY of ADDITIVITYCONJECTURES

- minimal output entropy- χ-capacity– convex closure/ constrained χ-capacity/EoF

Additivity of the minimal output entropy

)()(

)A()()()(

))((min)(

2121

HnH

HHH

HH

n

?

Rényi entropies and p-norms

1,p For

)(R)(R)(R

oftivity Multiplica

p1

p))((RR

)H()(R 1,p When

1p p

R

2p1p21p

p1

p1pH

p

p

pp

)A()A(

)A(ΦΦΦ

Φ

ΦlogΦmin)(

;logTr1

1)(

p

p

)(

Rényi entropies for p<1

)(R :0p For

norm!-p No

1,p For

)H()R 1,p When

1p0 p

R

0

p

pp

)(rankmin

)A()A(

;logTr1

1)(

p

(

The χ-capacity

xxx

p

xxx

xxx

p

HpH

HppHC

xxx

xx

))((min))((max

))(())((max)(,

ensemble average conditional output entropy

output entropy convex closure

Convex closure EoF

EoF ofivity superaddit

HHH

isometryg Stinesprin

ensembles (finite)

VVEHpH Fxxp xx

)A()()()(

*)())((min)(

2112 2121

Constrained capacity

}E :{A constraint linear

A subset compact :constraint

H-))(H(AC

ΦA

constTr:

)(

)(

)]([max),(

H

Additivity with constraints

ly individual

)((C(C

AC)A,(C)AA,(C

χχ

2χ1χ21χ

)A()(A

)A()(CA

)Φ,Φ)

)(CA),(ΦΦΦ

χ

χ

χ2121

H

?

Equivalent forms of (CA )

)()()()A(

;,)(CA

)(CA

)(CA

2112

21χ

χ

χ

22

HHH -

AAarbitrary with-

;A,Aarbitrary with -

;A,A lineararbitrary with -

:equivalentare following the

Φ,Φ channels For

11

21

21

21

21

THM

Partial results

• Qubit unital channel (King)

• Entanglement-breaking channel (Shor)

• Depolarizing channel (King)

Lieb-Thirring inequality:

))(dim( IId 2,H

MN xx

x Tr)(

0;TrTr BA, 1;p BA(AB) ppp

dIpp )(Tr)1()(

Recent work on special channels (2003-…)

Alicki-Fannes; Datta-Fukuda-Holevo-Suhov;

Giovannetti-Lloyd-Maccone-Shapiro-Yen;

Hayashi-Imai-Matsumoto-Ruskai-Shimono;

King-Nathanson-Ruskai; King-Koldan;

Matsumoto-Yura; Macchiavello-Palma;

Wolf-Eisert,…

ALL ADDITIVE!

Transpose-depolarizing channel

AH)-(Werner 4,783)p 3,(d 2d p,large for BREAKS

2,p1 for holds

unitaries) all-(Usymmetric highly

P IPd

;Id

g

asymasymT

))(A),A(()(A

1

2)(

~Tr

1

1)(

χp

Numerical search for counterexamples

Breakthrough 2007

Multiplicativity breaks:• p>2, large d (Winter);• 1<p<2, large d (Hayden);• p=0, large d (Winter); p close to 0.Method: random unitary (non-constructive)

It remains 0<p<1 and p=1 (the additivity!)... And many other questions

Random unitary channels

2.p RRR

ddOn d

dd

I-)(

:grandomizin- is y probabilit high With

unitary i.i.d. random -U

U Un

ppp

d

j

j

n

j j

),()()(

)log(,

)( *

1

1

The basic Additivity Conjecture

remains open

GLOBAL EQUIVALENCE

of additivity conjectures

(Shor, Audenaert-Braunstein, Matsumoto-

Shimono-Winter, Pomeranski, Holevo-

Shirokov)

(EoF)(H

ty of eradditivisup

Φ )ˆ

)A(

C of additivity )()(A χ

)(H of additivity

)A(

AC of additivity ),()(CAχ

“Global” proofs involvingShor’s channel extensions

Discontinuity ofIn infinite dimensions

)( C

)(A )(CA globally

.A,A sconstraint all and

Φ ,Φ all for holds )(CA then

,Φ ,Φ channels all for holds )(A If

21

21χ

21χ THM

Proof: Uses Shor’s trick: extension of the original channel which has capacity obtained by the Lagrange method with a linear constraint

Channel extension

ETrrate the at bits d sendsrarely but

, as actsmostly 0,q When

IE0 dqE

idle

E measures :q prob th wi

bits classical d

:q-1 prob th wi

bits classical d of inputs of Inputs

log

ˆ

);,,(ˆˆ

log:ˆ

logˆ

1

0

Lagrange Function

capacity- dconstraine

cE

multiplierLagrange -

E )ddE(C lim

const d, q, dq Let

d inuniformly

O(1) q E) dq-q)( )(C ρ

Tr:)(max

Tr)(max),log/,(ˆ

log0

Trlog()(1maxˆ

Hdim

• Set of states is separable metric space, not locally compact• Entropy is “almost always” infinite and

everywhere discontinuous

BUT• Entropy is lower semicontinuous• Entropy is finite and continuous on “useful”

compact subset of states (of bounded “mean energy”)

Hdim

The χ-capacity

x

xxx

xxp

xx

HppHC

x

xx

))(())((sup)(

)(

,

ensemble average conditional output entropy

output entropy

Generalized ensemble (GE)=Borel probability measure on state space

) dim,(CA) dim,(A χχ HH

s.constraintarbitrary and

channels ldimensiona-infinite

all for holds )(CA then

channels, ldimensiona-finite

all for holds )(A If

χ

χ THM

In particular, for all Gaussian channelswith energy constraints

Gaussian channels

Canonical variables (CCR) Gaussian environment Gaussian states Gaussian states Energy constraint

PROP For arbitrary Gaussian channel with energy constraint an optimal generalized ensemble (GE) exists.CONJ Optimal GE is a Gaussian probability measure supported by pure Gaussian states with fixed correlation matrix. (GAUSSIAN CHANNELS HAVE GAUSSIAN OPTIMIZERS?)

Holds for c-c, c-q, q-c Gaussian channels

------------------------------------------------------------

RRE

RKKRRR

T

ee

'

CLASSES of CHANNELS

Complementary channels(AH, Matsumoto et al.,2005)

Observation: additivity holds

for very classical channels;

for very quantum channels

Example:

Id

0

0

00 )(Tr)(~

)(

Complementary channels

dilationg StinesprinThe

VV)(

*VV)(

:Visometry

B

V A

C

B

C

A

A

CBA

*Tr~

Tr

~

H

H

HHH

Complementary channels

21

21

2121

~

~,

~)A()A(

,)A()A(THEOREM

~~)()(

)~

()(

for holds ) (resp.

for holds ) (resp.

to is

HH

HH

arycomplement

Entanglement-breaking channels

dephasing"" -channelsary Complement

breaking)-ntentangleme is channel(the

0 BA BA))(Id(

)HH(arbitrary and 2,...d for (ii)

quantum);classicalquantum is channel(the

MN with tionrepresentaa isthere (i)

:equivalentare

conditionsfollowing The Shor) Ruskai, ki,(P.Horodec

MN

kkk

kk12d

d12

xx

xx

x

,;

0,0(*)

THEOREM

(*)Tr)(

Entanglement-breaking channels-- additivity

1

ppp

21

channelsdephasing for hold properties

additivitythe all arity,complementBy

BA, BA(AB)

:inequalityThirring -Liebthe

onbasing King,by -p

Shorby destablishe ) fact, (in

arbitrary breaking,-ntentangleme For

~

0;TrTr

,1),(A

)A()(A),A(

p

χ

Symmetric channels

nonunital qubit ,

Kingby provedp

dIpp

channelng depolarizi (ii)

2);H (

channels unital qubit symmetricbinary (i)

)(H-(I/d))H()(C

: then e,irreducibl -U if

Gg *;)V(V*)ρUU

χ

g

gggg

)A(

)(A,1),(A

)(Tr)1()(

dim

)A()(A

Φ(

χp

χ

?