entropic characteristics of quantum channels and the additivity problem
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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 historyof 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
CCCCC
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 Classicalxstate Classical
ext
projection dim-1state pure pointExtreme setconvex compact
space State matrix density state Quantum
)](diag[
;}:)({)(
}1Tr,0:{)(
2
HH H
H
pure! are
subsystems of states Partial
state Pure
product Tensor
12121,2
12
2121
21
NOT
entangledHextHextHHext
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(
)(C1lim)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 Multiplicap1
p))((RR
)H()(R 1,p When
1p p
R
2p1p21p
p1
p1pHp
p
pp
)A()A(
)A(ΦΦΦ
Φ
ΦlogΦmin)(
;logTr1
1)(
p
p
)(
Rényi entropies for p<1
)(R :0p Fornorm!-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
xxxp
xxx
xxxp
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 linearA 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
12)(~Tr
11)(
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 RRRddOn d
ddI-)(
:grandomizin- is y probabilit high With
unitary i.i.d. random -U
U Un
ppp
d
j
jn
j j
),()()()log(,
)( *
1
1
The basic Additivity Conjecture
remains open
GLOBAL EQUIVALENCEof 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/,(ˆlog0
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 andchannels ldimensiona-infinite
all for holds )(CA thenchannels, 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 provedpdIpp
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
?