Accardi L., Freudenberg W., Obya M. (Eds.) Quantum Bio-informatics IV: From Quantum Information to Bio-informatics

80

4.3.

Quantum

noisy

and

noiseless

channels

Definition

4.2.

A quantum channel is

described

on

the

output algebra

of

Bob,

lE,

by

a

linear

normal

unital

completely

positive

map

(UCP)

A :

lE

----+

lEO

into

the

same

or

another

input

algebra

lEo

<:;;

B(~O).

The

algebra

lEo

in

the

input

state

<;0

should

not

be

identified

with

the

"Alice"

algebra

A

in a

state

f2

but

rather

with

the

transposed

algebra

lEO

= A which

can

be

the

same

as A

but

with

the

opposite

products

a*a

to

aa* E A

and

the

transposed

state

<;0

=

72.

Every

channel

A

admits

the

Kraus decomposition

A(b)

=

LVkbV~

E

lEo

Vb

E

lE,

k

where

Vk

are

contractions

~

----+

~o,

Lk

VkV~

=

1~o

and

is

dual

A =

A:~

to

map

AT

IlE~

==

AT

on

the

predual

space

lE~

into

lET

decomposed

as

ATW)

=

LVk~oV;

V~o

E

lE~,

k

where

V

k

=

J~

V~J~o

= Vk.

For

example,

quantum

noiseless

channel

in

the

case

lE

=

B(~),

lEo

=

B(~O)

is

described

by

any

single

coisometry

V :

~

----+

~o,

VV*

=

1~o,

as A(b) =

VbV*.

A noisy

quantum

channel

sends

input

pure

states

f20

=

<;0

on

the

algebra

lEo

=

B(~O)

into

mixed

states

<;

=

<;0

A

described

by

the

output

densities

~

=

AT

(~O)

given

by

the

predual

map

AT

=

AT

IlE~

to

the

mixed

normal

UCP

map

A :

lE

----+

lEo.

It

can

always

be

written

as A(b) = Trf+

[VbV*],

where

V is a

stochastic

coisometry, i.e. a

linear

operator

~

----+

~o

Q9

f+

with

the

partial

trace

Trf+

[VV*]

=

1~o

on

a

separable

Hilbert

space

f+

of

quantum

noise

in

the

channel.

Each

input

state

<;0

is

transmitted

into

an

output

state

<;

=

<;0

A given for

any

~o

E

lE~

by

the

density

operator

4.4.

Entanglement

as

quantum

encoding

Definition

4.3.

A

quantum

encoding

ofthe

input

state

<;0

for

the

channel

A

with

an

alphabet,

or

Alice

probe

algebra

A is

any

generalized

entangling

CP

map

K : A

----+

lEO

normalized

as

K(

1

p)

=

~o.

The

proper

quantum

encodings

correspond

to

the

proper

entanglements,

while

the

semi-quantum

encodings

are

described

by

the

commutative

(classical)

alphabet

algebras A = c (A).

81

The

set

of

all

quantum

encodings is convex,

denoted

by

K

q

,

or

Kq

(

<;0)

if

the

normalization

<;0

is fixed.

It

includes

the

standard

entanglements

(J0

(a) =

RaR,

corresponding

to

AO

=

llllo,

as pure quantum encodings,

while

the

convex

subset

of

all

semi-quantum

encodings,

denoted

respectively

by

K

c

,

or

Kc(

<;0),

excludes all

proper

entanglements.

Every

encoding K, E

Kq

(<;0)

is a

composition

K, =

(J0

0 K

of

the

pure

quantum

encoding

(J0

an~

normal

UCP

map

K : A

---->

AO

into

the

sufficient alphabet algebra

AO

=

llllo

antiisomorphic

to

llllo.

The

transposed

CP

map

K, T =

KT

0

pO

represents

quantum

encoding as a

transmission

process

by

KT

:

A~

---->

AT

inducing

the

alphabet

state

e = K,T

(1~)

from

the

transposed

input

state

eO

=

¢o

E

A~

maximally

entangled

by

(JOT

=

pO

to

the

input

state

<;0

on

the

antipode

llllO

of

the

algebra

A ° .

Each

encoding K, E

Kq

(<;0)

induces a

compound

state

wO(a@bO):=

Trf_[vo*(a@bO)vO]

==

(a@bO,wO).

given

on

the

input-probe

algebra

A@llllo

by

the

density

operator

WO

=

vO*vo

of

the

encoding

K,(a) = ,1[(a@

l~o)wO].

5.

Quantum

Channel

Capacities

and

their

Additivity

5.1.

The

quantum

and

semiclassical

capacities

The

channel A

transmits

the

probe-input encoding

K,

into

the

probe-output

encoding

AT

K,

entangling

the

output

state

via

this

channel

to

e =

1T(

1~)

by

K,

T A = 1T =

KT

A.

The

mutual

entangled

information,

transmitted

via

the

channel for

quantum

encoding

K"

is therefore

( 41)

where A =

pO

A is

the

standard

maximal

input

entanglement

pO

(b) =

#b

#

transmitted

via

the

channel

A.

Definition

5.1.

Given a

channel

A :

llll

---->

llllO

and

a

subset

K

<;;;;

K

q

,

the

channel information capacity is defined

by

(42)

The

channelled semiclassical

and

the

proper quantum information capaci-

ties

are

defined respectively as

( 43)

82

( 44)

Lemma

5.1.

Let A(b) =

VbV*

be

a unital completely positive

map

A:

lBl

--->

lBlo

(Noiseless channel.) Then

Jq(C;0,

A)

= Hrrdc;O),

Jc(C;°,

A)

= SlIdc;O)

andC(A)

=

lndim~o,

Q(A)

=

lndimlBlO

for the logarithmic capacities

of

any type

I12,lffi

=

Il~)

,

5.2.

The

true

quantum

capacity

Theorem

5.1.

The channelled entangled information achieves the value

( 45)

where A = 'ir

0

A is given

by

the optimal

input

entanglement 'ir

0

pO

of

the channel

input

state

C;O

to the transposed state

(/

=

~

induced on the

minimal

sufficient alphabet

A~

=

lBl~

,

Proof.

Use

the

monotonicity

of

IA,lffi(KTA)

with

respect

to

KT

and

the

commutativity

of

the

following

diagrams

0

Note

that

we have explicitly

three

types

of

such channeled informa-

tion

~:j

(C;O

,

A)

for each

contrast

function l

and

obviously,

~:~)

(C;O

,

A)

:S

~:~)(c;O,A),

QjA)

(A):S

QjB)

(A), where

Qj') (A) =

sup

{Ii',~o

(AT

0

'ir,o)

:

C;O

E

(3

(lBlO)}

,

It

is

interesting

to

compute

and

compare

the

Renyi

and

the

usual log-

arithmic

informations

~:j(c;O,A),

~:;(c;O,A)

and

the

capacities Qj')(A),

cF)

(A) for

the

particular

types

for some special, say

the

quantum

Gaussian

channel A

and

input

states

c;o,

5.3.

Block

encoding

for

quantum

product

channels

Here

we

consider

the

products

Q~n)

=

®i=l

c;i

and

transposed

states

c;~n)

=

(j~n)

respectively

on

the

tensor

products

A~n)

=

®i=l

lBl

i

and

lBl~n)

=

A~n)

and

for

the

notational

simplicity

we

implement

the

convention

A~

=

lBli,

f)i =

,0

and

A ® = A

(n)

f)® = f)(n) such

that

,-C

n

)

=

,®

is

the

product

t::o

""2

0

0,

t::'o

t:O

-:'0

":to

state

on

the

transposed

input

algebra

lBl~n)

=

lBl~

defining

the

standard

83

entanglement

(J@

=

/O,n

(Ji of

the

input

state

n(n)

=

n@

on

A

(n)

=

A@

to

o

"<:Yt=l

0

~o

t:o

0 0

the

probe

product

state

~~

=

(j~

on

lE~

=

A~.

Let

A @ = 0;'=1

Ai

denote

the

product

channel.

Definition

5.2.

A quantum block-encoding is a

normal

CP

map

f£(n) :

A(n)

---+

A~T

on

an

alphabet

algebra

A(n)

such

that

f£(n)

(l(n))

=

0~1~~

for

the

given

states

~i

E

lEiT.

Obviously,

the

compound

density

w~n)

E

A~n)

0

A~T

for

the

product

block encoding

f£@

=

0;'=f£i

is

the

product

w~

= 0;'=1 Woi

of

the

densities

for

the

input

compound

states

Woi =

wi,

each

entangled

by

tri

= f£I

on

Ai

0lEi.

However,

in

general

it

may

not

be

true,

despite

of

f£(n)

(1

(n))

=

?!~,

since

the

input

entangled

state

win)

on

A(n)

o

Ai

n

)

may

not

be

the

product

0;'=1

wi

even if A

(n)

is

the

product

alphabet

algebra

A@

=

0~1

Ai as is

A~n)

=

A~

but

f£(n)

(a@)

i- 0;'=1

f£i

(ai) for some

a@

= 0;'=1 ai E

A@.

This

is

due

to

one-to-one correspondence

w~n)

f--t

f£(n) :

5.4.

The

additivity

problem

for

quantum

capacities

In

the

normalized case

'P

(1) = 1 = w (1)

the

informational

divergence

vf;!

=

-Sl~)

as

negaentropy

of

any

type

has

the

multiplicative additivity

property

with

respect

to

every finite

product

reference

'P@

= 0;'=1

'Pi.

It

was shown

for (A)

and

(B)

types

entropy

in (24),

and

for

the

type

(C)

it

comes from

em

=

0e

mi

and

'P

=

0'Pi

in

the

transform

(29)

with

m =

2:

1

@(i-1)

0

mi

0

l@(n-i).

The

logarithmic

informations

I'JJ.,p,

(7r(n))

are

additive

iff

7r(n)

:

93

---+

m

T

is

product

entanglement

7r@

:

lE@

---+

A~,

and

the

logarithmic capacities

:ftc

((}~n),

A@

),

including

Hk)

((}~n))

=

sup

Il~

,'JJ.

(f£(n))

=

:f~.)

((}~n),

Id@)

",(n)

Etc

(e~n»)

are

additive

iff

they

are

achieved

on

product

encodings f£(n) =

f£@.

The

latter

holds if

the

constraint

K

on

f£

is

appropriate,

or, as we will prove be-

low, if

there

is no

additional

constraint

on

the

set

Kq

(

(}~n)).

On

the

other

84

hand,

the

semiclassical

constraint

K =

Kc

((}~n))

in

the

case

of

noncommu-

tative

A

o

seams

inappropriate,

although

it

is known

that

this

constraint

is

appropriate

for

the

logarithmic capacities

of

type

A in

the

category

of

the

commutative

Ao,

in

which

it

trivial

constraint.

In

particular,

the

semiclas-

sical Holevo

constraint

K =

Kc

is

not

appropriate

for

the

true

quantum

channels since as

it

is now known,

the

logarithmic capacities

J}A)

({}

~,

A 0)

and

C(A)

(A

0), known as

the

Holevo bounds

15,

are

not

additive for

the

general

quantum

channels

A.

5.5.

Optimal

true

quantum

block

encoding

The

following

Lemma

is valid for

any

type

of

quantum

entangled informa-

tion,

and

in

particular,

for

any

Renyi information.

Lemma

5.2.

Let

A~

:

A~T

---+

E~,

(}~n)

=

0i=1

{}~

==

{}

~

.

Then

Proof.

Take

AiT

:

EiT

---+

EiT

and

~~

E

EiT

with

Ei

=

A~

and

c:;i

=

{}~

evaluating

the

densities

~i

of

the

output

states

C:;i

=

{}~Ai

as

AiT

((}~).

Due

to

Theorem

1 we

can

decompose

the

quantum

block encodings

",en)

: A en)

---+

A0 as ",(n) = p0

K(n)

where

K(n)

.

A(n)

---+

A,.0

is a

normal

UCP

map

and

OT

0'

• 0

p~

(b~)

=

0i=

1

/?lb~/?l

is

the

standard

entanglement

on

A,.

~

:3

b~

to

(}~n)

=

0~

1{}~.

Note

that

although

the

range

of

K(n)

is

product

algebra

A,.0

the

domain

A(n)

is

any

algebra

For H(n) = K(n)o-0

A0

with

r = n

o , . T 0

"'!I

0 t:'o'

we

maximize

the

information

IA(

n)

Ja®

(H(n)) over

the

maximal

convex

set

of

quantum

encodings Kq

({}~n)):

'

J

q

((}~n),

A 0) =

sup

{I

21

,

'l:l

(K~n)o-

~

A 0 ) :

K~n)

0

o-

~

E

Kq}

.

Since

~~

=

l!~n)

is fixed,

the

supremum

should

be

taken

only over all chan-

nels

K~n)

with

fixed

domain

A,.~T

but

arbitrary

range

A~n).

Thus,

the

re-

quired result (46) follows by monotonicity

argument

(Theorem

2) for

any

type

of

quantum

channelled l-information J

q

({}

~,

A O

).

D

5.6.

The

additivity

of

true

quantum

capacities

Theorem

5.2.

The logarithmic q-informations Jq(.) are additive,

Jq(·)({}

~,

A

0

)

=

LJq(-)({}~,Ai)

85

and

so

is the logarithmic proper quantum capacity

Q(')

(A

0) =

sup

J

q

(')

((/

t!,

A

0)

= L

Q(

') (Ai) .

eo

Proof.

Follows

immediately

from

Lemma

3

by

additivity

argument

of

the

logarithmic

function l = In . 0

Note

that

this

proof

does

not

work for

any

type

of

the

logarithmic

semi-

quantum

information

J e(')

(Q~,

A 0)

and

the

capacity

C(·)

(A

0) which have

obviously

the

semiadditivity

property

The

lower

bound

achieved

on

the

product

encodings

",(n)

=

"'0

might

not

maximize

I~:'13

(K~n)

a ~

A 0)

under

the

constraint

that

A

(n)

must

be

com-

mutative

for

the

semiclassical encodings satisfying

",(n)

(1

(

n»)

=

Q~.

But

the

above semiclassical capacities

Je(A)

and

C(A)

are appropriate

for

semiclassical channels

A corresponding

to

the

commutative

input

algebra

A

o

=

lEo.

The

logarithmic capacities for such channels

of

the

type

A where

first

introduced

by

Levitin

under

the

name

of

the

defect

of

entropy

Je(A)

(Qo,

A)

=

SE

(Q

o

A)

-

SEIA

(",~A),

C(A)

(A)

=

sup

Je(A)

(Qo,

A),

eo

where SEIA

(",~A)

and

SE

(QoA)

are

conditional

and

unconditional

von-

Neumann

semiclassical entropies

corresponding

to

the

standard

classical

input

encoding

In

general,

these

capaciti

es

are

small

er

than

the

logarithmic

semiclassical

capacities

J}B)

(Qo,A)

= e

[goAT

In

(A~l~)J

C(B)

(A)

=

SUPJ}B)

(Qo,A)

eo

of

the

type

B. Here

AT

is

the

contravariant

v-density v

(AT

b)

= A (b)

of

the

semiclassical

channel

AT

:

go

f-+

J-l

(goAT)

mapping

the

classical J-l-desities

go

into

the

quantum

v-densities

~

=

AT

(g),

and

e =

J-l

Q9

v is

the

semiclassical

trace

on

lB~

®lE

T .

86

6.

Conclusion

Quantum

channel

capacities

can

have several different formulations

when

considering

to

send

classical

information

or

quantum

information,

one-way

or

two-way

communication,

prior

or

via

entanglement,

etc., however

they

all

can

be

considered as

the

capacities

achieved

under

the

different

constraints

on

the

encoding

class

of

true

quantum

entangling

codes K

q

.

Anyway,

the

op-

erational

quantum

channel

capacity

and

its

asymptotical

equivalence

with

the

additive

quantum

capacities

under

the

appropriate

constraints

is still a

big

open

and

challenging research

problem

in

quantum

information

theory.

Another

natural

problem

in

this

direction

is

to

compare

proper

quan-

tum

entropies

and

additive

capacities

of different

type

in

quantity

for some

interesting

quantum

channels,

such

as

Gaussian

channels,

and

other

smaller

capacities

of

different

types

under

the

well

known

constraints,

such

as

semiclassical

constraint,

entanglement-assistant

capacity, etc.,

and

find for

which class of

channels

they

assymptotically

coincide.

Generally

how

to

access

those

capacities, using physically

implement

able

operations

for encodings

and

decodings,

such

as

quantum

channel

capacity

for one-way

communication

via

entanglement,

is

of

course

an

important

open

problem

in

quantum

information

and

quantum

computation.

All

those

problems

wait

forthcoming

papers

in

the

future.

7.

Appendix:

The

standard

pairing

Let

lL

and

M

be

complex linear spaces

put

in

duality

via

bilinear form

(-,.;

: M x

lL

--->

C.

Let

M

be

closed

with

respect

to

an

associative

but

generally

non-commutative

binary

operation

M2

:3

(x, z)

f--+

XZ

E M (e.g.

pointwise

function

or

matrix

mUltiplication)

and

involution

*

as

a selfinverse

antilinear

map

reversing

the

multiplication

order,

(x*z)* = z*x, so

that

M

is a *-algebra

with

the

positive cone

of

x*x

generating

M.

Not

that,

if M

has

the

imaginary

unit

i =

-i

*,

then

it

the

case as

any

x E M is

the

linear

combination

of

four positive

elements

(x + i n)*(x +

in)

for n =

0,1,2,3.

However, we

do

not

require

that

M is

unital

but

will

assume

that

lL

has

the

identity

1*

= 1 defining a reference weight

rJ

(x) =

(x,l;

as a

strictly

positive

rJ

(x*x) > 0 for all nonzero x E M linear

functional

on

M.

Let

the

pairing

be

regular

such

that

for

any

left

multiplication

x

f--+

zx

on

M

there

exist

an

transposed

operator

y

f--+

z'y

on

lL

defined

by

(zx, y;

>=

(x,

z'y;,

and

that

lL

is closed

under

the

transposed

involution

denoted

also

by

*:

(x, y*; = (x*, y; *

(Every

separating

pairing

on

finite

dimensional

M

and

lL

is obviously regular.)

Then

the

following

theorem

hold:

87

Theorem

7.1.

The dual space

lL

of

M with respect to the regular pairing

is left module with respect

to

the transposed left action

of

M on

lL

such

that

(xz)'

=

z'

x',

and

it

is also the right module with respect to the right

transposed action

of

M on

lL

given

by

y

1-+

yz*'*,

where

(

*'*)

(*

*,

*) *

(*

* *) * - ( )

x,

yz

:=

x

,z

y = z x

,y

= xz, y .

Moreover,

if

Y = y* is positive element

of

lL

in

the sense that (x*x, y)

2:

0

for all x E

M,

then

z*'yz'*

is also positive for every z E

M.

The positive

maps

y

1-+

z*'yz'*

in

fact are completely positive (CP)

in

the sense that for

every semipositive

matrix

Y =

[Yi,k]

in

the sense

(x*x,

Y)

:=

L (X;Xk' yi,k)

2:

0

VXj

E

M,j

=

1,2,

...

,

i,k:;::l

the matrix z*'Yz'* = [z*'

I:

yi,k

z

'*]

remains semipositive.

The

pairing is called central,

or

tracial, if

the

left

and

right transposi-

tions

act

identically

on

1:

z*'1

==

z

==

1z'* for all z E M such

that

f)

(zz*) = (z, 1z'*) = (z, z*'1) =

f)

(z*z)

z E M .

The

antilinear invertibe

map

z

1-+

Z, intertwining

the

invo~utions

in

M

and

lL,

represents

the

*-algebra M

in

lL

as

the

opposite algebra M

with

respect

to

the

induced

product

z*z

=

z*z

such

that

z = z*_ becomes

the

transposition

z*z

=

z*z

of

M

onto

the

weakly dense

domain

M in

lL.

Moreover,

the

dual

space

lL

becomes two-sided *-module over M by identifying z

with

left

and

right

transpositions

as z'* = z =

z*'

which

adds

the

identity

1 E

lL

to

~.

Note

that

if

lL

is

taken

as minimal

predual

MT

completing

the

*-algebra M

with

respect

to

the

dual

to

C*-algebra

norm

on M,

then

lL

may

not

have

the

identity

as

the

density

of

a

not

finite

trace

f),

but

then

the

identity

can

be

added

to

M

to

represent

the

norm-trace

e (y) =

(1,

y)

on

lL

=

MT

uniquely

extending

the

transposed

trace

e (x) = (1, x) = f) (x) from

M.

The

basic example

of

such

central

pairing, called tilde-tracial pairing, is

given by

the

standard

trace

(x, y)

:=

Tr

(xy)

==

T (yx) ,

or

by

any

other

trace

e

on

lL

as

the

linear positive functional e (y) = (1,

y)

such

that

e (xy) = e (yx).

Note

that

the

standard

symmetric

tracial

pairing (x .

z)

= T (x . z) iden-

tifies

the

dual

space

of

M

with

the

complex conjugation

M*

=

j[

of

the

predual

space

lL

= M

T

.

In

this

preadjoint

space

M*

the

left multiplication

88

z'

is identified

not

with

the

left

multiplication

by

z

but

with

the

right

mul-

tiplication

by

z.

The

preadjoint

space

is obviously two-sided

M-module

containing

the

algebra

M itself

as

~a

dense subspace, which,

of

course, coin-

cides

with

M if M is

symmetric,

M = M.

Acknowledgment

This

work was

supported

by

QBIC

grant

of

Japanese

Ministry

of

High

Ed-

ucation

and

by

British

Council

PMI-2

grant

for

UK-Korea

collaboration.

The

author

would like

to

thank

for

warm

hospitality

of

Professor

Ohya

at

the

Science University

of

Tokyo

and

of

Professors So Young

Kim

from

Korean

Institute

for

Advanced

Studies

and

Un

Cig

Ji

from

Chungbuk

Na-

tional

University for

their

encouragement

to

prepare

these

lecture

notes

for

publication.

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Accardi L., Freudenberg W., Obya M. (Eds.) Quantum Bio-informatics IV: From Quantum Information to Bio-informatics

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