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

200

illustrate

our

investigations

with

several examples.

2.

Detecting

and

measuring

entanglement

In

this

paper,

we focus ourselves

on

bipartite

entanglement

which describes

a

sort

of

quantum

correlations

between

two

quantum

systems

defined in

their

m-dimensional

Hilbert

spaces

HA

and

HE, respectively.

The

pure

bipartite

state

1'1/1

E

HA

(>9

HE

of

the

composite

system

is called separable

if

it

can

be

written

in a

product

form

in

terms

of

two

pure

states

I¢

A

and

l<p

E

of

the

subsystems

A

and

B,

respectively.

Otherwise

the

state

1'1/1

is entangled.

The

problem

of

detecting

and

quantifying

pure

entangled

states

has

been

solved long ago using so-called

Schmidt

decomposition

of

1'1/1

1,2,

which

enables

to

construct

two

orthonormal

basis

{Iak

} E

HA

and

{Ib

k

} E HE,

such

that

m

(1)

k=l

where

Mk

are

Schmidt

coefficients. Hence, a

pure

state

1'1/1

is

entangled

if

and

only if

at

least two

Schmidt

coefficients

are

nonzero. Moreover,

using decomposition (1) we

can

introduce

the

following two

measures

of

entanglement:

•

entropy

of

a

reduced

system

P A =

Tr

E

(1'1/1

('1/1

I)

defined as

and

• concurrence

[(1'1/1

)

and

its

generalization

to

mixed

states

called entanglement

of

for-

mation

(EOF)

is considered as

the

most

natural

measure

but

it

is

hard

to

compute

and

even

to

estimate

for

most

mixed

states.

Therefore,

it

is

more

convenient

to

use concurrence even

if

it

has

no direct connection

with

entropy. For

pure

states

both

measures

are

simple functions

of

Schmidt

coefficients,

E(I

1/J

) = - L

f.tk

log2

f.tk

,

k

C(I1/J)

= 2 L

f.tkf.tZ·

k,l#k

201

For mixed

states

the

situation

is much more complicated. Let us re-

call

that

a mixed

state

P is entangled if it cannot

be

written

as a convex

combination of

product

mixed

states

of

the

subsystems, i.e.

P=

LPkPk

A

)

0

Pk

B

)

,

Pk

> 0 ,

LPk

= 1 ,

k k

where

Pk

A

)

and

Pk

B

)

are some mixed

stat

es of subsystems A

and

B, respec-

tively.

There

is still no universal

method

of detecting entanglement for any

mixed

state.

In

particular

, entangleme

nt

in so-called positive

partially

transpos

ed

(PPT)

states

(for m

2':

3)

is

hard

to

recognized. Entangle-

ment if

detected

should also

be

quantified. One expects

that

a measure

M(p) should fulfill

at

least

the

following requirements:

(1)

M(p)

2':

0

and

M(p) = 0 for separable P (sometimes

M(Pm

ax

) = 1

for maximally entangled

state

Pma

x),

(2)

invariance

with

respect

to

local

unitary

operation,

(3)

convex function

with

respect

to

convex combination of mixed states,

M(

LkPkPk)

~

LkPkM(Pk)·

A

method

extending measures initially defined for

pure

states

to

a mixed

state

setting

and

which guarantees fulfilling (1)-(3) is

the

convex roof con-

struction.

The

procedure consists in a decomposition of P into a convex

combination of I-dimensional projectors

P = LPkl1/Jk

(1/Jkl,

LPk

=

l.

k k

and

then

taking

infimum over all such decompositions, i.e. one defines

C(p) =

inf

LPkC(I1/Jk) '

{Pk.!1f

k}

k

E(p) = inf

LPkE(I1/Jk)'

{Pk.!1fk}

k

as concurrence

and

EOF

for mixed

stat

es, respectively.

(2)

202

The

problem arises when

carrying

out

the

extremalisation

procedure in

(2)

-

exact

re

sults

ar

e known only

in

very limited cases.

The

most

famous

of

them

are

two-qubit sy

stems

and

states

exhibiting some

symm

e

try

(like

isotropic

or

Werner

stat

es).

In

particular,

for two

qubits

there

has

been

shown long ago 8

that

(3)

where

)'1

~

),2

~

),3

~

),

4

are

singular values

of

a

matrix

Tkl

=

(Vk

100y

181

0"

Y

Ivi

with

IVk denoting eigenvectors

of

p,

and

0" y

stands

for

th

e

Pauli

matrix.

Simultaneously,

EOF

is a function

of

concurrence,

where

S2(X) =

-x

log2 X - (x

-1)

log2(x

-1)

is

the

binary

entropy

function.

For isotropic

states

ind

exed by a fidelity f E

[0,

1]

and

defined as

where

1'ljJ+

is

the

maximally entangled

state,

it

is well-known

21

that

{~

C(Pf) = m(m -

~

(mf-1),

1

- < f < 1

m

f:::;

11m.

EOF

is also known

22

but

the

relation is much

mor

e complicated.

Unfortunately,

the

gene

ral

algorithm

of calculating concurrence,

EOF

and

other

measures based

on

the

convex roof

construction

is still miss-

ing. Therefore, in

what

follows

we

will focus ourself

on

methods

leading

to

estimations

of

concurrence.

3.

Estimations

of

concurrence

There

are

seve

ral

methods

of

estimating

concurrence.

In

fact, almost every

criterion

of

entanglement

can

be

reformulated in such a way

that

it

can

serve as a source for

appropriate

estimation

of concurrence. Let us recall

here

the

methods

based

on

trace

norms

of

a realigned

matrix

or

partially

transposed

matrix

of

the

state

19,

on

local

uncertainty

relations,

on

geo-

metrical

methods

20

and

others.

We focus here

on

the

following

Br

euer results

24:

203

Proposition

3.1.

Let

f(p)

be

a convex functional

of

the state p

ob

eying

f(l

1/!

(1/!

I)

~

L

.Jf-tkf-tl

k#l

for

every pure state

I1/!

described by

Schmidt

coefficie

nts

yTik. Then

C(p)

::::

J

m(m

2

_

1)

f(p)·

In

what

follows,

we

ar

e going

to

define

f(p)

in

terms

of

the

EW

detecting

p. Let us recall

that

a he

rmitian

ope

rator

W

is

called

an

entanglement

witness (EW) for a

state

p, if

Tr(pW)

< 0 while

Tr(aW)

::::

0 for all

separable

stat

es

a.

Th

ere are

many

examples 25 of entanglement measures

M

(p)

(concurrence, negativity,

robustn

ess, etc.), which

can

be

the

expectation

value of some entanglement witness.

In

the

case of concurrence

one has

th

e following proposition due

to

Breuer

24

Proposition

3.2.

Let W

be

an entangle

ment

witn

ess such that

-(1/!IWI1/!

~

L .Jf-tif-tj

(4)

i,

jf.i

for

every pure state (1). Then for an arbitrary

mi

x

ed

state p detected

by

the witness W

C(p)

::::

J

m(::

_

1)

ITr(pW)I·

(5)

Proposition 3.2 is a simple consequence of Proposition 3.1 if one chooses

f(p)

=

-Tr(pW).

4.

Main

result

Proposition 3.2 distinguishes a class of witnesses satisfying condition (4).

Suppose now

that

W does not satisfy

this

co~dition.

Clearly, for any a > 0

the

rescaled

operator

a

-I

W still defines

an

EW.

Does

a-I

W satisfy (4)?

To answer

this

question let us observe

that

for

I1/!

= L::1

$ilai,

b

i

the

expectation

value of W reads as follows

(1/!

IWI

1/!

= L

.Jf-tkf-tIA~"'()(

1/!

)

,

k,l

where

the

1/!-dependent

matrix

A~"'()

is

defined by

A~"'(

)

(

1/!

)

= Re (

ak

' b

k

IWlal, b

l

(6)

(7)

204

Note,

that

A

(W)

kk

(ak'

bk

IWlak, b

k

is a

mean

value

of

W

on

separable

states,

hence

(8)

It

is clear

that

Ak~)

(1/J)

encodes

the

entire

information

about

W. Moreover,

the

condition

(4) is equivalent

to

(9)

k,l

Let

us observe

that

the

space

of

normalized

vectors

defines a

compact

set

and

hence one

may

define a positive

number

)'(W)

by

the

following proce-

dure

-)'(W)

:=

min

A(W\1/J) .

1jJ

k#l

kl

(10)

It

provides a new

characteristics

of

W.

Now, comes

the

main

result

Theorem

4.1.

For any a

;::

)'(W)

the rescaled entanglement witness

a-I

W does satisfy (4).

The

proof

is

almost

trivial.

One

has

k,l

k

+ L

v'J-LkJ-Ll(Ak~)(1/J)

+

1)

k,l#k

;::

1 + L

v'J-LkJ-LI(Ak~)(1/J)

+ 1) ,

k,l#k

where

we have

used

(8). As a consequence,

if

A(W)(ol,)

>

-1

kl

'f/

- ,

(11)

for all k, l, k

-I-

l

and

every

normalized

1/J,

then

W does satisfy (9),

hence

(4).

Suppose

now

that

the

above

condition

is

not

satisfied.

It

is therefore

clear

that

for

the

rescaled

witness

We<

:=

a-I

W

with

a

;::

),(W),

one

has

(12)

which proves

our

theorem.

It

should

be

stressed

that

the

best

estimation

is

provided

by

the

witness

corresponding

to

a = ),(W),

because

in

this

case

we achieve

equality

in (12)

at

least

for

some

k, l.

205

Summing

up,

we

propose

the

following simple

proc

e

dure

which provides

the

estim

a

tion

of

concurrence of P if

there

is some

EW

W

detecting

p:

W ------t A

k~)

------t

).

(W)

------t W A ------t

Tr

(W

A

p)

------t C

(p

) .

Tr

(WAP)

gives

the

estimation

for concurrence

C(p)

2:

-

~(2

) Tr(WAP).

V~

5.

Examples

Example

5.1.

Let

HA

=

HB

=

em

a

nd

consider

the

flip

operator

m

p = L Ii

(j

I 0

1j

(i I

i,j=l

where

{Ii}

is

the

computational

basis in

em.

Simple calculation gives

Akf)

= (ak' b

k

IFlal,

bl

=

(ak

Ibl)(bk

lal).

Now, evidently

Ak~)

=

I(ak

I b

k

)

12

2:

0 a

nd

for k

-=I-

l

Akf)

= Re((aklbl)(bklal)) >

-1

according

to

orthonormality

of

both

basis.

Example

5.2.

Let

us consider isotropic

states

in

em

0

em

and

let us consider a

EW

26

satisfying

Tr[WiSOpjl =

~

-

f,

m

(13)

(14)

that

is, W is

o

detects

isotropic

state

with

fidelity f > 11m. Since l

'lfi

+

('lfi+

I =

~

pTA

th

e previous example implies for i

-=I-

j

(W

iSO

) 1

A >

--

2J

-

m'

206

which shows

that

,\,(Wiso) = 11m. As a consequence,

the

optimal

W

iso

,

in

the

sense

of

(12), is W

iso

=

mWiso.

Now,

(15)

and

it

turns

out

that

the

estimation

(5) gives

an

exact result,

V

2m

(1

-

~)

=

C(Pf)·

m-1

m

Example

5.3.

In

23

we

have investigated

an

E

-f

amily

(10

> 0)

of

states

in

(C3

0

(C3

where

P3+

denotes

a

maximally

entangled

state

,

di,i+l =

10,

and

the

normalization

factor

1

d

i

,i

+2

=

-,

(mod

3)

10

1

Ne;=-----=-

1

+E+

C

1

(16)

It

turns

out

that

p(E)

is

entangled

if

and

only if

10

i=

1.

Moreover,

its

e

ntanglement

is

detected

by

the

entanglement

witness

1

...

-1

.

-1

1

..

-1

1

.

-1

1

1 .

-1

..

-1

1

(17)

207

for E < 1

and

1

-1·

.

-1

. 1

1

W

2

-1·

1

-1

(18)

..

1

-1·

..

1

for E > 1. To

make

pictures

more

transparent

we replaced all zeros by dots.

Interestingly,

WI

corresponds

to

the

celebrated

Choi positive indecompos-

able

map

and

W

2

to

its

dual.

Numerical

calculations show

that

indeed

A~ii)(1jJ);:::

-1

for i =

1,2.

Hence

one

obtains

the

following

estimation

for concurrence

based

on

the

above

EWs

{

E(E-l)

C(

(E))

> _

~

1 + E +

E2

P - J3

l-E

1 + E +

E2

O<E<1

(19)

E>1

We stress, however,

that

this

estimation

is weaker

than

the

one

obtained

from

the

trace

norm

of

realigned

matrix

(see Fig. 1).

c

Figure

1.

Two

estimations

of

concurrence

as a

function

of

c.

The

dashed

line is for

the

estimation

based

on

IIR(p(c))lll'

The

solid

line is for

the

estimation

based

on

(19).

208

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Quantum

Bio-Informatics

IV

eds. L.

Accardi,

W.

Freudenberg

and

M.

Ohya

World

Scientific

Publishing

Co.

(pp.

209- 222)

CLASSICAL

WAVE

MODEL

OF

QUANTUM-LIKE

PROCESSING

IN

BRAIN

A.

KHRENNIKOV

International

Center

for Mathematical Modelling

in

Physics and Cognitive Sciences

Linnaeus

University, S-35195, Viixjo, Sweden

We

discuss

the

conjectur

e

on

quantum

-like

(QL)

processing

of

information

in

the

brain.

It

is

not

based

on

the

physical

quantum

brain

(e.g.,

Penrose)

-

quantum

physical

carriers

of

inform

at

ion.

In

our

approach

the

brain

crea

ted the

QL

rep-

resentation

(QLR)

of

information

in

Hilb

ert

space

.

It

uses

quantum

information

rules

in decision

making.

The

existence

of

such

QLR

was

(at

least

pr

eliminary)

confirm

ed

by

experimental

data

from

cognitive

psycholog

y.

The

violation

of

the

law

of

total

probability

in

these

experiments

is

an

important

sign

of

nonclas

sicality

of

data.

In

so

called

"constructive

wave

fun

ct

ion

approach"

such

data

can

be

repre-

sented

by

com

plex

amplitudes.

We

present

ed

1,2

the

QL

model

of

decision

making.

In

this

paper

we

speculate

on

a

possible

physical

realization

of

QLR

in

the

brain:

a classical wave

model

producing

QLR

.

It

is

based

on

variety

of

time

scales in

the

brain

.

Each

pair

of

scales (fine -

the

ba

ckgro

und

fluctuations

of

electromagnetic

field

and

rough

-

the

cognitive

image

scale)

induces

the

QL

representation

.

The

background

field

plays

the

crucial

role

in

creation

of

"superstrong

QL

corr

elations

"

in

the

brain.

1.

Introduction

Many

authors

(e.g., Roger Penrose

and

Stuart

Homeroff) advertized

the

model of

the

quantum

brain, i.e.,

quantumness

of

the

brain

is a conse-

quence of

its

composition

of

quantum

systems.

Of

course, such a

brain

is a

processor

of

quantum

information.

This

approach

induces numerous com-

plicated questions

on

physics

of

such brain, e.g., "Is

the

brain

too

hot

to

be

quantum?"

Therefore,

although

we do

not

deny completely

this

interesting

model, we do

not

couple

our

mathematical

model

of

quantum

information

processing

in

the

brain

to

the

"physical

quantum

brain."

In

2001 I

pointed

3

,4

to

coupling between violation of

the

law

of

total

probability (LTP)

and

interference

of

probabilities

in

quantum

mechanics,

e.g., in

the

fundamental

two slits experiment. Interference

(both

classical

and

quantum)

implies

the

violation

of

LTPi

moreover, violation of

LTP

209

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

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