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

330

probability

densities;

they

can

be

considered

as

densities

of

collections of

points

in

the

phase

space.

Remark

4.3.

Under

the

assumption

that

all functions

grl,

...

,r

n

(t)

are

sym-

metric

for each t,

the

system

of

equations

in

Theorem

4.1 implies a

system

of

equations

similar

to

the

classical Bogolyubov system;

in

particular

the

original Bogolybov

system

can

also

be

obtained.

5.

Wigner

measures.

In

this

section

we

discuss some

quantum

analogs of

the

preceding results

using

an

infinite-dimensional version of

the

Wigner

representation

of

the

quantum

states

(the

finite-dimensional

Wigner

representation

is

introduced

in

[3]

and

is developed

in

[4]

(see also

[5,

6]). We define infinite-dimensional

pseudo

differential

operators

with

Weyl symbols

in

the

spirit

of

the

Hida

White

noise calculus

[16].

Those

operators

play

the

role of

quantum

observ-

abIes

and

hence

the

pass from symbols (which

are

classical observables)

to

operators

can

be

called Schrodinger-Weyl

quantization.

We pose

the

Plank

constant

is

equal

to

one. We

introduce

the

notion

of

the

Wigner

measure,

which

substitute,

for infinite-dimensional systems,

the

Wigner

function, for-

mulate

the

equation

which describes

the

evolution of

the

Wigner

measure

and

show

that

the

results

of

the

preceding sections

can

be

extended

to

the

case

of

the

Wigner

measure.

Within

this

section E = Q x

P,

where

the

LeS

Q

and

P

are

such

that

P =

Q*

and

Q = P*; hence E* = P x Q

and

the

mapping

J :

E

--+

E*,

(q,

p) f---+ (p,

q)

is

an

isomorphism.

Let

also

the

mapping

I :

E*

--+

E

be

defined

by

I(p,

q)

=

(q,

-p).

The

LeS

Q (resp.

P)

is called

the

configuration

space

(resp.,

the

momentum

space)

ofthe

Hamiltonian

system

(E,

I,

H).

If

ql,

q2

E Q,PI,P2 E P

then

the

value

that

the

linear functional

(PI,

qd

= J (ql,

pd

takes

at

the

element

(q2,

P2)

is

denoted

by

PI

q2

+

q1P2·

We assume

that

the

Hilbert

space H

of

the

corresponding

quantum

system

is

the

complex space

L2

(Q,

f.L)

where

f.L

is a

P-cylindrical

measure

on

Q;

in

order

to

define infinite-dimensional pseudodifferential

operators

we assume

that

this

is a

Gaussian

measure

and

use

the

ideology of

the

Hida

White

noise analysis calculus. Nevertheless

this

measure

does

not

appear

in

the

final formulae.

Let

the

symbol T

denote

the

von

Neumann

density

operator

(=trace

class positive

operator

in H whose

trace

is

equal

to

one)

that

defines

a

state

of

the

system,

and

let

PT

denote

the

integral

kernel

of

the

density

operator.

If

TJ

is a (X*)-cylindrical

measure

on

a

LeS

X

and

D(TJ)

is

the

collection

331

of all vectors along which

the

measure

r;

is differentiable

then

the

generalized

density

of

r;

is a scalar function

F7)

on

D(r;) whose

logarithmic

derivative

along

any

h E D(r;) is

equal

to

j37)(h,

.).

Even

for

Gaussian

measures

the

generalized

density

is defined only

up

to

a multiplicative

constant.

One

can

show

that

if

r;

is

the

Gaussian

measure

whose Fourier

transform

ii

is defined

by

ii(z) =

exp(-~(zB(z))),

where B is a linear

mapping

of

X*

into

X,

then

F7)(x)

=

Cexp(-~(xB-1(x))).

This

result

shows

that

the

Gaussian

measure

can

be

defined

by

its

generalized density. Below

we

use

the

generalized

density

of

the

Gaussian

measure

in

order

to

define

pseudo differential

operators

in

L2

(Q,

fJ).

Let

the

measure

fJ

be

the

Gaussian

measure

defined

by

its

general-

ized density

as

follows:

Ff.L(q)

=

exp(-~(qB-1(q)))

where B E

L(P,Q)

and

let

l/

be

a Q-cylindrical

Gaussian

measure

on P defined

by

Fl/(p) =

exp(-~(q(B*)-l(q))).

It

is well known

that

if

Q

and

P are

Hilbert

spaces

then

fJ

and

l/

are

a-additive

if

and

only

if

B is a (positive)

trace

class

operator.

For

each "good enough"

scalar

function (we will

not

formulate

the

cor-

responding

analytical

assumptions)

H

on

E(=

Q x P)

the

symbol F de-

notes

the

pseudodifferential

operator

in

L2

(Q,

fJ)

(which is

supposed

to

be

essentially selfadjoint), whose Weyl symbol is

H.

This

means

that

if

rp

E domH(c L

2

(Q,fJ)),

then

(Hrp)(q) = r r H(q1 +q,p)e-ip(ql-q)rp(qd

JpJQ

2

x

(Ff.L

(q)) -

~

(Ff.L

(q1))

-

~

(Fl/(p))

-lfJ(

dqdl/(

dp).

The

integral

ar

r.h.s. is defined as follows:

where

c;;:l = r r e-ip(ql -q)

(Ff.L(

q))

-

~

(Ff.L

(qd) -

~

(Fl/(p))

-1

fJ(

dqdl/(

dp)

JPn

J

Qn

and

Qn

x P

n

=

Fk1,

...

,k

n

;

we

assume

that

for

any

n

the

subspace

Fk1,

...

,k

n

is

contained

in

the

domain

of

the

integrands

of

the

latter

finite-dimensional

332

integrals

and

use

the

regularisations of

finite=dimensional

integrals which

are

defined

by

the

following way.

If

f E Lioc(IRn),

then

we say

that

the

integral

JlRn

f(x)dx

exists

if

for

any

rp

E V(IRn), for which

rp(O)

=

1,

the

limit lima-+oo

JlRn

rp(ax)f(x)dx

exists

and

then

by

definition

JlRn

f(x)dx

= lima-+oo

JlRn

rp(ax)f(x)dx

(the

definition does

not

depend

on

the

choice of

rp).

For

any

h E E

the

symbol

h

denotes

the

pseudodifIerential

operator

in

L

2

(Q), whose Weyl

symbol

is

Jh(E

E*)i

in

particular

if h =

q+p(=

(q,p))

then

h = P +

g.

Let

us

mention

that

if

qo

E Q

then

go

is

the

operator

of

the

momentum

in

the

direction

of

qo

(but

not

of

the

coordinate).

Remark

5.1.

Let

E =

{h

:

hE

E}i

then

the

mapping

P-

:

Jh

f--+

h,

E*

-+

E is a liner isomorphism (we

assume

that

E is

equipped

with

the

natural

structure

of

a vector space).

The

extension of

P-

to

a linear

mapping,

of

the

space

generated

by

E*

and

the

function

on

E whose values

at

each

points

are

equal

to

one,

into

the

space

of

operators

in

L2

(Q,

JL),

defined

by

the

assumption

that

the

image

of

this

function is

the

multiplication

by

i, is called

the

Scrodinger

representation

of

the

canonical

commutation

relations.

Definition

5.1.

The

Weyl

operator

W(h)

generated

by

h E E is defined

by:

W(h)

=

e-

ih

.

The

Weyl function

WT

corresponding

to

the

state

(=den-

sity

operator)

T

of

the

quantum

system,

whose

Hilbert

space

is L2

(Q,

JL),

is

the

function

on

E,

defined

by

WT(h)

=

trTW(h).

Definition

5.2.

The

Wigner

measure

WT

generated

by

the

state

(=density

operator)

T

of

the

quantum

system,

whose

Hilbert

space is L2

(Q,

JL),

is

the

image,

with

respect

to

the

mapping

J-

1

:

E*

-+

E,

of

the

E-cylindrical

measure

on

E*

whose Fourier

transform

is

the

Weyl function.

This

means

that

r e

i

(P,Q2+Q,P2)W

T

(dq1

d

pd

= W

T

(q2,P2).

lQxP

Remark

5.2.

One

can

show

that

the

Wigner

measure

can

also

be

defined

as

the

integral

kernel

of

the

linear functional P

f--+

trT

F

on

the

vector

space

of

the

Weyl symbols

of

(bounded)

pseudodifIerential

operators

in

L

2

(Q,JL).

This

means

that

for

any

such Weyl's

symbol

P

the

following

identity

holds

trTF

= tk

P(q,p)WT(dqdp)

(1)

333

(see (cf.[IO] where only finite-dimensional spaces

are

considered

when

the

Wigner's

measure

can

be

substituted

by

its

density, which is called

the

Wigner

function).

Remark

5.3.

The

measure

wi?

on

Q defined

by

Wi?(dq) =

Jp

WT(dqdp)

is

the

(cylindrical)

probability

on

Q describing

the

distribution

of

results

of

measurements

of

the

coordinates.

To

formulate

the

equation

describing

the

evolution

of

the

Wigner

mea-

sure

we

need a definition

of

what

one could call a function

of

the

Poisson

bracket. Below

we

use some topological

tensor

powers

of

the

phase

space

but

we

do

not

discuss

the

topologies

of

them.

We use

the

assumptions

and

definitions of sections 3

and

4.

Let, for

any

n E

N,

the

symbol

I@n

denotes

the

mapping,

of

a

proper

subspace

of

Bn(E),

into

E@n

generated

by

n-th

tensorial power

of

I (here

E@n

is a topological

tensor

product

of n copies

of

E).

Let

moreover, for

any

two scalar functions F

and

G

on

E,

and

Cr;) H(x) = {F, G}(n)(x)

(of course, C

G

# Cr;»).

Finally

let, for

any

a >

0,

the

operator

(sin)aC

c

be

defined

by

00

2n-l

(

. )

r

'"'

a

r(2n-l)

sm

al..-C =

~

(2n _1)!l..-c

(we

do

not

discuss now

in

which sense

the

series converges)

and

let

(sin)aCb

be

the

operator

in

a space

of

E*

-cylindrical

measures

on

E,

which is

adjoint

to

(sin)aCc.

Theorem

5.1.

The dynamics

of

the Wigner measure is governed by the

following equation:

The

proof

can

be

obtained

by

combination

of

technique of

the

theory

of

differentiable measures

and

some

methods

of

developing

equations

describ-

ing

the

evolution

of

the

Wigner

function

[4], [5],

[6].

334

Theorem

5.2.

Let

z;

E

Mg(E),

{ek"

..

. ,ekn}, {e

rp

.

..

,e

r

",}

C B

and

let

the

Hamiltonian

function

H

be

Gr" .

..

,r

Tn

-cylindrical.

If

{e

rp

..

. , e

rTn

} C

{ek" ... ,ek

n

},

then

f

2

(sin

H

.c;,.c

v

\x)

=

kl

J·

..

,kn

J

2(sin)~£H

(f{

vr r }U

{k

k })( ... xs" ...

,x

s

...

)dxs,

..

. dxs .

2 T l , ·

..

,T;n

1 ,

···

,

rn

1,

·

··

, n P P

Theorem

5.3.

A

function

W(·)

taking values

in

Mg(E)

is a solution

of

the equation governing the

dynamics

of

the

Wigner

measure

if

and

only

if

the

functions

t

f--'

gr"

...

,r

n

(t), defined by the equalities

gr"

..

.

,r

n

(t) =

fr

2,(

,

s.'.·n.,r

)

n~.c:

H(

W

(

t ))

fi

satisfy

the following

in

nite

system

of

differential equa-

tions:

g~"

. . ,kn (t)(·)

= L J

2(sin)~£Hr",rTn

(g{

r"

...

,r=}U{k"

...

,kn}(t))

where the

summation

is

over

all

finit

e sets

{rl,

... ,

rm}

of

natural

numbers

for

which

{rl,

... , r

m}

n

{k

1

,

...

, k

n

}

-f.

0,

and,

in

accordance with what has

been

said

above,

if

{rl,

... ,

rm}

n

{k

1

,

...

, k

n

} =

{rl,

... ,

rm},

then

the integral

sign

is

assumed

to

be

missing.

This

follows from

the

theorems 5.1

and

5.2.

Suppose

that

the

operator

(sin)£'H is defined on

MOO(E).

Theorem

5.4.

Let

,

for

eve

ry

finite set

{rl'"''

r

n}

of

natural

numb

ers,

gr"

...

,r

n

be a

function

of

a real argume

nt

taking values

in

the space

of

(bounded continuous)

functions

on

Fr"

...

,r

n

, and,

for

each t the

family

of

functions

gr"

...

,r

n

(t) is compatible

and

determin

es a unique measure

w(t)

E MOO(E). Then, the

function

w(·) is a solution

of

the equation

governing

th

e

dynamics

of

the

Wigner

measure,

if

the

family

of

functions

335

grl,

...

,r

n

is a solution

of

the

system

of

equations

x (.) +

1

""

lim

--

~

N-.oo

Nm-p

{jl

,,,.,

jp

}={

rl

,,,. ,

r",}

n {

kl

,,,.

,k

n

}

If

it

is

not

assumed that the Plank constant is equal to one then

it

is neces-

sary

to substitute

"2(sin)~"

by

"~(sin)~

",

where fi is the Plank constant.

Remark

5.4.

From

Theorem

5.4, which is similar

to

theorem

4.1, one

can

deduce a

quantum

version

of

the

classical Bogolyubov

system

of equations

and

also some

other

similar systems of equations.

It

is

also

worth

noticing

that

the

integrals

Jp

w(t)(dq,

dp) are not probabilities

and

hence (Remark

5.3)

the

measures

w(t)

are

not

Wigner measures.

6.

Generalization

of

Poincare's

model.

In

this

section

we

formulate a proposition (Proposition 6.1 below)

that

improve

the

similar proposition related to

the

Poincare model ([1],

[8],

[9])

of irreversible

but

symmetrical

with

respect

to

time

evolution. Actually in

the

original model one assumes

that

the

initial probability

distribution

, on

the

phase space,

is

the

product

of identical one-dimensional distributions;

in

Proposition 6.1

we

consider some general

distribution

on

the

phase space

and

prove

that

this

distribution

tends

(both

when time

tends

to

+00

and

to

~oo)

to

the

same limit as in

the

case when

the

initial distribution

is

the

product

of

the

identical distributions.

Th

e similar improvement

is

valid for

the

quantum

version

[10]

of

the

Poincare model.

Proposition

6.1.

Let

E = Q x

P,

E'

= P x

Q,

I(p,

q)

=

(q,

~p),

and let,

2

in

natural notations, 'H(q,p) =

2:

f,;:;

(this Hamiltonian

system

describes

noninteracting particles). Let

v(-)

be

a solution

of

the Liouville equation

for probability measures such that the finite-dimensional projections

of

the

measure

v(O) on E satisfy the conditions

of

Theorem

4·1

in

flO).

If

Qn is

a finite-dimensional subspace

of

the space Q and,

for

each t,

P(t,')

is the

density

of

the projection

of

the measure

v(t)

on

Qn, then, for any compact

336

subsets

Kl

and K2

of

Qn! the following holds:

JK,

P(t,

q)dq

mesK

1

J

--->

, t

--->

±oo,

K2

P(t,

q)dq

mesK

2

where

mes

denotes the Lebesgue measure on Qn (so one could say that

the probability distribution

on

the configuration space tends to the

uniform

distribution) .

The

proof

uses

Theorem

3.2

and

is similar

to

the

proof

of

Theorem

4.1

from

[10].

This

proposition

somewhat

strengthens

a result

of

[8]

which goes back

to

Poincare

[1].

In

[8]

it

was

actually

proved in

the

special case when

the

initial probability measure

v(O)

is

the

product

of

copies

of

a one-dimensional

probability measure.

A completely similar

proposition

is valid for

quantum

systems because

for systems

with

quadratic

Hamiltonian

functions

the

equations

for

the

Wigner

measure coincide

with

the

Liouville equations

(cf.

[10]).

References

1.

H.Poincare.

J.de

Physique

theorique

et

appliquee,

4 serie, 5,

(1906),369-403.

2.

N.N.Bogolyubov.

Problems

of

dynamical

theory

in

statistical

physics.

Moscow, 1946 (in

Russian;

there

exists

an

English

translation).

3.

E.Wigner.

Phys.Rev.,

40, (1932), 749.

4.

J.

E. Moya!.

Quantum

mechanics

as a

statistical

theory.

Proc.

Cambridge

Philos. Soc., v. 45, (1949), 99-24.

5.

G.

B.

Folland.

Harmonic

Analysis

in

Phase

Space.

(Princeton

Univ.

Press,

1989) .

6.

Kim

Y. S., Noz M.

E.

Phase-Space

Picture

of

Quantum

Mechanics.

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Approach.

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Radu

Balescu,

Equilibrium

and

nonequilibrium

statistical

mechanics,

vo!'l

(John

Wiley

and

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V. V. Kozlov,

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9. V.V.Kozlov. Reg.

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V.V.Kozlov,

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Vo!' 51,

1,

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O.G.Smolyanov,

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12.

O.G.Smolyanov,

H.von

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T.

321,

ser.

1.

(1995), 103-108.

13.

N.Bourbaki

,

Integration,

Chapitre

6

(Springer,

2007).

337

14.

O.G.Smolyanov,

H.v.Weizsacker.

Smooth

probability

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associ-

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Inf. Dimens.

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Probab.

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Relat.

Top. V.2,

1,

(1999),51-78.

15. L.

Accardi

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O. G. Smolyanov.

Generalized

Levy

Laplacians

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Cesaro

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Doklady

Mathematics,

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1,

(2009), 1-4.

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T.Hida,

H.H.Kuo,

J.Pothoff,

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White

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An

infinite

dimensional

calculus.

Kluwer

Academic, 1993.

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Quantum

Bio-Informatics

IV

eds. L.

Accardi

,

W.

Freudenberg

and

M.

Ohya

World

Scientific

Publishing

Co.

(pp.

339-354)

ANALYSIS

OF

SEVERAL

CATEGORICAL

DATA

USING

MEASURE

OF

PROPORTIONAL

REDUCTION

IN

VARIATION

KOUJI

YAMAMOTO,

KOUJI

TAHATA, NOBUKO MIYAMOTO

AND SADAO TOMIZAWA *

Department

of

Information

Sciences, Tokyo University

of

Science,

Noda

City, Chiba 278-8510, Japan

* E-mail: tomizawa@is.noda.tus.ac.jp

For

a

two-way

contingency

table

with

nominal

row

and

column

variables,

the

mea-

sures

which

describe

the

proportional

reduction

in

variation

(PRV)

from

the

mar-

ginal

distribution

of

one

variable

to

the

conditional

distribution

given

the

other

variable

are

proposed

by

Goodman

and

Kruskal

(1954),

Theil

(1970),

and

Freeman

(1987, p. 101).

Tomizawa,

Seo

and

Ebi

(1997),

and

Miyamoto,

Usui

and

Tomizawa

(2005)

proposed

the

generalization

of

those

measures.

Tomizawa,

Miyamoto

and

Yajima

(2002),

and

Yamamoto

and

Tomizawa

(2009)

proposed

the

PRV

measures

for a

nominal-ordinal

contingency

table

and

for

an

ordinal-ordinal

contingency

table,

respectively.

The

present

paper

(1)

reviews

these

PRY

measures

and

(2)

an-

alyzes

and

compares

between

several

categorical

data

using

these

PRY

measures.

Keywords:

Concentration

coefficient;

Measure,

Proportional

reduction

in

vari-

ation

;

Square

contingency

table;

Total

uncertainty

coefficient.

1.

Introduction

The

data

in Table 1

taken

from

the

Meteorological Agency in

Japan

are ob-

tained

from

the

daily

temperatures

at

Nagasaki City,

Japan,

in two years,

2001

and

2002, using

three

levels, (1) below normals, (2) normals

and

(3)

above normals (see

Tahata,

Takazawa

and

Tomizawa, 2008).

The

observa-

tions, say,

Iij,

in

the

(i,

j)-th

cell indicate

that

for each

of

Iij

days

in

365

days (i.e., from 1

January

to

31

December),

the

temperatures

in

two years

are

i

in

2001

and

j

in

2002.

Table 2 is

taken

from Tallis (1962)

and

constructed

from

the

cross-

classified

data

of

Merino ewes according

to

the

numbers

of

lambs

born

in

consecutive years, 1952

and

1953 (also see Bishop, Fienberg

and

Holland,

1975, p. 288; Miyamoto, Niibe

and

Tomizawa, 2005).

339

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