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150
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
x
Figure
1.
The
graph
of
D
EN
for (/1 (x)
The
marginal
states
p,
(J
of
8
1
(c:)
can
be
calculated
as
1
p =
(J
= 3
(eoo
+
ell
+
e22)
Therefore, we have
(
)
3
[3
3c:
1
3/
c:
J
D 8
1
(c:):
p,(J =
-A
log A +
clog
j[
+
~logA
+log3
,
(16)
(17)
with
A defined in (12).
Actually
, one
can
classify
the
state
of
8
1
(c:)
by
the
value
of
c:
(see Ref. 17).
Theorem
3.1.
(1) 8
1
(c:)
is separable
iff
c:
= 1;
(2) 8
1
(c:)
is both
PPT
and entangled for
c:
cJ
l.
The
corresponding
graph
of
DEN
for 8
1
(c:)
is shown
in
Fig.
1.
The
maximal
value
corresponds
to
c:
= 1
and
is given
by
D
(8
1
(1) :
p,
(J)
=
~
log 3
~
0.7324.
151
3.2.
Horodecki
state
Let us consider
another
example of circulant
states
introduced
in Refs. 15,
16:
(0
~
0:
~
5),
where
1
1/J=
J3(eo0eo
+
e10
e
1
+
e20
e
2),
1
8+
=
"3
(eoo
0
ell
+
ell
0
e22
+
e22
0 eoo),
1
8 - =
"3
(ell
0
eoo
+
e22
0
ell
+
eoo
0 e22) ,
and
hence
221
0
0 0
2
0 0
0
2
21 21
0
<>
0 0 0 0 0
0
0
21
0 0
5-<>
0 0 0 0 0 0
----:2l
0 0 0
5-<>
0 0 0
0
0
----:2l
2
0 0 0
2
0 0
0
2
21
21
21
0 0 0 0 0
<>
0 0 0
21
0 0 0 0 0 0
<>
0 0
21
0 0 0 0 0 0 0
5-a
0
----:2l
2
0 0 0
2
0 0 0
2
21
21
21
(18)
(19)
(20)
(21)
(22)
The
marginal
states
p
and
(J
are
the
same as (16).
The
von
Neumann
entropy of 8
2
(0:)
is calculated
to
be
2 2
0: 0:
5-0:
5-0:
5
8(8
2 (0:)) =
-7
log
7
-7
log
7
-
-7-
log
-7-
+ 7 log
3.
(23)
Therefore, we have
2 2
0:
0:
5-0: 5-0:
2
D(8
2
(0:):
p,(J) =
-7
log
7
-7log7
-
-7-log-7-
-7log3.
(24)
As in
the
case of
the
circulant
state
8
1
(E),
it
is
known
that
the
state
8
2
(0:)
can
be
classified by
the
values of
0:
(see Refs.
[15,
16]).
Theorem
3.2.
(1) 8
2
(0:)
is separable
if
and
only
if
0:
E [2,3];
(2) 8
2
(0:)
is both entangled
and
PPT
if
and
only
if
0:
E [1,2) U (3,4];
(3) 8
2
(0:)
is
not
PPT
if
and
only
if
0:
E [0,1) U (4,5].
152
0.8
0.75
0.7
1:::
1;' 0.65
'"
g;
0.6
~
i
0.55
C')
OJ
o 0.5
~
c::
X 0.45
.;,
f 0.4
X
0.
0.35
OJ
o
~
0.3
~
0.25
C>
o
~
2
2.5
3
3.5
4 4.5
5
x
Figure
2.
The
graph
of
DEN
for (l2(X)
Now, since 8
1
(10)
and
8
2
(a) have common marginal states,
we
can
com-
pare
DEN
for them. One
has
for example
D(8
1
(1)
:
p,O")
~
0.7324 < D(8
2
(3.1)
:
p,O")
~
0.7587. (25)
Note, however,
that
8
1
(1) is separable while 8
2
(3.1) is entangled.
This
example gives negative answer
to
our
original question.
The
corresponding
graph
of
DEN
for 8
2
(a)
is
shown in Fig.
2.
4.
Bell
diagonal
states
In
this
section, we analyze Bell diagonal
states
for d = 3 (see Refs.
[4,
5,
6, 7,
8, 9]). Consider
th
e Hilbert space 1t =
re
3
with
the
standard
basis
{eo,
e1,
e2
}. We set
(26)
and
define flk,e for any
k,
C (0
:S
k,
C
:S
2)
as
flk
,e
= (Wk
,e
® 1
3
)fl
o
,o
,
(27)
153
where
Wk,e
means
the
circle
action
given
by
(i=0,1,2).
(28)
Finally, we define
l-a-p
P(
)
e(
a,
p) = 9
I3
0
I3
+ aIOo,o)
(0
0
,0 1
+
2"
1
0
1,0)
(0
1
,01
+ 1
0
2,0)
(0
2,0 1
(29)
Note
that
e(
a,
p)
can
be
written
in
the
form
1+2(0:+(3)
0
0 0
20:-(3
0
0 0
20:-(3
9
-6-
-6-
0
1-0:-(3
0 0 0 0 0
0 0
-g-
O
0
1-0:-(3
0 0 0 0
0
0 0
0 0
0 0 0
20:-(3
0
1+2(0:+(3)
0 0 0
200-(3
-6-
9
-6-
0 0
0 0
1-00-(3
0 0
0
0 0 0
0 0 0
0 0
0 0 0 0
1-00-(3
0
-g-
20:-(3
0 0
0
200-(3
0 0 0
1+2(00+(3)
-6-
-6-
9
Note
that
tr
e(a,
p)
= 1
and
the
eigenvalues
of
e(a, p)
are
given
by
l-a-p
9
-2a
+ 7p + 2
18
8a
- p + 1
9
(30)
Therefore, e(a, p) defines a
state
if
and
only
if
the
parameters
a,
p satisfy
a+p:::;
1,
2a
-7p:::;
2,
-8a
+ p:::;
1.
(31)
On
the
other
hand,
let
us
ask
for
the
condition
when
e(a, p)
can
be
positive
under
partial
transpose.
The
partial
transpose
Te(a, p)
of
e(a, p) is given
by
1+2(00+(3)
0 0
0 0 0 0 0 0
9
0
1-00-(3
0
20:-(3
0 0 0 0 0
-g-
-6-
0 0
1-0:-(3
0 0 0
200-(3
0 0
-g-
O
00-
0
1-00-
0 0 0 0 0
6
-g-
O 0 0 0
1+2(0:+(3)
0 0 0 0
9
0 0 0 0 0
1-00-(3
20:-(3
0
0 0
0 0
0
0
0
0 0
200-(3
1-00-(3
0
-6-
-g-
O 0 0 0 0 0 0
1+
2
(00+(3)
9
154
From
this
one
can
check
that
the
trace
is equal
to
1
and
the
eigenvalues
are
1+2(a+,8)
9
-8a
+,8
+ 2
18
4a
-
5,8
+ 2
18
Hence,
the
PPT
condition holds for
a,
,8
such
that
1
a +
,8
?:
-"2'
8a
-
,8
::;
2,
-4a
+
5,8
::;
2.
(32)
(33)
On
the
set
which consists of
a,
,8
satisfying
both
(31)
and
(33),
the
state
is
either
separable
or
bound
entangled. However,
it
was shown
7
that
for
this
class all
PPT
states
are
separable.
Let
us consider for example
the
line
,8
= 3/
5.
One
has: B(a, 3/
5)
is entangled if
and
only if
a E [0,1/4) U (13/40,2/5] ,
(34)
and
it
is
separable
iff
a E
[1
/ 4,13/
40]
.
(35)
The
marginal
states
p
and
a
are
calculated
to
be
the
same
as (16). Hence
we
obtain
DEN
for e(a,
3/5)
(
(
/
)
. )
_
lOa-4
l
(2
-
5a)_
31
-
10a
l
(31-lOa)
DBa,3
5 . p, a -
15
og
45
45
og
90
40a + 2 1
(40a
+
2)
1 3
- og - og .
45 45
(36)
By
a simple calculation,
it
easily be verified
that
D(e(a,
3/5)
p,
a)
is
monotonically decreasing if
a E [1/20,2/5). Hence,
th
e values
of
DEN
for
a E
[1
/
20,1/4)
are
greater
than
the
ones for a E
(1
/4,13/40) (see Fig. 3).
This
gives also
the
negative answer
to
our
original question.
5.
Conclusions
We provided several examples
of
bipartite
quantum
states
for which one
can
easily
compute
DEN. Moreover, since
they
possess
the
same
marginal
states
(maximally mixed) one
can
compare
th
e corresponding degree of entangle-
ment.
It
turned
out
that
separable
state
can
hav
e
stronger
correlation
(with
respect
to
DEN)
than
an
entangled
state
.
This
observation is in-
consistent
with
the
conventional
understanding
of
quantum
entanglement.
Consequently, we propose
that
the
meaning
of
DEN
should
be
changed
to
express
the
intensity
not
of
entanglement
but
of
correlation.
The
details
will
be
discussed
in
the
forthcoming
paper
19.
155
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
·0.1
0 0.05 0.1 0.15 0.2 0.25
0.3 0.35
0.4
x
Figure
3.
The
graph
of
DEN
for B(x,
3/5)
Acknowledgments
This work was
partially
supported
by
the
Polish
Ministry
of
Science
and
Higher
Education
Grant
No
3004/B/H03/2007/33.
References
1.
M. A. Nielsen
and
I.
L.
Chuang,
Quantum
computation and quantum
infor·
mation,
Cambridge
University Press,
Cambridge,
2000.
2.
R. Horodecki, P. Horodecki, M. Horodecki
and
K.
Horodecki:
Quantum
en·
tanglement, Rev. Mod.
Phys.
81
(2009).
pp
865-942.
3. L. Accardi,
T.
Matsuoka,
M. Ohya:Entangled Markov chains are indeed
en·
tangled, Infin. Dimens. Anal.
Quantum
Probab.
Relat.
Top., Vol. 9 (2006),
pp
379-390.
4.
R. A.
Bertlmann,
and
Ph.
Krammer:
Simplex
of
bound entangled multipartite
qubit states,
Phys.
Rev. A 78(2008), 10pp.
5. R. A.
Bertlmann,
and
Ph.
Krammer:
Geometric entanglement witnesses and
bound entanglement,
Phys.
Rev. A 77, 024303 (2008), 4pp.
6.
B.
Baumgartner,
B. Hiesmayr,
and
H. Narnhofer:
State
space
for
two qutrits
has a phase space structure
in
its cone,
Phys.
Rev. A 74 (2006),
l4pp.
156
7.
B.
Baumgartner,
B. Hiesmayr,
and
H. Narnhofer: A special simplex
in
the
state space
for entangled qudits,
J.
Phys.
A:
Math.
Theor.
40,
7919 (2007).
8. B.
Baumgartner,
B.C.
Hiesmayr
and
H. Narnhofer: The geometry
of
biparti-
cle qutrits including bound entanglement,
Physics
Letters
A,
372
(2008),
pp
2190-2195.
9. D. Chruscinski, A. Kossakowski,
T.
Matsuoka,
K. Mlodawski, A class
of
Bell
diagonal states and entanglement witnesses,
to
apper
in
Open
Systems
and
Inf.
Dynamics
10. V. P. Belavkin
and
M. Ohya:
Quantum
entropy and
information
in
discrete
entangled state,
Infin. Dimens. Anal.
Quantum
Probab.
Relat.
Top., Vol. 4
(2001),
pp
137-160.
11.
V. P. Belavkin
and
M. Ohya: Entanglement,
quantum
entropy and
mutual
information,
Proc.
R. Soc.
London
A 458 (2002),
pp
209-231.
12. D.
Chruscinski
and
A. Kossakowski: Circulant states with positive partial
transpose,
Physical
Review A
76
(2007), 14 pp.
13. D.
Chrusciilski
and
A.
Pittenger:
Generalized Circulant Densities and a Suffi-
cient
Condition for Separabil ity,
J.
Phys.
A:
Math.
Theor.
41
(2008) 385301.
14. D.
Chruscinski
and
A. Kossakowski, Multipartite Circulant States with Pos-
itive Partial Transpose,
Open
Sys.
Information
Dyn.
15
(2008)
pp
189-212.
15. P. Horodecki, M. Horodecki,
and
R. Horodecki:
Bound
entanglement can
be
activated,
Phys.
Rev.
Lett.
82,
(1999),
pp
1056-1059.
16. M. Horodecki, P. Horodecki
and
R. Horodecki: Mixed state entanglement
and quantum condition,
In
Quantum
Information,
Springer
Tracts
in
Modern
Physics 173 (2001),
pp
151-195.
17.
J.
Jurkowski, D. Chrusciilski
and
A. Rutkowski: A class
of
bound entangled
states
of
two qutrits,
Open
Syst. Inf. Dyn., 16 (2009), no 2-3,
pp
235-242.
18. M.
Ohya
and
T.
Matsuoka:
Quantum
entangled state and its characterization,
Found.
Probab.
Phys.
no 3, 750 (2005),
pp
298-306.
19. D.
Chrusciilski, Y.
Hirota,
T.
Matsuoka
and
M. Ohya:
Quantum
correlation
and essential q-entanglement,
in
preparation.
Quantum
Bio-Informatics
IV
eds.
L.
Accardi,
W.
Freudenberg
and
M.
Ohya
© 2011
World
Scientific
Publishing
Co.
(pp.
157-171)
A
COMPLETELY
DISCRETE
PARTICLE
MODEL
DERIVED
FROM
A
STOCHASTIC
PARTIAL
DIFFERENTIAL
EQUATION
BY
POINT
SYSTEMS
KARL-HEINZ
FICHTNER\
KEI INOUE
2
AND MASANORI OHYA
3
1 Friedrich-Schiller- Universitiit Jena,
Fakultiit
fur
Mathematik
und
Informatik,
Institut
fur
Angewandte Mathematik,
07737 Jena, Germany.
E-mail: kaTi-heinz.jichtner@uni-jena.de
2 Department
of
Electrical Engineering,
Tokyo University
of
Science, Yamaguchi,
Sanyo-Onoda, Yamaguchi
756-0884, Japan.
E-mail: inoue@ed.yama.tus.ac.jp
3
Department
of
Information Sciences,
Tokyo University
of
Science,
Noda
City, Chiba 278-8510, Japan.
ohya@rs.noda.tus.ac.jp
Several
scientific
and
technical
problems
can
be
described
by
a
stochastic
partial
differential
equation.
The
solution
of
the
equation
could
be
considered
as
the
limit
of
a
suitable
discrete
particle
model.
The
existence
of
such
a
kind
of
approximation
was
discussed
in
5.
A
completely
discrete
particle
model,
which
is
constructed
to
simulate
by
computer,
is
considered
in
3.
In
this
paper
we give
proofs
of
some
lemmas
which
are
used
to
prove
the
main
theorem
in
3.
1.
Introduction
Several scientific
and
technical
problems
can
be
described
by
partial
differ-
ential
equations
of
the
type
av
1
-=-~v+f.
at 2
(1)
A
more
precise
model
is
to
take
into
account
stochastic
disturbances,
i.e.,
to
the
right
hand
of
(1)
there
have
to
be
added
corresponding
source
terms.
Often
this
disturbance
occurs
as
the
so-called
white
noise. So we
regard
the
stochastic
partial
differential
equation
av
1
at (x, t) =
2~v(x,
t) +
f(x,
t) +
O"(x,
t)
~(x,
t).
(2)
157
158
Note
that
(2) is connected
with
the
idea
of
diffusion
and
generation
of
particles in
random
spatial-temporal
points. Therefore
the
solution
of
(2)
could
be
considered
as
the
limit
of
a
suitable
discrete
particle
model.
The
existence
of
such a
kind
of
approximation
was discussed in
5,
and
the
related
problems were considered in
4,6.
In
3 we
construct
a completely discrete
particle
model
approximating
a continuous
system
being differe
nt
from
the
limit
considered in
5.
In
this
paper
we
give proofs
of
some
lemmas
which
are
used
to
prove
the
main
theorem
in
3.
2.
Basic
notations
The
Poisson
process
Following
7,
we
introduce
the
notions
and
notations
in
this
section.
Let
G
be
a Polish space, i.e., a
separable
topological space for which
there
is
a
complete
m
etr
ic. We
denote
the
(I-algebra
of
the
Borel
sets
in
G by ®
and
the
ring
of
the
bounded
Borel
sets
of
®
by
.e.
In
the
case G = R d we
use
notations
® = ®d, .e =
.e
d
.
Further,
M
denotes
th
e
set
of
all integer-
valued measures
on
® being finite
on.e.
Let
9Jt
be
the
smallest (I-alge
bra
of
M-subsets
which makes
the
function
<I>
f---+
<I>(X),
<I>
E M
measurable
for each X
from.e.
A
measurable
mapping
from a
probability
space [n, F,
P]
into
[M,9Jt]
is called a
random
point
system
in
G,
the
dis-
tribution
on
[M,9Jt]
generated
by
such a
random
point
system
is
said
to
be
a
point
process
with
the
phase
space G.
To
each
measure
H on
[M,9Jt],
a
measure
PH
on
®, called
the
intensity
measure
of
H,
is assigned
by
PH(X):=
J <I>(X)H(d<I», X E ®.
Let
Xl"
'"
Xm
be
elements
of
®.
The
meas
urable
mapping
<I>
f---+
[<I>(Xd,
...
,
<I>(Xm)]
from
Minto
Nm
transforms
each
measure
H
on
[M,9Jt]
with
H({<I>:
<I>(X
i
)
= oo}) =
0,
i =
1,
.
..
,m
into
a
measure
Hx,oo.x=
on
[Nm
,')'(Nm)].
If
H is a
point
process,
the
measures Hx,.oox=
are
called
the
finite dimensional
distributions
of
H.
159
To
any
'c-finite measure
f.L
on
®
we
can
associate
the
Poisson process
Pep)
with
an
intensity measure
f.L
characterized by
p(p)X,
...
x=
= xz;,(7rp(Xi)
for all finite sequences
(Xi)~l
of
pairwise disjoint
sets
from'c.
Here 11")"
A > 0 denotes
the
Poisson
distribution
with
expectation
A.
For all 'c-finite
measure
f.L
on
®
it
holds
(p(p)t
* =
p(np)
(3)
where
(p(p)t*
is
the
n-th
convolution-power
of
p(J-t) (cf.
7).
l7-additive
processes
Definition
2.1.
A
random
process
7]
= (7](B);
BE
,c)
on
[n,
F,
Pl
is called
a-additive
if for all sequences
(Bi)~l
of pairwise disjoint elements from
,c
such
that
U~l
Bi
E
,c,
we have
Definition
2.2.
A
a-additive
process
7]
= (7](B); B E
,c)
is called de-
composable if for all finite sequences
(Bi)~l
of pairwise disjoint
bounded
measurable sets
the
random
variables 7](Bd,
...
,
7](Bm)
are
independent
(cf.
2).
If
\fF
is a
random
point
system
in G
then
the
family
\II
:=
(\fF(B); B E
,c)
represents a
a-additive
process. Moreover, if
the
distribution
of
\fF
is a
Poisson process
then
\II
is a decomposable process.
The
white
noise
based
on
the
Lebesgue measure
on
[Rd,
®dl
as well as
the
so-called generalized
Brownian
motion
represents a decomposable
proc
ess
in
this sense
(cf.
8).
Here U is called
to
be
a generalized Brownian
motion
on
G
with
noise
intensity
measure
f.L
if
(i) each
random
variable
U(B)
has
law N(O,
f.L(B))
,
(ii)
the
variables
U(B
1
),
...
, U(Bm)
are
independent
whenever
B
1
,
...
,Bm
are pairwise disjoint.
Definition
2.3.
For each n E N let
7](n)
and
7]
be
a-additive
processes
on
[n,
F,
Pl.
The
sequence
7](n)
is said
to
converge
toward
7]
if for
every finite sequence
(Bi)~l
of
elements from
,C
the
random
vector