Accardi L., Freudenberg W., Obya M. (Eds.) Quantum Bio-informatics IV: From Quantum Information to Bio-informatics
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260
Proposition
3.2.
Let
~
: M
2
(C)
-+
M
2
(C)
181
M
2
(C)
be
a linear operator
with
a H
aar
state
T.
Then
the bilinear
form
~
*
(-
181
.)
is positive
if
and
only
if
one
holds
(6)
From
the
proof
of
Proposition
3.1
and
the
last
proposition we
get
Corollary
3.1.
Let
R(f)
be
the corresponding
matrix
to
an
operator given
by
(4).
Then
IIIRIII
:'S
1
if
and
only
if
(6) is satisfied.
ReIllark
3.1.
Note
that
characterizations
of positive
maps
defined
on
M2
(<C)
were considered
in
11.
Characterization
of
completely positive
map-
pings from
M2
(C) into itself
with
invariant
state
T was established in
18.
Next
we would like
to
find some conditions for q.q.o.
to
be
the
Kadison-
Schwarz ones.
Let
~
:
M2
(C)
-+
M2
(C)
181
M2
(C)
be
a linear
operator
with
a
Haar
state
T,
then
it
has a form (4). Now we
are
going
to
find some conditions
to
the
coefficients
{bml,d
when
~
is a Kadison-Schwarz
operator.
Given
x =
Wo
+
wa
and
state
<p
E
S(M2(C))
let us
denote
Xm
= ((b
m1
, w), (b
m2
,
w),
(b
m3
,
w)),
fm
=
<p(a
m
),
(7)
aml
=
(Xm,Xl)
-
(Xl
,X
m
),
Iml
= [xm,xt] + [xm' Xl],
(8)
where
m,
l =
1,2,3.
Note
that
here
the
numbers
ami
are
skew-symmetric,
i.e.
ami
=
-ami.
By
7r
we shall
denote
mapping
{I,
2, 3,
4}
to
{I,
2,
3}
defined by
7r(1)
=
2,
7r(2)
=
3,
7r(3)
=
1,
7r(4)
= 7r(1).
Denote
q(f,
w)
=
((iJ(fh,
[w, w]),
(iJ(fh,
[w, w]),
(iJ(fh,
[w,
w])),
(9)
where iJ(f)m = (iJ(f)ml, iJ(f)m2, iJ(f)m3) (see (5))
TheoreIll
3.2.
Let
~
: M
2
(C)
-+
M
2
(C)
181
M
2
(C)
be
a
Kadison-Schwarz
operator
with
a H
aar
state
T,
then
it
has the
form
(4)
and
the coefficients
{bml,d
satisfy
the following conditions
3 3
IIwl1
2
:'S
i L
fm
a
1f(m),1f(m+l)
+ L
II
x
ml1
2
(10)
m = 1 m = 1
261
Ilq(f,
w)
-
ii;/ml'7r(m)
'
7r
C
m+l)
-
[Xm,
xmlll
::;
IIwl1
2
-
itfka7rCk)
,
7rCk+l)
3
-
LII
X
mI1
2
.
(11)
m=l
for
all f E
S,
wE
1[:3.
Here as before
Xm
=
((b
ml
,
w),
(b
m2
,
w),
(b
m3
,
w)),
b
ml
= (bml,l, b
ml
,2,
b
m
l,3)
and
q(f,
w), ami and l'ml are defined
in
(7)
,(8) .(9), respectively.
Remark
3.2.
The
provided characterization
with
10,18
allows us
to
con-
struct
examp
les of positive or Kadison-Schwarz
operators
which are
not
completely positive (see
next
section).
4.
An
Example
of
q.q.o.
which
is
not
Kadision-Schwarz
one
In
this
section we are going
to
provide
an
example of q.q.o. which does
not
satisfy
the
Kadision-Schwarz property.
Let us consider
the
following
matrix
{bij,d given by:
bll
,l =
E;
b
ll
,2 =
0;
b
ll
,3
=
0;
b
12
,1
=
0;
b
12
,2
=
0;
b
12
,3
=
E;
h3
,1 =
0;
b
13
,2
=
E;
h3
,3 =
0;
b
22
,1
=
0;
b
22
,2
=
E;
b
22
,3
=
0;
b
23
,1
=
E;
b
23
,2 =
0;
b
23
,3
=
0;
b
33
,1
=
0;
b
33
,2 =
0;
b
33
,3
=
E;
and
bij,k = bji,k.
Via
(4) we define a lin
ear
operator
D.
c
,
for which T is a
Haar
state.
In
the
sequel we would like
to
find some conditions
to
E which ensure positivity
of
D.
c.
For given
{bijd
one
can
easily find a form of
D.
c as follows
D.c(x) =
won
0 n +
EWIO'I
00'1
+
EW30'1
00'2
+
EW20'1
00'3
+EW30'2
0
0'1
+
EW20'2
0
0'2
+
EWl0'2
0
0'3
+EW20'3
0
0'1
+
EWl0'3
0
0'2
+
EW30'3
0
0'3,
(12)
where as before x =
wo
n +
WO'.
Theorem
4.1.
A linear operator
D.c
is a q.q.o
if
and
only
if
lEI
::;
~.
262
Proof.
Let
x =
won
+
wa
be
a positive element from
M2(C),
Let
us
show
positivity
of
the
matrix
b.c:(x).
To do it, we
rewrite
(12) as follows
b.c:(x)
=
won
+ EB,
here
B=
(
W3
W2
-
iWI
W2
-
iWI
WI
-
2iw3.-
WZj
Wz
+
iWI
-W3
WI
+
Wz
-W2 + ZWI
W2
+
iWI
WI
+
W2
-W3 -W2 +
iWI
WI
+
2iw3
-
W2
-W2 -
iWI
-W2 -
iWI
W3
where
positivity
of x yields
that
wo,
WI,
W2,
W3
are
real
numbers.
In
what
follows,
without
loss
of
generality, we
may
assume
that
Wo
= 1,
and
there-
fore
I/wl/
::;
1.
It
is known
that
positivity
of
the
matrix
b.c:(x)
is equivalent
to
positivity
of
the
eigenvalues
of
b.c:(x).
Let
us first
examine
eigenvalues
of
B.
A simple
algebra
shows us
that
all eigenvalues
of
B
can
be
written
as follows
Now
examine
maximum
of
the
functions
AI(W),
A2(W), A3(W),
A4(W)
on
the
ball
Ilwll
::;
1.
One
can
see
that
3 3
I
A
3(W)1
= I
A
4(W)1
::;
:L
IWkl
::;
V3:L
IWkl2
::;
V3
k=1
k=1
Now
let
us
rewrite
AI(W)
and
A2(W)
as follows
AI(W)
=
WI
+
W2
+
W3
+
~J3(wi
+
w~
+
w§)
-
(WI
+
Wz
+
w3)2
(13)
AZ(W)
=
WI
+W2 +W3 -
~J3(wi
+W§ +W§) -
(WI
+W2 +W3)2
(14)
On
the
other
hand,
we have
263
for
any
h E
lR
(k =
1,2).
Therefore,
the
functions Ak(W), k =
1,2
reach
their
maximum
on
the
sphere
wi
+
w~
+
w~
= 1 (i.e.
Ilwll
= 1). Hence,
denoting
t =
WI
+
W2
+
W3
from
(13)
and
(14)
we
introduce
the
following
functions
gl(t)
= t +
~~,
where
It
I
::::
V3.
One
can
find
that
the
critical values of
gl
are
t = ±1,
and
the
critical
value
of
g2
is t =
-1.
Consequently,
the
maximum
of
gl
and
g2
on
It
I
::::
V3
are
the
following:
max
Igl(t)1 = 3,
max
Ig2(t)1
=
3;
Itl:"=v'3
Itl:"=v'3
Therefore, we conclude
that
max
IAl(W)1
= 3,
max
IA2(W)1
=
3;
Iwl:"=l
Iwl9
It
is known
that
for
the
spectrum
of
II
+
EB
one
has
8p(ll
+ EB) = 1 + E8p(B)
Therefore,
8p(ll
+ EB) =
{I
+
EAk(W)
: k = 1,4}
So, if
1
lEI
< , k = 1,4
-
max
IAk(W)1
IIwl19
then
we have I+EAk(w)
2:
0 for allllwil
::::
1,
k = 1,4.
This
implies
that
the
matrix
II +
EB
is positive for all w
with
Ilwll
::::
1.
This
yields
the
required
assertion.
0
Theorem
4.2.
Let E = i then corresponding q.q.o. b.
E
is
not
KS-opemtor.
Proof.
It
is enough
to
show dissatisfaction
the
inequality (11)
at
some
value
w:
Ilwll:::: 1
and
f =
(h,
h,
h)·
Assume
that
f =
(1,0,0),
then
a little
algebra
shows
that
(11) reduces
to
the
following one
VA+B+C<D
(15)
264
where
A
=
IE(W
2W3 -
W3W2)
- iE2(2w2W3 -
21wll2
-
W2Wi
+
WiW2
-
WiW3
+ W3WiW
B =
IE(WiW2
-
W2Wt}
- iE2(2wiW2 -
21w312
-
WiW3
+
W3Wi
-
W3W2
+
W2
W
3W
C =
IE(W3Wi
-
WiW3)
- iE2(2w3Wi -
21w212
-
W3W2
+
W2W3
-
W2Wi
+
WiW2)1
2
D =
(1
-
3IEI2)(lWlI2
+
IW212
+
IW312)
-iE2(W3W2 -
W2W3
+
W2
W
i
-
WiW2
+
WiW3
-
W3
W
t}
Now choose w as follows:
1
Wi
= -
9;
5i
W3
= 27
Then
calculations show
that
Hence, we find
A = 9594 .
19131876'
1625
C = 3779136;
B = 19625 .
86093442'
D=~.
17496
9594 19625 1625 589
19131876
+ 86093442 + 3779136 > 17496
which means
that
(15)
is
not
satisfied. Hence,
.6.",
is
not
a KS-operator
at
10
=
1/3.
0
Now
we
are going
to
show
that
condition (6) is necessary for positivity
of
.6..
Let us again consider
the
operator
.6.",.
If
1101
:s:
~
is satisfied,
then
(6) holds. Indeed,
t,
1
iti
bi
j,kfiPj
12
= E
2
(lhpi
+
hp2
+ hp31
2
+
Ihpi
+
hp2
+ hP31
2
+Ihpi
+
hp2
+ hp31
2
)
:s:
E2((j~
+
fi
+
f~)(pi
+
p~
+
p~)
+(j~
+
fi
+
ff)(pi
+
p~
+
p~)
+(pi
+
p~
+
p~)(fi
+
f~
+
f~))
:s:
10
2
(1
+ 1 +
1)
=
310
2
:s:
l.
From
this
and
Theorem
4.1 we conclude
that
if
10
E
(~,
~)
th
en
the
operator
.6.",
is
not
positive, while (6) is satisfied.
265
Acknowledgement
The
first
author
(F.M.) would like
to
thank
Professor
Noboru
Watanabe
and
Professor Masanori
Ohya
for
their
kind hospitality
during
ICQBIC
2010.
This
work is
partially
supported
by
the
Malaysian
Ministry
of
Science,
Technology
and
Innovation
Grant
01-01-08-SF0079
and
by
HUM
Research
Endowment
Grant
B
(EDW
B 0905-303).
References
1.
S.
N.
Bernstein,
Uchen. Zapiski
NI
Kaf. Ukr. Otd. Mat. 1924, no. 1, 83115.
(Russian)
2.
L.
Boltzmann,
Selected works, Nauka, Moscow 1984.
(Russian)
3. O.
Bratteli,
D.
W.
Robertson,
Operator algebras and
quantum
statistical
me-
chanics. I, Springer, New YorkHeidelbergBerlin 1979.
4.
M-D. Choi, Lin. Alg. Appl.
10(1975),
285-290.
5. N. N.
Ganikhodzhaev,
F.
M.
Mukhamedov,
Uzb.
Matem. Zh. 1997, no. 3,
8-20. (Russian)
6.
N.N.
Ganikhodzhaev,
F.
M.
Mukhamedov,
Izv. Math.
65
(2000), 873-890.
7.
H.
Kesten,
Adv.
in
Appl. Probab. 1970, no.
2,
182, 179228.
8. Yu. I. Lyubich,
Russian
Math. Surveys 26(1971).
9.
Yu. I. Lyubich, Mathematical structures
in
population genetics, Springer,
Berlin
1992.
10.
W.A.
Majewski, M.
Marciniak,
J.
Phys.
A:
Math. Gen.
34
(2001) 5863-5874.
11.
W.A.
Majewski, M.
Marciniak
,
arXiv:0705.0798
12.
W.A.
Majewski,
J.
Phys.
A:
Math. Gen.
40
(2007) 11539-11545.
13.
F.M.
Mukhamedov,
Method
of
Funct. Anal. and Topology, 7(2001), No.1,
63-75.
14.
F.M.
Mukhamedov,
Izvestiya Math. 68(2004), 1009-1024.
15. M.A. Nielsen, I.L.
Chuang,
Quantum
Computation
and
Quantum
Informa-
tion,
Cambridge
University
Press,
Cambridge,
2000.
16. A. G.
Robertson,
Math.
Z.
156(1977),
205-206.
17. A. G.
Robertson,
Math. Proc. Camb. Philos. Soc.
94(1983),
291-296.
18. M.B. Ruskai,
S.
Szarek,
E.
Werner,
Lin. Alg. Appl.
347
(2002) 159-187.
19.
E.
Stormer,
Acta
Math.
110(1963),
233-278.
20.
S.
M. Ulam, A collection
of
mathematical problems, Interscience, New York-
London
1960.
This page intentionally left blankThis page intentionally left blank
Quantum
Bio-Informatics
IV
eds. L.
Accardi,
W .
Freudenberg
and
M.
Ohya
© 2011
World
Scientific
Publishing
Co.
(pp.
267-278)
ON
PHASE
TRANSITIONS
IN
QUANTUM
MARKOV
CHAINS
ON
CAYLEY
TREE
LUIGI ACCARDI and FARRUKH MUKHAMEDOY* MANSOOR
SABUROY+
Centro Interdisciplinare Vito Volterra
II
Universita
di
Roma
"Tor Vergata"
Via Columbia 2, 00133
Roma
, Italy
E-email: accardi@volterra.uniroma2.
it
Department
of
Computational & Theoretical Sciences
Faculty
of
Science, International Islamic University Malaysia
P.O. Box, 141, 25710,
Kuantan
Pahang, Malaysia
* E-mail:
far75m@yandex.ru;farrukh_m
@iiu.edu.my
+ E-mail: msaburov@gmail.com
In
th
e
present
paper
we
continue
our
investigations
started
in [
Accardi
L.,
Ohno
,
H. ,
Mukhamedov
,
F.
,
Quantum
Markov
fields
on
graphs
, Inf.
Dim.
Analysis
,
Quantum
Probab.
Relat
ed Topics
(accepted)
arxi
v:
0911 . 1667].
In
[Accardi
L.,
Mukhamedov,
F.,
Saburov
M.
On
Quantum
Markov
chains
on
Cayley
tr
ee
and
associated
chains
with
XY-model
arxiv:
1004.3623]
we
provided
a
construction
of
forward
and
backward
Quantum
Markov
Chains
(QMC)
defined
on
the
C
ay
ley
tree
,
and
estab
lished
uniqueness
of
QMC
associated
with
XY
-
model
on
a
Cayley
tree
order
2.
In
the
pr
e
sent
paper
we
study
the
same
mod
el
on
a
Cayley
tree
order
3.
Surprisingly
in
this
case, we
establish
a
phase
transition
(i.e.
existence
of
two
distinct
quantum
Markov
chains)
for
the
considered
model
on
the
Cayley
tree
order
3.
K eywords:
quantum
Markov
Chain;
Cayley
tree;
phase
transitio
n.
1.
Introduction
Nowadays,
it
is know
that
Markov fields
play
an
important
role in clas-
sical probability, in physics, in biological
and
neurological models
and
in
an
increasing
number
of technological
problems
such
as
image recognition.
Therefore,
it
is
quite
natural
to
forecast
that
the
quantum
analogue
of
these
models will also
playa
relevant role.
The
quantum
analogues
of
Markov
processes were first
constructed
in
1,
where
the
notion
of
quantum
Markov
267
268
chain
on
infinite
tensor
product
algebras was
introduced.
Nowadays,
quan-
tum
Markov chains have become a
standard
computational
tool
in
solid
state
physics,
and
several
natural
applications
have emerged in
quantum
statistical
mechanics
and
quantum
information
theory.
The
reader
is re-
ferred
to
3,5,6,16,21
and
the
references
cited
therein,
for recent developments
of
the
theory
and
the
applications.
A first
attempts
to
construct
a
quantum
analogue
of
classical Markov
fields
has
been
done
in
4,6,9,17.
These
papers
extend
to
fields
the
notion
of
quantum
Markov state
introduced
in
8
as
a
sub-class
of
the
quantum
Markov chains
introduced
in 1 In 7
it
has
been
proposed
a definition of
quantum
Markov
states
and
chains, which
extend
a
proposed
one
in
20,
and
includes all
the
presently known examples.
Note
that
in
mentioned
pa-
pers
quantum
Markov fields were considered over multidimensional integer
lattice
7l,d.
This
lattice
has
so called
amenability
condition. Therefore,
it
is
natural
to
investigate
quantum
Markov fields over
non-amenable
lattices.
One
of
the
simplest
non-amenable
lattice
is a Cayley tree.
First
attempts
to
investigate
Quantum
Makov chains over such
trees
was done
in
12,
such
studies
were
related
to
investigate
thermodynamic
limit
of
valence-bond-
solid models on a
Cayley
tree
14.
It
was
constructed
finitely
correlated
states
as
ground
states
of
that
model.
The
mentioned
considerations
natu-
rally suggest
the
study
of
the
following problem:
the
extension
to
fields of
the
notion
of
generalized Markov chain. In
10
we
have
introduced
a hierar-
chy
of
notions
of
Markovianity
for
states
on
discrete infinite
tensor
products
of
C*
-algebras
and
for each
of
these
notions
we
constructed
some explicit
examples. We showed
that
the
construction
of
8
can
be
generalized
to
trees.
It
is
worth
to
note
that,
in
a different
context
and
for
quite
different
pur-
poses,
the
special role
of
trees
was
already
emphasized
in
17.
Note
that
a
noncommutative
extensions
of
classical Markov fields,
associated
with
Ising
and
Potts
models
on
Cayley tree, were investigated
in
18,19.
In
the
classical
case, Markov fields
on
trees
are
also considered in
22_26.
In
the
present
paper
we continue
our
investigations
started
in
10,11.
Using
the
construction
(see
11)
of
forward
and
backward
Quantum
Markov
Chains
(QMC)
defined
on
the
Cayley tree, we investigate
QMC
associated
with
XY-model
on
a Cayley
tree
order
3.
We establish a
phase
transition
for
the
XY-model
on
the
Cayley
tree
order
3 in a
QMC
scheme.
Note
that
classical
XY-model
have
been
investigated
by
many
authors
on
a Cayley
tree
13.
269
2.
Preliminaries
Recall
that
a Cayley
tree
rk
of
order
kQ:1
is
an
infinite
tree
whose each
vertices have
exactly
k + 1 edges.
The
vertices x
and
yare
called
nearest
neighbors
and
they
are
denoted
by
I
=<
x,
Y >
if
there
exists
an
edge
connecting
them.
A collection
of
the
pairs
<
x,
Xl
>,
... , <
Xd-l,
Y > is
called a
path
from
the
point
x
to
the
point
y.
The
distance
d(x,
y),
x,
Y E
V,
on
the
Cayley tree, is
the
length
of
the
shortest
path
from x
to
y.
If
we
cut
away
an
edge {x, y} of
the
tree
r
k
,
then
rk splits
into
connected
components,
called semi-infinite
trees
with
roots
x
and
y,
which will
be
denoted
respectively
by
rk(x)
and
rk(y).
If
we
cut
away from rk
the
origin °
together
with
all k + 1
nearest
neighbor
vertices,
in
the
result
we
obtain
k+1
semi-infinite
rk(x)
trees
with
x E So =
{y
E rk :
d(O,
y) =
I}.
Hence we have
rk = U
rk(x)
U
{a}.
xESo
Therefore,
in
the
sequel
we
will consider semi-infinite Cayley
tree
r~
=
(L,
E)
with
the
root
xo,
L is
the
set
of
vertices
and
E is
the
set
of
edges.
Now
we
are
going
to
introduce
a
coordinate
structure
in
r~
as follows:
every
vertex
x (except for xo)
of
r~
has
coordinates
(iI,
...
,in),
here im E
{I,
...
, k}, 1
:::;
m
:::;
n
and
for
the
vertex
xo
we
put
(0). Namely,
the
symbol
(0)
constitutes
level
0,
and
the
sites (i
l
,
...
,
in)
form level n ( i.e.
d(
xo,
x)
= n)
of
the
lattice.
Let
us set
n
m
Wn
=
{x
E L :
d(x,xo)
=
n},
An
= U
Wk,
A[n,mJ
= U
Wk,
(n < m)
k=O
k=n
co
En
= { <
X,y
>E
E :
X,y
E
An},
A~
= U
Wk
k=n
For
x E
r~,
x = (il,
...
,
in)
denote
S(x)
=
{(x,i):
1:::;
i:::;
k},
here (x, i)
means
that
(i
l
,
...
,
in,
i).
This
set
is called a
set
of
direct
suc-
cessors
of
x.
From
these
one
can
see
that
Am
= A
m
-
2
U ( U
{x
U
S(x)}),
XEW=-l
(1)
Em \
Em-l
= U U {<
X,y
>}.
(2)
XEW=_l
YES(x)