Accardi L., Freudenberg W., Obya M. (Eds.) Quantum Bio-informatics IV: From Quantum Information to Bio-informatics
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270
The
algebra
of
observables Ex for
any
single site x E L will
be
taken
as
the
algebra
Md
of
the
complex d x d matrices.
The
algebra
of
observables
localized
in
the
finite volume A c L is
then
given
by
EA
= ® Ex. As
xEA
usual
if Al c
A2
C L,
then
EAl
is identified
as
a
sub
algebra
of
EA2
by
tensoring
with
units
matrices
on
the
sites x E A
2
\
AI.
Note
that,
in
the
sequel,
by
E
A
,+
we
denote
positive
part
of
EA.
The
full
algebra
EL
of
the
tree
is
obtained
in
the
usual
manner
by
an
inductive
limit
One
can
see
that
the
shift
Ii
induces a
homomorphism
on
E
L
.
In
what
follows,
by
S(EA) we will
denote
the
set
of all
states
defined
on
the
algebra
EA.
Consider a
triplet
C c E c A
of
unital
C* -algebras. Recall
that
a quasi-
conditional expectation
with
respect
to
the
given
triplet
is a completely
positive
(CP)
identity
preserving linear
map
E : A
----*
E such
that
E(ca) = cE(a), a E A, c E
C.
(3)
Notice
that,
as
the
quasi-conditional
expectation
E is a real
map,
one
has
E(ac) = E(a)c,
a E A, c E
C.
as well.
Definition
2.1.
Let
cp
be
a
state
on
E
L
.
Then
cp
is called
(i) a
forward
quantum
d-Markov
chain
(QMC)
,
associated
to
{An},
on
EL
if for each An,
there
exist a quasi-conditional
expectation
EAi.
with
respect
to
the
triplet
(4)
and
a
state
such
that
for
any
n E N
one
has
(5)
and
(6)
in
the
weak-* topology.
271
(ii) a backward
quantum
d-Markov chain, associated
to
{An}, if
there
exist a quasi-conditional
expectation
EAn
with
respect
to
the
triple
BA
n
_
1
C
BAn
C
BAn+l
for each n E N
and
an
initial
state
Po
E
S(BAo) such
that
in
the
weak-* topology.
In
this
definition, forward
and
backward
QMC
<p
are
well-defined.
Remark
2.1.
Note
that
in
10
the
forward
QMC
has
been
called a gener-
alized
quantum
Markov
state.
We
think
the
present
definition is a more
adequate
than
the
generalized
quantum
Markov
state.
3.
Constructions
of
Quantum
d-Markov
chains
on
the
Cayley
tree
In
this
section, we
are
going
to
provide
constructions
of
forward
and
back-
ward
quantum
d-Markov chains.
Note
that
some
of
part
of
the
construction
has
been
outlined
in
10.
Assume
that
for each edge <
x,
y
>E
E of
the
tree
is assigned
an
operator
K<x,y>
E
B{x,y}'
We would like
to
define a
state
on
BAn
with
boundary
conditions
Wo
E
B(o),+
and
h =
{h
x
E B
x
,+
}xEL.
To
do
this, let
us
denote
Kn
=
w6/
2
K[0,ljK[1,2j
...
K[n-1,n]
II
h~/2,
(7)
xEW
n
here
by
definition
we
put
k k
K[O,l]
:=
II
K<xO,(xO,i»'
K[m-1,mj:=
II
II
K<x,(x,i»,
m?:
1
(8)
i=l
Since, in generally,
the
operators
K<x,y>
do
not
commute
(they
share
a
vertex)
and
therefore,
the
product
over i
of
operators
Kx,(x,i)
is
actually
ordered
product,
i.e. here we
are
taking
a
multiplication
of
those
operators
from left
to
right
in
the
given
coordinate
structure.
Define
(9)
It
is clear
that
Wnj,
Wnj
are
positive.
272
In
what
follows, by
TrA
: BL
-f
BA we
mean
normalized
partial
trace,
for
any
A
<:;;fi
n
L.
For
the
sake of shortness
we
put
Trnj
:=
TrAn'
Now
we
are
ready
to
define positive functionals
'PSn,b~
and
'PSn,~
on
BAn
0,
0 ,
by
'PS:,b~(a)
=
Tr(W
n
+
1
j(a ®
llw
n+J),
(10)
'P~
'
~(a)
=
Tr(W
n
+
1
j(a ®
llw
n+J),
(11)
for every a E BAn' where
llw
n+l = ®
ll.
Note
that
here,
Tr
is a
normalized
trace
on
B
L.
To get a
state
'P(c)
on
B
L
,
i.e.
on
the
infinite-volume tree, by means
of
'P(n
,c
h
) (here c =
b,
f)
such
that
'P(c) r
BA
=
'Pw(n
,c
h
),
we need
to
impose
wo, n 0 ,
some constrains
to
the
boundary
conditions {wo,
h}
so
that
the
functionals
{'P(n
,c
h
)}
satisfy
the
compatibility
condition, i.e.
WQ,
(n+1
,c) r =
(n,c)
'PwQ,h
BAn
'PwQ,h'
(12)
Theorem
3.1.
10
Let
the boundary conditions
Wo
E B(
o)
,+
and
h =
{h
x
E
B
x
,+ }xEL
satisfy
the following conditions:
Tr(woh
o
)
= 1 (13)
k k k
Tr{x}
(g
K <x,(x,i» g h(x
,i)
g
K~x
'(X'k+
1
-i»
)
= h
x
,
(14)
for
every x E
L.
Then
the
functionals
{'PS:
'
~~}
satisfy
the compatibility
condition
(12). There is a unique
state
'P(b)
h
on
BL such
that
(n(b)
h
wo,
rwo,
r
(n,b)
W -
1m
n
->oo
'P
wQ
,h .
If
hx is invertible for all x E L,
then
'P(b)
h is a backward
QMC
on
B
L
.
WQ,
On
the
other
hand,
it
is known 8
that
{'P~'
,
~}
satisfy
the
compatibility
condition if a sequence
{Wnj}
is projective
with
respect
to
Trnj,
i.e.
Tr
n
-
1
j
(W
n
j)
= W
n
-
1
j,
Vn E
N.
(15)
One
has
the
following
Theorem
3.2.
11
Let
the boundary conditions
Wo
E
B(o)
,+
and
h =
{h
x
E
Bx,+}
xEL
satisfy
(13)
and
(14).
Then
{Wnj}
is a projective sequences
of
density operators, i.
e.
there is a unique forward Q M C
'P
(f)
h
on
B L such
WQ,
that
(n
U
)
= W - lim
(n
(n,J).
rWQ
,h n---+oo
Ywo
,h
273
Definition
3.1.
We say
that
there
exists a phase transition for a family
of
operators
{K<x,y>}
if (13)
and
(14) have
at
least two
(wo,
{hx}xEL)
and
Cwo,
{hx}xEL)
solutions such
that
the
corresponding
QMC
'Pw
hand
'Pw
fi
0,
0,
are
disjoint. Otherwise, we say
there
is no phase transition.
4.
Quantum
d-Markov
chains
associated
with
XY-model
In
this
section,
we
prove
the
existence of a
phase
transition
of
the
quantum
d-Markov
chain
associated
with
XY-model
on
a Cayley
tree
of
order
three.
Let
us consider a semi-infinite Cayley
tree
r~
= (L,
E)
of
order
3.
Our
starting
C*
-algebra
is
the
same
E L
but
with
Ex
=
M2
(q
for x E L.
By
O"~u),
O"£u)
,
O"~u)
we
denote
the
Pauli
spin
operators
for
at
site
u E L. Here
(u)=(OI)
O"x
10'
(u)
=
(0
-i)
O"y
i ° '
(u) =
(1
° )
O"z
0-1'
(16)
For
every edge < u, v
>E
E
put
K<u,v> =
exp{,6H<u,v>},
,6
> 0,
(17)
where
1
H = -
(O"(u)O"(v)
+
O"(u)O"(v))
<u,v> 2 x x y y .
(18)
Now
taking
into
account
the
following equalities
H
2m
_
H2
_ 1
(n
(u)
(v))
<u,v> - <u,v>
-:2
-
o"z
O"z
,
H
2m-l
H
<u,v>
==
<u,v>,
mEN,
one finds
K<u,v> = n + sinh,6H<u,v> + (cosh,6 -
I)H~u,v>'
We
are
going
to
describe all solutions h =
{h
x
}
and
Wo
of
the
equations
(13),(14).
Furthermore,
we
shall assume
that
hx
=
hy
for every
x,
y E W
n
,
n E
N.
Hence, we
denote
h~n)
:=
hx,
if x E W
n
. Now from (17),(18)
one
can
see
that
K<u,u> =
K~u,v>'
therefore,
the
equation
(14)
can
be
rewritten
as follows
(
K K
h
(n)h(n)h(n)K
K K ) -
h(n-l)
Trx K<x,y>
<X,z>
<x,v> y z v <x,v>
<X,z>
<X,y>
- x ,
(19)
for every x E
L.
After a
little
algebra
the
equation
(19) reduces
to
the
following
system
2
11
2
11
12
-
11
(20)
{
B
(a
(n))3
+ A a(n)la(n)1
2
-
a(n-l)
Bllai~)I(ai~))2
+
Allai~)13
=
lai~-I)1
274
where
Al
=
sinh
3
;3
cosh;3,
BI
= sinh;3 cosh
2
;3(1
+ cosh;3 + cosh
2
;3), (21)
A2
= sinh2
;3
cosh
2
;3(1
+ 2 cosh ;3),
B2
= cosh
6
;3,
(22)
(
(n)
(n))
hen)
=
hen)
=
hen)
=
all
a
l2
y z v
(n)
(n)
.
a
21
a
22
Remark
4.1.
Note
that
according
to
positivity
of
h~n)
and
ai';)
=
a~~)
we
conclude
that
ai~)
>
lai;)
I for all n E N.
Now we
are
going
to
investigate
the
derived
system
(20). To do this, let
us define a
mapping
f :
(x,
y) E
lR~
--+
(Xl,
yl) E
lR~
given by
{
B2(XI)3
+ A
2
x
'
(y')2 = X
BI
(Xl?yl +
Al
(yl)3 = Y
(23)
Furthermore,
due
Remark
4.1, we restrict
the
dynamical
system
(23)
to
the
following
domain
Denote
,6. =
{(x,y)
E
lR~
: x > y}.
Pg(t) = t
g
-
t
S
-
t
7
-
t
6
+ 2t4 + 2t
3
-
t -
1,
1
E
:=
sinh2;3 cosh
2
;3(1
+ 2 cosh;3) + D cosh
6
;3'
D:=
A2
-
AI.
B
I
-B2
Further,
we will need
the
following auxiliary fact:
(24)
(25)
(26)
Lemma
4.1.
Let
AI,
B
I
, A
2
,
B
2
,
D
be
numbers defined by (24), (22), (26)
and
Pg(t)
be
polynomial given
by
(24), where
;3
>
O.
Then
the following
statements
holds true:
(i) The polynomial Pg (t) has only tree positive roots
1,
t*,
and
t*
such
that
1.05 <
t*
< 1.1
and
1.5 <
t*
< 1.6. Moreover,
if
t E (1, t*) u
(t*,oo) then Pg(t) > 0
and
t E (t*,t*)
then
Pg(t) <
O.
Denote
by
;3*
= cosh
-1
t*
and;3* = cosh
-1
t*;
(ii) For any
;3
E
(0,
(0)
we have
Al
< A
2
;
(iii)
If;3
E (0,;3*] U [;3*,(0)
then
BI
:s;
B2
and
If;3
E (;3*,;3*)
then
BI
> B
2
;
(iv) For
any
(3
E (0,00) we have
Al
+
BI
<
A2
+ B
2
;
(v)
If
(3
E
((3*,
(3*)
then
D > 1
and
E >
0;
275
(vi) For
any
(3
E (0,00) we have
AIA2
<
BIB2
and
AIB2
< A
2
B
I
;
(vii)
If
(3
E
((3*,
(3*)
then
A2BI
<
AIA2
+
3A
I
B
2
+
BIB2
and
2AIA2
+
3A
I
B
2
< A
2
B
I
·
Let us first find all of
the
fixed
and
periodic points of (23).
Theorem
4.1.
Let
f
be
a dynamical
system
given by (23).
Then
the fol-
lowing assertions hold true:
(i)
If
(3
E (0,
(3*]
U [(3*,00)
then
there is a unique fixed
point
(cos~3
f3'
0)
in
the
domain
6.;
(ii)
If
(3
E
((3*,
(3*)
then
there are two fixed points
in
the
domain
6.,
which are
Cos~3f3'0)
and
(VDE,VE).
(iii) For
any
(3
E (0,00) the dynamical
system
f does
not
have
any
k
periodic points, where k
;:::
2.
Now let us formulate results concerning limiting behavior of f.
Theorem
4.2.
Let
f : 6.
---+
lR~
be
the dynamical
system
given by (23)
and
(3
E (0,
(3*]
U [(3*,00).
Then
the following assertions hold true:
(i) ify(O) > 0
then
the trajectory
{(x(n),y(n))}~=o
of
f
starting
from
the
point
(x(O),
y(O))
is finite.
(ii)
if
yeO)
= 0
then
the trajectory
{(x(n),y(n))}~=o
starting
from
the
point
(x(O),
yeO))
has the following
form
{
3\1
x(O)
cosh
3
(3
x(n) =
---n---
cosh
3
(3
yen)
= 0,
and
it
converges to the fixed
point
(cos~3
f3
' 0).
Theorem
4.3.
Let
f : 6.
---+
lR~
be
the dynamical
system
given by (23)
and
(3
E
((3*,
(3*).
The following assertions hold true:
(i) There are two
invariant
lines w.r.t. f defined by h =
{(x,
y) E 6. :
y =
O}
and
12
=
{(x,y)
E
6.:
y =
]v};
(ii)
if
an
initial
point
(x(O),
yeO))
belongs to the
invariant
lines lk'
of
dy-
namical
system
(23),
then
the trajectory {(x(n) ,
y(n))}~=o,
starting
from the
point
(x(O),
yeO)),
converges to the fixed
point
which belongs
an
invariant
line lk' where k =
l,2;
276
(iii)
if
an initial point
(x(O),
yeO))
satisfies the following condition
yeO)
(
1)
x(O)
E
0,
y75 ,
then the trajectory {(x(n),
y(n))}~=o,
starting from the point
(x(O)
,
y(O)),
converges to the fixed point
(cos~3
f3
,0) which belongs an
invariant line
h;
(iv)
if
an initial point
(x(O),
y(O))
satisfies the following condition
yeO)
(1
)
x(O)
E
y75,1
,
then the trajectory
{(x(n),y(n))}~=o,
starting from the point
(x(O)
,
y(O)),
is finite.
Let
13
E
(0,13*]
u
[13*,00).
From
Theorem
4.2,
we
infer
that
equation
(19)
has
a lot
of
parametrical
solutions
(wo
(a), {h
x
(a)}) given
by
(
3\1
a cosh
3
13
h~n)
(a) = cosh
3
f3
°
wo(a) =
(27)
for every
x E
V.,
here a is
any
positive real
number.
The
boundary
conditions
corresponding
to
the
fixed
point
of
(23) is give
1
by
(27)
at
value
of
ao = 3
in.
Therefore, further, we
denote
such
cosh
13
operators
by
Wo
(ao)
and
h~n)
(ao).
Let
us consider
the
states
<p~~'~l),h(a)
corresponding
to
the
solutions
(wo(a),{h~n\a)}).
One
can
show
that
<p(n,(b))h() =
<p(n,(b)
)h(
)'
for
Wo
Q:,
Q
Wo
0::0,
0:0
any
a.
From
the
definition
of
Quantum
d-
Markov
chain
we find
that
<P~~(a),h(a)
=
<P~~(ao),h(ao)'
which yields
that
the
uniqueness
of
backward
Quantum
d-Markov
chain
associated
with
the
model (17).
Similarly,
due
to
Theorem
3.2,
we
may
consider
the
states
<P~~'(2),h(a)
corresponding
to
the
solutions (wo(a),
{h~n)(a)}).
Using
the
same
argu-
ment
as
shown above, we
can
get
that
<p(n(
)
h(
) =
<p(n(
)
h(
)'
which
Wo
ex,
Q:
Wo
ao,
ao
means
that
the
uniqueness
of
forward
Quantum
d-Markov
chain
associated
with
the
model
(17). Therefore we have
the
following
Theorem
4.4.
Let
13
E
(0,13*]
u
[13*,00).
There is a unique backward and
forward quantum d-Markov chains, respectively, for the model
(17) on the
Cayley tree
of
order
3.
277
Taking
into
account
Theorems
4.1
and
4.3 one
can
establish
the
following
Theorem
4.5.
Let
(3
E
((3*,
(3*),
then there is a phase transition for the
model
(17)
on the Cayley tree
of
order
3.
We
note
that
in
11 we have considered
the
model (17)
on
the
Cayley
tree
of
order
2
and
established uniqueness of
the
backward
and
forward QMS
associated
with
this
model.
Acknowledgement
The
authors
would like
to
thank
Professor
Noboru
Watanabe
and
Professor
Masanori
Ohya
for
their
kind
hospitality
during
ICQBIC
2010.
The
present
study
have
been
done within
the
Malaysian
Ministry
of
Higher
Education
Grant
FRGS0308-91.
The
authors
(F.M.
and
M.S.) also acknowledge
the
Malaysian
Ministry
of
Science, Technology
and
Innovation
Grant
01-01-08-
SF0079.
References
1.
Accardi
L.,
On
the
noncommutative
Markov
property,
Func. Anal. Appl., 9
(1975) 1-8.
2.
Accardi
L.,
Cecchini's
transition
expectations
and
Markov
chains
in:
Quan-
tum
Probability
and
Applications
IV
Springer
LNM
N.
1396
(1987) 1-6
3.
Accardi
1.,
Fidaleo
F.,
Entangled
Markov
chains.
Annali
di
Matematica
Pum
e Applicata,
184(2005),
327-346.
4.
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Quantum
Bio-Informatics
IV
eds. L.
Accardi,
W. Freudenberg
and
M.
Ohya
© 2011
World
Scientific
Publishing
Co.
(pp.
279-289)
SPACE(-TIME)
EMERGENCE
AS
SYMMETRY
BREAKING
EFFECT*
IZUMIOJIMA
Research
Institute
for
Mathematical Sciences,
Kyoto
University,
Kyoto
606-8502, Japan
The
microscopic
origin
of
space(-time)
geometry
is
explained
on
the
basis
of
an
emergence
process
associated
with
the
condensation
of
infinite
number
of
micro-
scopic
quanta
responsible
for
symmetry
breakdown,
which
implements
the
basic
essence
of
"Quantum-Classical
Correpondense"
and
of
the
forcing
method
in physi-
cal
and
mathematical
contexts,
respectively.
From
this
viewpoint,
the
space(-time)
dependence
of
physical
quantities
arises
from
the
"logical
extension"
6
to
change
"constant
objects"
into
"variable
objects"
by
tagging
the
order
parameters
asso-
ciated
with
the
condensation
onto
"constant
objects";
the
logical
direction
here
from a
value
y
to
a
domain
variable
x
(to
materialize
the
basic
mechanism
behind
the
Gel'fand
isomorphism)
is
just
opposite
to
that
common
in
the
usual
definition
of
a
function
I : x
f----t
I(x)
from
its
domain
variable
x
to
a
value
y =
I(x).
1.
Outline
of
the
Problem
Before going
into
the
main
context,
a
comment
would
be
necessary
on
the
parenthesis
in
"Space( -
Time)
Emergence" in
the
title: in
sharp
contrast
to
the
case
of
space,
the
emergence of time axis seems now doubtful. To
justify
this
suspicion, we need re-examine
its
consistency
with
the
space-
time picture
essential
in
special
and
general theories of relativity, which is
not
undertaken
yet
here.
This
is
the
reason for
the
expression "Space(-
Time)".
1.1.
"Theory
of
Everything"
vs.
Duheme-Quine
thesis
In
search for a new
theory
to
incorporate
the
old
and
standard
one as a
special case,
one
usually
attempts
trial-and-error
searches in a heuristic
way which seems
to
be
unavoidable. How
and
to
which
extent
can
this
be
made
systematic
by
the
method
for solving
"inverse
problem"?
In
this
*
Talk
at
ICQBIC
(10 - 13
March
2010)
at
Tokyo
University
of
Sciences,
Noda
279