Accardi L., Freudenberg W., Obya M. (Eds.) Quantum Bio-informatics IV: From Quantum Information to Bio-informatics
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250
construction
analytically
with
the
use of
Mathematica
and
we have applied
the
witness W
to
a family
of
equilibrium
states
of
(7),
(f3
B
D)
= exp(f3H)
(}
, ,
Z'
1
f3
=
kT'
Z =
Trexp(-f3H)
,
(13)
for a
range
of positive
temperatures.
For T close
to
0
the
mixture
(}
is
dominated
by
the
terms
corresponding
to
the
lowest energies, say
>'1
<
A2
< ... ,
(14)
and
so
(}
inherits
mostly
the
entanglement
of
l?,I;l)'
Therefore,
it
is
detected
by
W which,
by
construction,
is
most
sensitive precisely
to
the
entanglement
of
l?,I;l)
type.
However, one
type
of
entanglement
often
present
in
thermal
equilibrium
states
cannot
be
detected
by
witnesses
of
this
kind.
The
separability
of
the
ground
state
l?,I;l)
of
H,
while
making
our
witness
construction
faulty,
does
not
exclude
that
the
state
(13)
can
nevertheless
be
entangled
for a
range
of
positive
temperatures.
This
phenomenon,
called
thermally
induced
entanglement,
happens
when
the
first excited
eigenstate
1?,I;2)
is
entangled
and
its
admixture
shows
up
in
(}
via
the
second
term
in (14) for
certain
T>
Tmin.
In
Fig. 7 we collect
the
results
of
our
tests
of W
H.
The
plots
(left
column) show
the
value
max
{
-Tr(W
H
(})
,
o}
.
for a
range
of
Band
T
parameters.
Separate
plots
correspond
to
different
values
of
D,
see also Fig. 8 for
the
location
of
these
values
on
the
eigenvalue
level crossing
diagram.
We shall describe now how H
can
be
used
in
the
construction
of
a
genuine
entanglement
witness.
In
this
case, b. is
the
set
of
biseparable
states.
The
argument
that
has
previously led us
to
the
simplification
(9-
10) of
the
extremalization
problem
for H is
no
longer valid, for
the
local
components
l<Pi)
in
(8)
cannot
be
considered
independent
and,
hence,
any
parametrization
of
the
set
b.
must
necessarily
be
much
more
involved.
We will
apply
the
approximation
(5-6).
To
this
end,
we
need
to
reex-
amine
the
structure
of
"phase
diagram"
of
Fig. 6
in
more
detail
to
identify
the
ordering
of
lowest
energy
levels
in
distinct
sectors
of
the
(B, D) plane.
This
refined
spectrum
structure
is shown in Fig. 8. Shades
of
gray, as be-
fore,
represent
the
sectors where
the
ground
state
energies
are
A1,
A2
and
A3.
Additional
solid lines
correspond
to
level crossings
at
higher energies.
251
Figure
7.
Left
column:
positive
values
of
-
Tr(W
He)
plott
ed for a
range
of
Band
f3
=
k~
for
several
fixed D
values
(see also
Fig.
8).
Right
column:
similarly, for
the
genuine
entanglement
witness
WHo
Specifically, we have
the
following ordering
of
eigenvalues
III
the
sectors
252
Figure
8.
The
refinement
of
the
level crossing
diagram
of
Fig.
6 for 3 lowest energies of
H
(d.
(15)
and
the
surrounding
text).
Horizontal
dashed
lines
correspond
to
the
values
of
D
used
in
the
plots
of
Fig.
7.
(a)-(e):
(a)
(b)
(c)
(d)
( d')
(e)
A2
<
A3
<
AI,
A2
<
Al
< A3,
A2
< Al < A4,
Al
<
A2
< A4,
Al
<
A2
< A3,
Al
<
A4
< A2,
(15)
where
the
eigenvalue
A4
=
-4
-
4D
corresponds
to
the
GHZ-type
state
11{>4)
=
11010)
-10101).
In
order
to
find a as
in
(5),
we
need
to
estimate
the
maximal
overlap
of
the
ground
state
11{>2),
and
respectively
11{>1)
in
sectors
(d)-(e),
with
biseparable vectors.
This
is
done
by considering all
bipartite
splits
of
the
entire
system:
1-
234, 2
-134,
...
,
12
-
34,
13
-
24,
...
etc.
and,
in each case,
writing
down
the
Schmidt
form
of
11{>2)
with
respect
to
this
split.
Then
the
square
of
the
largest
among
all
Schmidt
coefficients
obtained
in
this
process
is
the
desired value
of
a.
For example,
the
normalized
state
11{>2)
can
be
253
written
as
with
respect
to
the
1- 234 split.
On
the
other
hand,
taking
the
splitting
12
- 34,
we
have
Now, using
the
matrices
A = [aij]
and
B = [bij] as above,
[
00000
0
O-~]
000
1 0
_1
1 0 '
2 2 2
[
0 0 0 0
1
o 0
O-~
o 0 0 1
2
o
-~
~
0
the
respective
squared
Schmidt coefficients
are
the
nonzero eigenvalues
of
A*
A
and
B*
B,
i.e.
{~,
t}
and
{~,
~}.
It
is easy
to
see
that
due
to
the
sym-
metry
of
11J'!2)
other
bipartite
splittings
of
the
system
space yield precisely
the
same
Schmidt coefficients. Hence,
we
can
take
as
the
first approxima-
tion
(i
=
P'2
+
t>'3
in
the
region (a)
and
(i
=
~A2
+
tAl
in
(b)-(c).
Let
us analyze
whether
this
estimate
can
be
improved by (6). Since
11J'!3)
is separable, its
maximal
overlap
with
biesparable
states
in
~
is
1,
so
(i
cannot
be
improved
any
further
in
the
region (a). Similarly,
the
maximal
squared
Schmidt coefficient over all biesparable expansions of
11J'!1)
is
1
+D
+ D'
,=
2D'
where
D'
=
JD2
+
2D
+
9.
Since,
> t for D
~
-2,
also
in
the
region
(b) - (c) no improvement
of
(i
is possible.
In
(d)
and
(d')
our
approximation
reads
(i
=
,AI
+
(1-,)A2
and
in (e)
(i
=
,AI
+
(1
-,)A4'
Again
the
fact
that,
> t for D
~
-2
excludes
any
improvement
of
(i
in
(d)
or
(d'). Finally, as
the
maximal
squared
overlap
of
11J'!4)
with
states
in
~
is
~
and
,
~
~
in (e), no
better
(i
can
be
found as
well in
this
case.
Fig. 7 summarizes
our
computational
results.
Its
left column, as men-
tioned
earlier, contains plots
of
-Tr(WHI?)
for
the
entanglement
witness
W
H,
and
the
right
column
- plots for
the
approximate
genuine
entan-
glement witness W H
just
constructed.
The
exemplary D values used
are
1.0,0.25,0.0,
-0.5.
The
reader
may
relate
these
plots
with
the
location
of
D
values
on
the
level crossing
diagram
of
Fig.
8:
edges
in
the
plots correspond
to
transitions
between different sectors (a) - (e)
with
the
change
of
B value.
254
Acknowledgment
The
author
is
greatly
indebted
to
Professor Masanori
Ohya
and
the
QBIC
Centre
for
support
and
hospitality
during
his visit
at
Tokyo University of
Science in March 2010.
References
1.
Michalski M., [in
:]
Quantum Bio-Informatics
III,
L.
Accardi,
W.
Freuden-
berg, M.
Ohya, eds.,
Quantum
Probability
and
White
Noise Analysis XXVI,
p. 217, World Scientific,
2009.
2.
Bennett
Ch.,
et
al., Phys. Rev.
Lett.
82,
5385 (1999).
3.
Collins D.,
et
al., Phys. Rev.
Lett.
88,
170405 (2002).
4.
Diir
W.,
J.
1.
Cirac,
R.
Tarrach,
Phys. Rev.
Lett.
83,
3562 (1999).
5.
Acin A.,
et
al., Phys. Rev.
Lett.
87,
040401-1 (2001).
6.
Bourennane
M.,
et
al., Phys. Rev.
Lett.
92,
087902-1 (2004).
7.
Horodecki M., P. Horodecki,
R.
Horodecki, Phys.
Lett.
A
223
, 1 (1996).
8.
Lewenstein M., B.
Kraus,
J.
1.
Cirac, P. Horodecki, Phys. Rev. A
62,
052310
(2000).
9.
Terhal
B., Phys.
Lett.
A
271
, 319 (2000).
10.
T6th
G., Phys. Rev. A
71
, 010301(R) (2005).
11. Horodecki M., P. Horodecki, R. Horodrecki, Phys.
Lett.
A
283
,1
(2001).
12.
Eisert
J.,
F. G.
S.
L.
Brandao,
K.
M.
Audenaert,
New
J.
Phys. 9,
46
(2007).
13.
Eisert
J.,
D. Gross, Multi-partite entanglement,
[in]
Lectures
on
quantum
information, D.
Bruss
and
G. Leuchs Eds., Wiley-VCH, Weinheim, 2006.
14. Horodecki R., P. Horodecki, M. Horodecki,
K.
Horodecki, Rev. Mod. Phys.
81,
865 (2009).
15. Acin A. ,
et
al., Phys. Rev.
Lett
.
85,
1560 (2000);
quant-ph/0003050.
16. Acin A., A. Andrianov, E.
Jane,
R.
Tarrach,
J.
Phys.
A:
Math.
Gen.
34
,
6725
(2001);
quant-ph/0009107.
17. Michalski M., [in:] Quantum Bio-Informatics
III,
L.
Accardi,
W.
Freuden-
berg, M.
Ohya, eds.,
Quantum
Probability
and
White
Noise Analysis XXVI,
p. 231, World Scientific,
2009.
18. Jamiolkowski A., Rep.
Math
. Phys. 3, 275 (1972).
19.
Carteret
H. A., A. Higuchi, A. Sudbery,
J.
Math.
Phys.
41
, 7932 (2000).
20.
Verstraete
F., J. Dehaene, B. De Moor, Phys. Rev. A
68,
012103 (2003).
21.
Verstraete
F.,
J.
Dehaene, B. De Moor, H. Verschelde, Phys. Rev. A
65
,
052112 (2002).
22. Diir
W.,
J.
1.
Cirac, Phys. Rev. A
61,
042314 (2000).
23. Zhao M.-J., Z.-X. Wang, Rep.
Math.
Phys.
63
, 409 (2009).
24.
Brukner
C., V. Vedral, arXiv:quant-ph/0406040v1 (2004).
25. Vedral V.,
Open
Sys.
Information
Dyn.
16
, 287 (2009).
Quantum
Bio-
Informatics
IV
eds. L.
Accardi,
W.
Freudenberg
and
M.
Ohya
© 2011
World
Scientific
Publishing
Co.
(pp.
255-265)
ON
KADISON-SCHWARZ
PROPERTY
OF
QUANTUM
QUADRATIC
OPERATORS
ON
M
2
(C)
FARRUKH
MUKHAMEDOY*
ABDUAZIZ
ABDUGANIEY+
Department
of
Computational
€3
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: azizi85@yandex.ru
In
the
present
paper
we first
describe
quantum
quadratic
operators
(q.q.o)
acting
on
the
algebra
of
2 x 2
matrices
M2(1C).
Moreover,
we
provide
necessary
conditions
for q.q.o.
with
Haar
to
satisfy
Kadison-Schwarz
condition.
By
means
of
such
a
description
we give
an
example
of q.q.o.
which
is
not
the
Kadision-Schwarz
operator.
Keywords:
quantum
quadratic
operators;
Kadison-Schwartz
operator.
1.
Introduction
It
is known
that
one of
the
main
problems
of
quantum
information
is
characterization
of
positive
and
completely positive
maps
on
C* -algebras.
There
are
many
papers
devoted
to
this
problem
(see for
example
4,10,18,19).
In
the
literature
the
most
tractable
maps,
the
completely positive ones,
have proved
to
be
of
great
importance
in
the
structure
theory
of
C*-
algebras. However, general positive (order-preserving) linear
maps
are
very
intractable
1o
,12.
It
is therefore of
interest
to
study
conditions
stronger
than
positivity,
but
weaker
than
complete
positivity. Such a
condition
is called
Kadison-Schwarz property, i.e a
map
¢ satisfies
the
Kadison-Schwarz
prop-
erty
if
¢(a)*¢(a)
:0:::
¢(a*a) holds for every
a.
Note
that
every
unital
com-
pletely positive
map
satisfies
this
inequality,
and
a famous
result
of
Kadison
states
that
any
positive
unital
map
satisfies
the
inequality
for self-adjoint
elements
a.
In
17
relations
between
n-positivity
of
a
map
¢
and
the
Kadison-
Schwarz
property
of
certain
map
is established. Some a nice
property
of
the
Kadison-Schwarz
maps
were investigated
in
16.
The
present
paper
is devoted
to
the
Kadison-Schwarz
property
of
certain
255
256
class of
operators
on
M2(C),
It
is known
that
the
theory
of
Markov processes is a
rapidly
develop-
ing field
with
numerous
applications
to
many
branches
of
mathematics
and
physics. However,
there
are
physical
systems
that
can
not
be
described
by
Markov processes.
One
of such
systems
is given
by
quadratic
stochastic
operators
(see
2),
which relates
to
population
genetics.
The
problem
of
studying
the
behavior
of
trajectories
of
quadratic
stochastic
operators
was
stated
in
20.
The
limit
behavior
and
ergodic
properties
of
trajectories
of
quadratic
stochastic
operators
were
studied
in
7,8,9.
However, such
kind
of
operators
do
not
cover
the
case
of
quantum
systems. Therefore, in
5,6
quantum
quadratic
operators
acting
on
a von
Neumann
algebra
were de-
fined
and
studied.
Certain
ergodic
properties
of such
operators
were
studied
in
13,14.
In
the
present
paper
we
are
going
to
study
quantum
quadratic
operators
(q.q.o.)
with
the
Kadison-Schwarz property.
Note
that
operators
with
this
property
no
need
to
be
completely positive.
Aim
of
the
paper
is
to
find
some necessary conditions for
the
trace-preserving
quadratic
operators
to
be
the
Kadison-Schwarz ones. Since
trace-preserving
maps
arise
naturally
in
quantum
information
theory
(see e.g.
15)
and
other
situations
in which
one
wishes
to
restrict
attention
to
a
quantum
system
that
should
prop-
erly
be
considered a
subsystem
of
a
larger
system
with
which
it
interacts.
Therefore,
in
Section 3
we
describe q.q.o.
with
Haar
state
(invariant
with
respect
to
trace),
namely
certain
characterizations
of
q.q.o,
the
Kadison-
Schwarz
operators
are
given.
By
means
of
such a
description
in Section
4, we shall provide
an
example
of
q.q.o. which is
not
a Kadision-Schwarz
operator.
It
is
worth
to
mention
that
characterizations
of
positive
and
com-
pletely positive
maps
defined
on
M2
(C) were considered
in
10,11,19
and
18,
respectively.
2.
Preliminaries
In
what
follows,
by
M
2
(C) we
denote
an
algebra
of
2 x 2
matrices
over
complex filed
C.
By
M
2
(C) 0 M
2
(C) we
mean
tensor
product
of
M
2
(C)
into
itself. We
note
that
such a
product
can
be
considered
as
an
algebra
of
4 x 4
matrices
M
4
(C) over C. In
the
sequel
II
means
an
identity
matrix,
i.e. n =
(~~).
By
S(M
2
(C)) we
denote
the
set
of
all
states
(i.e. linear
positive functionals which
take
value 1
at
ll)
defined
on
M2(C)'
Definition
2.1.
A linear
operator
~ : M
2
(C)
----+
M
2
(C) 0 M
2
(C) is
said
257
to
be
(a) - a quantum quadratic operator
(q.
q.
0.) if
it
satisfies
the
following
conditions:
(i)
unital,
i.e.
~ll
=
II
®
ll;
(ii)
~
is positive, i.e.
~x
~
0 whenever x
~
0;
(b) - a K adison-Schwarz operator
(KS)
if
it
satisfies
~(x*x)
~
~(x*)~(x)
for all x E M2(C)' (1)
Note
that
if
~
is
unital
and
KS
operator,
then
it
is a q.q.o. A
state
hE
S(M
2
(C)) is called a Haar state for a q.q.o.
~
if for every x E M
2
(C)
one
has
(h ® id) 0
~(x)
= (id ® h) 0
~(x)
= h(x)ll.
(2)
Remark
2.1.
Let
U : M
2
(C) ® M
2
(C)
--+
M
2
(C) ® M
2
(C)
be
a linear
operator
such
that
U(x
® y) = y ® x for all
x,
y E M2(C)'
If
a q.q.o.
~
satisfies U ~ =
~,
then
~ is called a quantum quadratic stochastic operator.
Such a
kind
of
operators
were
studied
and
investigated in
13,14.
3.
Quantum
quadratic
operators
with
Kadison-Schwarz
property
on
M
2
(C)
In
this
section
we
are
going
to
describe
quantum
quadratic
operators
on
M
2
(C) as well as find necessary conditions for such
operators
to
satisfy
the
Kadison-Schwarz property.
Recall 3
that
the
identity
and
Pauli
matrices
{ll,
0"1,0"2,
0"3}
form a basis
for
M
2
(C), where
(
0
1)
(0
-i)
(1
0 )
0"1
= 1 0
0"2
= i 0
0"3
= 0
-1
.
In
this
basis every
matrix
x E M
2
(C)
can
be
written
as
x = woll +
wa
with
Wo
E C, w =
(WI,
W2,
W3)
E C
3
,
here
WO"
=
WIO"I
+
W20"2
+
W30"3·
Lemma
3.1.
18
The following assertions hold true:
(a) x is self-adjoint
iffwo,w
are reals;
(b)
Tr(
x)
= 1
iff
Wo
= 0.5, here
Tr
is the trace
of
a
matrix
x;
(c) x > 0
iff
Ilwll
::;
wo, where Ilwll =
JIW112
+
IW212
+ I
W
312.
258
Note
that
any
state
t.p
E
S(M
2
(C))
can
be
represented
by
t.p(woll
+
wO")
=
Wo
+ (w, f),
(3)
where f =
(11,12,
h)
E
IR3
with
Ilfll
:;
1.
Here
as
before (-,.)
stands
for
the
scalar
product
in
CC
3
.
Therefore,
in
the
sequel we will identify a
state
t.p
with
a
vector
f E IR3.
In
what
follows
by
T we
denote
a
normalized
trace,
i.e. T(X) =
~
Tr(x),
x E
M2(C)'
Let
~
: M
2
(C)
-+
M
2
(C) 0 M
2
(C)
be
a q.q.o.
with
a
Haar
state
T.
Let
us
write
the
operator
~
in
terms
of a basis in M
2
(C) 0 M
2
(C) formed
by
the
Pauli
matrices. Namely,
~ll
= ll 0 ll;
333
~(O"i)
= bi(ll 0 ll) +
2:);;)(ll
0
O"j)
+ :2);;)(O"j 0 ll) + L
bml,i(O"m
00"d,
j=l j=l
m ,
l=l
where
i =
1,2,3.
By
means
of
the
Haar
equality
with
T (see (2)) we find
that
b
i
=
0,
b
el)
- b(2) - 0 f
..
{I
2 3}
ij
-
ij
-
or
every z, J E " .
Positivity
of
~
implies
that
all
numbers
{bij,d
are
real ones. Hence,
one
can
prove
the
following
Theorem
3.1.
Let
~ : M
2
(C)
-+
M
2
(C) 0 M
2
(C)
be
a g.g.o. with a Haar
state
T,
then
it
has the following form:
3
~(x)
= woll 0 II + L (b
m1
,
w)a-m
0
0"1,
(4)
m,l
=l
where x =
Wo
+
WO",
b
m1
=
(bm1,1,
b
m
l,2, b
m1
,3).
Let
us
turn
to
the
positivity
of~.
Given
vector
f =
(11
,
12,
h)
E
IR3
put
3
(3(f)ij = L
bki,jfk.
(5)
k=l
Define a
matrix
JB(f)
= ((3(f)ij)rj=l.
By
IIJB(f)
II
we
denote
a
norm
of
the
matrix
JB(f)
associated
with
Euclid-
ean
norm
in
CC
3
.
Put
S = {p = (Pl,P2,P3) E
IR3:
pi
+
p~
+
p~
:;
I}
259
and
denote
111lH\111
=
sup
11lH\(f)
II·
fES
Now given a
state
cp
by
E<p
we
denote
the
canonical
conditional
expec-
tation
defined
by
E<p(x
® y) = cp(x)y, where x, y E M
2
(C).
Proposition
3.1.
Let
~
be
a q.q.o. with a Haar state
T,
then
111lH\111
:S
1.
Proof.
From
(4) we find
3
E<p(~(x))
= woll + L (bi,j,
W)fiaj
= woll + lH\(f)wa
i,j=l
where
cp(x)
=
Wo
+ (f, w), f = (il,
12,
h)
E S,
and
we have
used
cp(ai)
= fi
and
3
L(bi,j,
W)fi
= (lH\(f)w)j
i=l
Positivity
of
x yields
that
E<p(~(x))
is positive, for all
states
cp,
since
E<p
is a
conditional
expectation.
Hence,
according
to
Lemma
3.1
positivity
of
E<p(~(x))
equivalent
to
11lH\(f)wll
:S
Wo
for all f
and
w
with
Ilwll < Woo
Consequently, one finds
that
11lH\(f)11
=
sup
11lH\(f)wll:S
1,
which yields
the
Ilwll:<::;l
assertion.
o
Let
~
: M
2
(C)
----+
M
2
(C) ® M
2
(C)
be
a linear
operator
with
a
Haar
state
T.
Then
due
to
Theorem
3.1 ~
has
a form (4). Take
arbitrary
states
cp,
'ljJ
E S(M2(C))
and
f,
pES
be
the
corresponding
vectors
(see (3)).
Then
one
finds
that
3
~*(cp®'ljJ)(ak)=
Lbij,kfipj,
k=1,2,3.
i,j=l
Thanks
to
Lemma
3.1
the
functional
~
*
(cp
®
'ljJ)
is a
state
if
and
only if
the
vector
satisfies
Ilf~'(<p,,p)11
:S
1.
So, we have
the
following