Accardi L., Freudenberg W., Obya M. (Eds.) Quantum Bio-informatics IV: From Quantum Information to Bio-informatics
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220
The
operator
D
-
is positively defined if E >
w -
W*
W*\[r
+ Ef
(
WW*
+ Ef W )
o is large enough. Hence,
it
determines
uniquely
the
Gaussian
measure
on
the
space
of
classical fields. Suppose now
that
¢(w) is a
random
vector
with
the
covariance
operator
Dw.
Then
(5)
This
relation
for averages
and
relation
(4) provide coupling
between
PCSFT
and
QM.
Quantum
statistical
quantities
can
be
obtained
from correspond-
ing
quantities
for classical
random
field: "irreducible quantum randomness
/I
is reduced to randomness
of
classical prequantum fields.
Vacuum
fluctuations:
The
additional
term
given
by
the
unit
operator
in
the
diagonal blocks
of
the
covariance
operator
of
the
prequantum
vector
field corresponds
to
the
field of
the
white
noise
type.
Such a field
can
be
considered
as
vacuum
fluctuations,
vacuum
field.
PCSFT
induces
the
following
picture
of reality:
Fluctuations
of
the
vacuum
field
are
combined
with
random
fields rep-
resenting
quantum
systems. Since we
cannot
separate,
e.g., electron from
the
vacuum
field,
we
cannot
separate
totally
any
two
quantum
systems.
Thus
all
quantum
systems
are
"entangled"
via
the
vacuum
field.
WHITE
NOISE
is
the
basis
of
everything
in
Nature
- Hida's Dream.
QM
as
renormalization
formalism:
Averages given
by
the
math-
ematical
formalism of
traditional
QM
are
obtained
as
renormalizations
of
classical averages, see (5).
Thus the
QM-formalism
can
be
considered
as
a
method
of
renormalization
of averages
with
respect
to
vacuum
fluctuations:
it
cancels
the
contribution
of
the
vacuum
field. Such a
renormalization
is
especially
important
in
the
case
of
observables of
the
nontrace
class. Here
the
contribution
of
the
background
field is infinite.
Thus
it
should
be
sub-
tracted
from
the
classical average, cf.
with
renormalization
procedures
of
QFT.
Superstrong
quantum
correlations:
In
PCSFT
such correlations
(violating Bell's inequality)
are
due
to
the
presence
of
the
vacuum
field.
The
off-diagonal
term
W
can
be
so large only if
the
diagonal
terms
are
completed
by
the
contribution
of
the
vacuum
filed.
Mathematics
tells us
this.
Thus they
are
so
strong,
because
the
vacuum
field really couple
any
two systems;
they
are
in
the
same
fluctuating
space.
Space is a huge
random
wave;
quantum
systems
are
spikes
on
this
wave;
221
they
are
correlated
via
this space-wave.
Thus
quantum
correlations have
two contributions: 1) initial
preparation
; 2) coupling
via
the
vacuum
field.
The
picture
is
pure
classical...
In
this
model
the
vacuum
field is
the
source
of
additional
correlations.
It
seems
that
this
classical
vacuum
field
is
an
additional
(purely classical)
quantum
computational
resource.
5.
QL
processing
in
the
brain
Consider two
time
scales (te, t
pe
) : tpe « t
e
,
cognitive
and
precognitive.
Take a signal
in
the
brain
oscillating
on
the
time
sca
le
tpe' We speculate
that
the
brain
has
an
integration
device
integrating
these fluctuations over
the
interval
te.
The
result
of
such
int
egrat
ion is considered as a cognitive
image.
In
this
paper
we do
not
present a concrete
proc
e
dure
of
integration
and, hence, creation
of
cognitive images from
random
fluctuations.
5.1.
Multiplicity
of
time
scales
in
brain
and
cognitive
QLR
The
main
lesson from
the
experimental
and
theoretical investigations
on
the
temporal
structure
of processes in
brain
is
that
there
are
various
time
scales.
Th
ey correspond
to
(or least
they
are
coupled with) various aspects
of cognition. Therefore
we
are
not
able
to
determine
once
and
for ever
the
cognitive
time
scale
te
("psychological
time").
There
are
a few such scales.
We shall discuss some evident possibilities.
It
is well known
that
there
are
well established
time
scales corresponding
to
the
alpha,
beta,
gamma
, delta,
and
theta
waves.
Let
us consider
these
time
scales as different cognitive scales.
For
the
alpha
waves we choose
its
upper
limit frequency,
12
Hz,
and
hence
the
te,a
~
0.083 sec. For
the
beta
waves we consider (by
taking
upper
bounds
of
frequency ranges)
three
different
time
scales: 15 Hz, t
e
,
(3,
low
~
0.067 sec. - low
beta
waves, 18Hz, t
e
,
(3
~ 0.056 sec. -
beta
waves,
23
Hz
t
e
,
(3,
high
~
0.043 sec. - high
beta
waves. For
gamma
waves we
take
the
characteristic frequency 40 Hz
and
hence
the
time
scale t
e
"
~
0.025 sec.
5.2.
Precognitive
time
scale
Our
choice
of
the
precognitive (very fine)
time
scale tpe will
be
motivated
by so called Taxonomic
Quantum
Model, see proposed by Geissler
et
aI,
see, e.g.,
paper
16
, for
representation
of
cognitive processes in
th
e
brain
(which was developed
on
the
basis of
the
huge
experimental
research
on
time-mind
relation.
They
found
that
information processing in cognitive
222
tasks
is based
on
time
scales Qq = q x Qo. where spc = Qo =
4.6ms.
We
choose
Qo as
the
unit
of
the
precognitive
time
scale.
This
corresponds
to
frequencies
~
220 Hz.
Under
such
an
assumption
about
the
precognitive
scale
we
can
find
the
measure
of
QL-ness for different
EEG
bands
. For
the
alpha
scale, we have
"'a.
= tQo
~
0.055. For
the beta
scales, we have:
"'c
f3
low
=
-t
Qo
~
0.069;
"'c
f3
~,atQo
~
0.082;
"'c
f3
high
=
-t
Qo
~
0.107.
,
'c
,j3,
low
'
c,/3
' ,
C,{3,h
lgh
For
the
gamma
scale
we
have:
"',
= tQo
~
1.84. Smaller
'"
correspond
to
c,-,
larger
integration
time
in
the
process
of
creation
of cognitive images; less
images
can
be
created
and
processed.
"Thinking
through
the
alpha
waves"
is essentially less advanced
than,
e.g.,
"thinking
through
the
gamma
waves"
I would like
to
thank
M. Ohya, L. Accardi, M. Asano, K.-H.Fichtner,
W.Freudenb
er
g for fruitful discussions.
References
1.
A.
Khr
en
nikov, Ubiquitous quantum structure: from psychology to finance,
Spring
er, Heidelberg- Berlin-New York, 2010.
2.
M. Asano, A.
Khrennikov
, M. Ohya, Foundations
of
Physics,
to
be
published
2010.
3. A.
Khr
en
nikov,
J.
Phys.A: Math. Gen.
349965-9981
(2001).
4.
A. Khrennikov, Contextual approach to quantum formalism
(Fundamental
Th
eories of Physics). Springer, Heidelb
erg
- Berlin-New York, 2009.
5. A. Khrennikov,
BioSystems
84
, 225- 241 (2006).
6.
A.
Khr
ennikov, Open
Systems
and
Information
Dynamics
11
(3), 267-275
(2004) .
7.
A. Khrennikov,
J.
Phys.
A:
Math. Gen.
38
, 9051-9073, 2005.
8.
A.
Khrennikov, Physics Letters A,
372
, 6588-6592 (2008).
9.
A.
Khr
enni
kov, Europhysics Letters
88,
40005 (2009).
10.
A.
Khr
enn
ikov,
J.
Consciousness Studies
15,
10-25 (2009).
11. A.
Khr
ennikov,
Interpr
etations
of
probability,
VSP
International
Science
Pub-
lishers,
Utrecht
(1999); second
addition
(completed) De
Gruyter,
Berlin,
2009.
12. L. Accardi, A.
Khrennikov
, M.
Ohya
, Open
Systems
and Information Dy-
namics
,
16,
441-443 (2009).
13.
K.-H.Fichtner,
L.Fichtner,
W.Freudenberg
and
M.Ohya,
On
a
quantum
mod
el of
the
recognition process. QP-PQ:Quantum Prob. White Noise Analy-
sis
21,
64-84 (2008).
14. A.
Einstein
and
L.
Inf
eld, Evolution
of
Physics: The Growth
of
Ideas from
Early Concepts to Relativity and Quanta,
Simon
and
Schuster, 1961.
15. A.
Khr
en
nikov, Theoretical and Mathematic
al
Physics
157,
1448-1460 (2008).
16. B. Schack, N.
Vath,
H.
Petsche,
H.
-G.
Geissler, E. Mller, Int.
J.
Psychophys-
iology
44,
143-163 (2002).
Quantum
Bio-Informatics
IV
eds. L.
Accardi
,
W.
Freudenberg
and
M.
Ohya
© 2011
World
Scientific
Publishing
Co.
(pp.
223-236)
ENTANGLEMENT
MAPPING
VS.
QUANTUM
CONDITIONAL
PROBABILITY
OPERATOR
DARIUSZ CHRUSCINSKI,
ANDRZEJ
KOSSAKOWSKI
Institute
of
Physics, Nicolaus Copernicus University
Grudziadzka
5/7,
87-100 Torun, Poland
TAKASHI
MATSUOKA
Faculty
of
Management
of
Administration and Information
Tokyo University
of
Science, Suwa
Toyohira 5000-1, Chino City, Nagano 391-0292, Japan
MASANORIOHYA
Department
of
Information Science, Tokyo University
of
Science
Yamazaki
2641, Noda City, Chiba 278-8501, Japan
The
relation
between
two
me
thods
which
construct
the
density
operator
on
com-
posite
system
is
shown.
One
of
them
is called
an
entanglement
mapping
and
another
one
is
called
a
quantum
conditional
probability
operator.
On
th
e
base
of
this
relation
we
discuss
the
quantum
correlation
by
means
of
some
typ
es
of
quantum
entropy.
1.
Introduction
In
quantum
information
theory
it
is crucial
to
make clear
the
following
two problems.
The
first one concerns
the
study
of
classifying
quantum
states
of
composite systems. In classical description
of
a physical
compound
system
its
correlation
can
be
represented by a
joint
probability
measure
or
a conditional probability measure. In
quantum
case a
quantum
state
of
a
composite system, i.e. a
compound
state
describes a correlation between its
marginal
states.
However, in
quantum
system,
it
is known
that
the
joint
probability
and
the
conditional
probability
do
not
generally exist, which
is
an
essential difference from classical
system
20
,22.
The
typical example
of
such difference is
the
existence
of
entangled
states
in
quantum
system.
The
difference means
that
one have
to
deal more carefully
not
only
with
the
correlation
of
entangled
state
but
also
that
of separable one.
From
223
224
this
point
of
view
to
give
the
proper
classification of
quantum
states
is
of
primary
importance. Hence, one needs
an
appropriate
measure
of
quantum
correlations.
We
start
with
notions
of
positivity:
the
so-called complete positivity
(CP)
and
complete co-positivity
(CCP).
Let A
and
B
be
C*-algebras
(with
a
unit).
A linear
map
X : A
1-+
B is
CP
if
and
only if
(1)
is positive for all
n.
Here
Mn
(A)
stands
for n x n matrices
with
entries
from
A. A linear
map
X : A
1-+
B is
CCP
if
and
only if
(2)
is positive for all n. Every n-co-positive
map
is positive
but
not
necessarily
complete positive even if
it
is complete co-positive, unless B is Abelian,
in which case a positive
map
is
both
complete positive
and
complete co-
positive.
Belavkin
and
Ohya
gave a rigorous
construction
of
quantum
compound
states
by means
of
a
CCP
map
called
entanglement
mapping
4
,5.
We
can
construct
a
compound
state
by applying
the
entanglement
mapping
to
Jamiolkowski's isomorphism.
On
the
other
hand
Kossakowski
et
al. in-
troduced
the
notion of
quantum
conditional probability
operator,
QCPO
for
short,
by means
of
a
CP
mapl.
If
one
of
marginal
states
is given,
then
we
can
also
construct
the
compound
state
by
operating
the
marginal
state
by
QCPO.
In
the
present
paper
we show
the
relation between
the
entan-
glement
mapping
and
QCPO,
i.e.
the
relation between
the
CCP
map
and
the
CP
map.
This
paper
is organized as follows:
In
sections 2
and
3
the
defini-
tion
of
entanglement
mapping
and
QCPO
are
reviewed.
In
the
middle
of
1990s Horodecki family
and
Peres
gave a classification of
compound
states
by
introducing
the
criteria
so called a
PPT
(Partial
Positive Transpose)
condition
l2
,2l. Here
we
also review
the
relation between
the
entanglement
mapping
and
PPT
condition
l5
,l8,
and
the
relation between
QCPO
and
PPT
condition 1 .
The
section 4 gives
the
relation
of
such two methods. We show
that
the
compound
state
by
entanglement
mapping
is
related
to
QCPO
via
the
similar
equation
to
classical Bayesian relation.
In
the
section 5
we
review
the
two
types
of
quantum
entropies, so called
the
quantum
mutual
entropy
and
the quantum
conditional entropy,
and
such entropies
are
applied
to
some examples
of
entanglement
mapping. Finally, from
the
observation
225
of
the
above
computational
examples
the
meaning
of
such entropies
are
discussed
on
the
base
of
quantum
Bayesian relation.
The
above
argument
can
be
formulated
at
general C* -algebraic
setting
4
,5,15,17,18.
However, for simplicity, only
the
finite-dimensional case
will
be
considered here.
2.
Entanglement
mapping
and
its
classification
Let
us
consider two
quantum
systems
(Hk, A
k
, S
(A
k
)h=I,2
where
Hk
is a
Hilbert
space given
by
Hk = (Cn,
Ak
= B (Hk)
stands
for
the
algebra
of
complex
k x k
matrices
and
S (Ak)
stands
for
the
set
of
all
states
on
A
k
.
Then
the
composite
system
(H,
A,
S) is given
by
H
=:0
HI
0H
2
, A
=:0
B (HI
0H
2
)
=
Al
0 A
2
, S
=:0
S (A) = S (AI 0 A
2
) .
Notice
that
S
::J
conv (S
(Ad
0 S
(A
2
))
.
It
was
mentioned
that
conv(S
(Ad
0 S
(A
2
))
are
called
separable
states
and
the
subset
of
states
S\conv(S
(Ad
0 S
(A
2
))
is called
the
set
of
entangled
states,
i.e. S =
Sent
U
Ssep'
We
are
interested
in
the
special
subset
of
S,
SpPT=:O{WES;w
o
(10T)
ES},
(3)
where T is
stands
for
transposition.
Such
states
are
called
partial
positive
transposition
(PPT)
states
I2
,21.
Clearly
S
::J
SPPT
::J
Ssep.
In
this
section
we
will review how
to
construct
the
compound
state
by
means
of
entanglement
mapping
and
we will show
that
the
entanglement
mapping
gives
the
same
classification
of
states
with
PPT
condition.
For a
normal
state
w E S
there
exist a
density
matrix
B such
that
w(a0b)
=
Tr(a0b)B,a
E
AI,b
E A
2
.
(4)
The
state
w
can
be
written
as
(5)
where ¢ is a linear
map
from A2
to
Al
given
by
¢
(b)
=:oTr
2
(10
b)
BE AI,
and
its
dual
map
¢* from
Al
to
A2
given
by
¢*
(a)
=:oTrI
(a
01)
B E A
2
•
In
(5)
the
partial
trace
with
respect
to
Hk
is
denoted
by
Trk (.).
It
is clear
that
the
marginal
densities
of
B
are
given
by
(6)
226
Belavkin
and
Ohya
revealed
that
such maps
can
be
reconstructed by
Hilbert-Schmidt
operator,
which
is
called
the
entangling operator,
and
they
showed
that
both
¢
and
¢* are complete co-positive,
but
not
always com-
plete positive
on
general C* -algebraic
setting
4
,
5.
For example, if w
is
a
pure
entangled
state
,
then
its
entanglement mappings are
not
CPo
Here we recall
the
necessary
and
sufficient condition of
CP
and
CCP
in
the
case of finite dimension.
Let
Mn
denote
the
algebra of complex n x n
matrices
and
let
I:
[Mn,
MmJ
be
the
complex space of linear maps from
Mn
to
Mm. We recall
the
isomorphism W :
I:
[Mn,
MmJ
f->
Mn
0
Mm
8,16
,24:
For each linear
map
X E
I:
[Mn,
MmJ
one defines
n
WIP'
+
(X)
=
"L)ek)
(ezl
0 X
(Iek)
(ezl)
= (1 0
X)
lP'
+ (7)
k,l
and
n
W
JJ
(X)
=
~)
e
k
)
(e11
0 X
(leI)
(ekl)
= (1 0
X)Jf
(8)
k,l
where
{Iek)
(ell}
is
an
orthonormal
base of
Mn.
The
map
(7)
is
called
n
the
Choi's isomorphism
8
with
lP'
+ = L
lek)
(ezl
0
lek)
(ell
which gives
the
k,l
maximally entanglement
state
(not normalized),
and
the
map
(8)
is
called
n
the
Jamiolkowski isomorphism
14
with
Jf
= L
leI)
(ekI
0 Ie
k)
(ezl.
k,l
One has
the
following
Lemma
2.1.
8,
16
,
24
For X E
I:
[Mn,
MmJ
the following
statements
hold:
(1) X is CP
iff
W
IP'+
(X)
2:
o.
(2) X is CCP
iff
WJJ
(X)
2:
o.
We
apply
Lemma
2.1
to
entanglement
mapping
¢
and
its
dual
¢*.
The
density
matrix
()
in (4)
can
be
represented by
matrix
elements as
n
()=
L eij:
a
(3
l
e
i)(ejI
0
I!a)(f
(3
1
(9)
i ,j ,a ,
/3
= l
where{lei) (ejl} (resp. {Ifa)
(f
(3
I}
) is a
standard
base of
A1
(resp.A
2
) ,
and
e
ij
:
a
(3
is
given by e
ij
:
a
(3
= (ei 0 fa ,
eej
0 f
(3
). Using
this
representation ¢
227
and
¢*
are
given
by
¢
(b)
=
Trd1
®
b)
e
n
= L eij:
a
(3
U(3,
bfa}
lei}
(ej
I,
(10)
a,(3=l
and
¢*
(a)
=
Trda
® 1) e
n
= L eij:
a
(3
(ej,aei)
Ifa}
U(3I·
(11)
i,j=l
So
that
we
have
n
W.Jf
(¢*)
= L (Iek) (ell) ®
¢*
(lei) (ekl)
k.l=l
n
L e
ij
:
a
(3l
e
i}
(ejl ® Ifa)
U(31
= e
(12)
i,j,a,(3=l
Namely
the
entanglement
¢*
reconstructs
its
density
matrix
e
via
the
Jamiolkowski
isomorphism.
This
fact
means
that
¢*
is always
CCP.
In
the
similar
way
we
can
show
also
the
complete
co-positivity
of
¢.
On
the
other
hand
we
apply
¢
to
the
Choi's
isomorphism.
Then
we
have
n
k.l=l
n
L e
ij
:
a
(3l
e
i}
(ejl ®
If(3)
Ual =
(1
®
T)
e (13)
i,j,a,(3=l
The
positivity
of
WIP'
+ (¢) is
equivalent
to
the
complete
positivity
of
¢
and
it
is also
equivalent
to
the
PPT
condition
of
e.
Theorem
2.1.
e E
SPPT
if
and only
if
the map ¢* is CF.
Remark
2.1.
Theorem
2.1
was
proved
at
C*-algebraic
general
setting
15
,18.
If
an
entanglement
mapping
¢*,
i.e. a
normalized
CCP
map,
is given,
then
one
can
construct
a
compound
state
e¢
as
n n
k.l=l k.l=l
228
3.
Quantum
conditional
probability
operator
and
its
classification
In
this
section
we
review
another
construction
of
compound
state
by Kos-
sakowski
et
all.
For a given
()
E S
with
p
=Tr
2
()
one
can
define
the
operator
(15)
which
has
the
following
properties
7r1!
~
0,
(16)
(17)
and
we
assume
that
p >
0,
that
is, p represents a faithful
state.
It
follows
from (16)
and
(17)
that
the
operator
7r1!
is
the
quantum
analogue
of
a clas-
sical conditional probability. Indeed,
in
the
case A is
an
Abelian alge
bra
,
7r1!
coincides
with
a classical conditional probability.
Definition
3.1.
An
operator
7r E A is called a
quantum
conditional prob-
ability
op
e
rator
(QCPO)
if 7r satisfies condition (16)
and
(17).
One
easily finds
the
following formula
n
7r<p
= L
'P
(Ifk)
Ull)
®
Ilk)
Ull
(18)
k.l=l
where
'P
is a
CP
unital
map
from A2
to
A
1
,
i.e.,
'P
(1)
=
11.
Using
Lemma
2.1
and
unitality
of'P
it
is easy
to
check
that
7r
<p
satisfies conditions (16)
and
(17). For a given 7r
<p
and
any
marginal
state
pES
(Ad
one
can
construct
a
compound
state
as
n
()
<p
= L
p~
'P
(lfk)
Ull)
p~
®
Ifk
)
Ull·
(19)
k .l= l
Then
we
apply
Lemma
2.1 (2)
to
the
map
A
(-)
==
p
~
'P
(.)
p~:
n
W.
dA)
= L
p
~
'P
(lfk)
Uti)
p~
®
1M
(
Ik
I =
(1
®
T)
()
<p'
(20)
k.l
= l
Hence
Corollary
3.1.
1()<p
E
SPP
T
if
and only
if
the map
'P
is CCP.
229
4.
The
relation
between
entanglement
mapping
and
QCPO
Belavkin
and
Dai
6
gave
the
following
decomposition
of
an
entanglement
mapping.
Lemma
4.1.
Every entanglement mapping ¢ with ¢ (1) = P has a decom-
position
where
zp
is a OP unital map to
be
found
as
a unique solution to
zpC)
=
p-!¢OT(')p-!,
Now we
can
show
the
following
(21)
(22)
Theorem
4.1.
If
a compound state
e</>
given
by
(14) has a marginal state
P
=
Tr2
e</>
, then
e</>
is represented
by
(23)
Proof.
We apply the decomposition (21) to
e</>.
Then
n
e</>
= L ¢ (Ifk) (!II) o If
I)
Ukl
k.l=l
n
= L p!zp 0 T (I!h) (11)
p!
o liz)
Ukl
k.l=l
n
= L p!zp(lfl)
Ukl)p!
o If
I)
(!hI
k.l=l
In
the opposite direction we have
n
(p!
(1)
1f<p
(p!
(1)
= L p!zp(lfk)
(11)p!
0lfk)
Ull
k.l=l
k.l=l
n
= L ¢ (liz) Ukl)
0lfk)
Ull
=
e</>.
o
k.l=l