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The space L2 (L2 (M + (G))) is called compound Fock space over G. In an
analogous way to (2.14) we can dene an exponential vectorEXP{#} 5
L2 (L2 (M + (G))) (# 5 L2 (M + (G)).
The complete information (knowledge/expected view of the world) is
represented by a certain signal which is characteristic for the individual (at
xed time-learning can change that). The memory consists of a large set
of ”copies" of that signal. Further, recognition is based on choosing one
signal from the memory. That should not aect essentially the state of the
memory. Taking into account the description of choosing a signal from the
memory (cf. 2,3 ) that motivates the choice of the state of the memory as
a coherent state (cf. 11 )
2.4. EEG-Measurements
Using EEG-device one measures the electric potential of congurations of
the excited neurons. Let x and y be the position of an excited neuron
and the position of an electrode placed on the scalp, respectively. Then
the measured potential of the neuron is a certain function u
˜(x y) of the
distance of these positions.
Now let * be the point system representing the positions of a set of
excited neurons. For any function u on G we dene a function gu on M (G)
putting
Z
X
u(x) = *(dx)u(x).
gu (*) =
x5*
Then the potential of * is given as gu˜ (*).
Now, the apparatus consists of r + 1 electrodes. Let (yk )rk=0 be the
positions of r + 1 electrodes where usually r = 2m , m 5 N. Putting
˜(x yk ),
uk (x) := u
k = 0, . . . , r,
(2.22)
we get the measured potentials of a conguration of excited neurons related
to the dierent electrodes:
Z
guk (*) := *(dx)uk (x), k = 0, . . . , r
(2.23)
REMARK 1 Usually specialists assume that G is a convex compact subset
of Rd equipped with Lebesgue measure. Observe that the electrodes are xed
on the scalp. Thus yk is not an element of G representing the physical body
of the brain. Nevertheless in the following we can assume that the functions
uk are continuous (consequently bounded).
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Now we put
u
ˆk := uk u0 ,
k = 1, . . . , r.
(2.24)
Using the apparatus the outcomes of the EEG-measurement are these differences of potentials
guˆk (*) = guk (*) gu0 (*),
k = 1, . . . , r.
(2.25)
Then the quantum measurement of the potential of the signal according
to the k-th electrode is represented by the operator Oguk of multiplication
corresponding to the function guk 2,9 . Consequently, the output of the
quantum EEG-measurement corresponds to the dierences of the operators
Ogˆuk = Oguk Ogu0 ,
k = 1, . . . , r.
(2.26)
Now, for any function u on G we dene a function hu on M (M + (G))
putting
Z
Z
Z
hu () := (d*)gu (*) = (d*) *(dx)u(x), 5 M (M + (G)).
(2.27)
Each signal is related to the set of excited neurons. The union of all these
sets represents the set of excited neurons supporting the memory. For that
reason, the outcomes of EEG measurements of the electric potentials of the
excited neurons supporting the memory are given by huˆk (k = 1, . . . , r).
3. Main Results
We consider the asymptotic behaviour of the internal noise caused by the
memory.
We consider the following pure state of the memory.
:= |EXP{c#}i
(3.1)
where
#=
X
j5J
cj | exp{afj } exp{0}i,
(3.2)
0 6= a, c 5 C, J N, (fj )j5J L2 (G) is an ONS, and (cj )j5J is a (nite)
family of complex numbers with
X
|cj |2 = 1.
(3.3)
j5J
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REMARK 2 From the assumption that (fj )j5J L2 (G) is an ONS, it
follows that
(| exp{afj } exp{0}i)j5J
is an ONS (contained in (L2 (G))) and
k#k = 1.
(3.4)
Furthermore we assume
hfj |uk fj i = hfj |u0 fj i ,
k = 1, . . . , r,
j 5 J.
(3.5)
EXAMPLE 1 Let (Gr )nr=1 be a measurable decomposition of G Rd . We
n
X
x f r 5 L2 (Gr ) such that
|f r |2 1 on G and dene
r=1
1
fj := |G| 2
n
X
arj f r
(3.6)
r=1
where aj = [a1j , . . . , anj ] 5 C, j 5 J with
|asj | = 1,
s = 1, . . . , n,
j 5 J,
and
n
X
s=1
a
¯sj1 asj2 kf s k2 = 0,
j1 6= j2 .
REMARK 3 Let (Gr )nr=1 be a “maximal” measurable decomposition of
G Rd in the sense that the dierent regions are responsible for dierent tasks. It can be motivated by the postulates 3,9 that related to each of
the regions Gr we consider the motion of a noninteracting quantum mechanical particle systems(“free motion”) related to von Neumann boundary
condition, i.e. the Hamiltonian operator is the second quantization of the
corresponding one-particle operator Hr . The total Hamiltonian operator is
given by the tensor product of these local Hamiltonian operators being again
a second quantization.
It is often technical convenient when the regions are parallelepipeds to
work with periodic boundary conditions. In order to avoid complicated formula in that case one can specialize d = 1, and Gr := [zr , zr + Lr ]. Further
154
we put ~ = 1. Then we have the following sequence of eigen functions of
Hr (r = 1, . . . , n)
1
fr,m (x) := ei(xzr )2mLr ,
m 0,
x 5 Gr
(3.7)
2
related to the eigen values (2mL1
r ) .
In general one should use a coherent function corresponding to a general superposition of these functions. But it is wellknown that the results of
measurement of frequencies are non-random (up to external noise depending on the device). For that reason one should consider a coherent state
related to a xed eigen function fr,mr in order to describe parts of the signal corresponding to the r-th region. Then inside of the regions the density
of the excited neurons would be constant (we have |fr,m (x)| 1). Further,
it is wellknown that the density of excited neurons supporting the signal in
the memory is constant on G, i.e. the size of the regions determines the
expectation of the number of excited neurons. Thus the densities do not depend on the regions. Summarizing we obtain the representation considered
in Example 1 putting
f r := fr,mr ,
r = 1, . . . , n
(3.8)
Our brain contains about 100 billions of neurons. Thus in reality the
number of excited neurons supporting one signal and the number of signals
stored in the memory are very large, i.e. the densities related to |a|2 and
|c|2 are very high. On the other hand the values of the functions us related
to the electric potential of one excited neuron are very low. In order to
re ect that we put
1
1
a := bm 4 , c := dm 4 ,
Us
us := s , s = 0, . . . , r.
m
(3.9)
(3.10)
(3.9) implies
s
|a|2 = |b|2 m,
s
|c|2 = |d|2 m.
(3.11)
Using (3.10) and (2.24) we get the representations
ˆk
U
u
ˆk = s ,
m
k = 1, . . . , r
(3.12)
where
ˆk := Uk U0 ,
U
k = 1, . . . , r.
(3.13)
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Now, from (3.9) it follows that the density of the excited neurons supporting the memory is increasing of the order m. Taking into account (3.10)
one can see that measuring the potential of all of the excited neurons supporting the signals stored in the memory by one electrode gives expected
s
values increasing of the order m (very large and strongly depending on
the individual). On the other hand concerning only one signal the expectation of the measured potential does not depend on m. For that reason
it makes no sense to use the values related to the single electrode in order to obtain information concerning the process of recognition of signals.
For that reason, as it was already done in 2.4, one uses the dierence of
the values related to the recording electrodes and the reference electrode
(sometimes specialists consider pairs of electrodes). Practice shows that
these dierences give information on the process of recognition disturbed
by an (internal) noise caused by the potential of the excited neurons supporting the signals stored in the memory.
One can use our main results (cf. Theorem 1) 12 in order to describe
the statistical behaviour of that noise.
The operators Ohuˆk commute. For that reason the output of the EEGmeasurement in the case of a pure state of the memory gives the classical
random vectors = [ 1 , . . . , r ] with probability distribution Pm depending
on the level m.
THEOREM 1 Let = [ 1 , . . . , r ] be a normal distributed with expectation
Ek = 0,
k = 1, . . . , r
Z
ˆk U
ˆs d,
|fj |2 U
and covariance matrix
E k s = |d|2 |b|2
X
j5J
|cj |2
k, s = 1, . . . , r.
Then the sequence (Pm ) of probability distribution converges weakly to P
of the random vector .
REMARK 4 The random vector can be represented by stochastic integration
Z
ˆk dB, k = 1, . . . , r
k := U
with respect to a generalized Brownian motion B with “noise intensity measure” being absolutely continuous with respect to with density
X
|d|2 |b|2
|cj |2 |fj |2 .
j5J
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We can conclude that concerning EEG-measurement the memory causes a
noise which can be asymptotically identied with the mentioned generalized
Brownian motion. Considering the case of Remark 3 we have to specialize |fj |2 |G|1 , and denotes the Lebesgue measure. In that case the
parameters |d|2 and |b|2 depend on the individual.
References
1. A. K. Engel and W. Singer, Temporal binding and the neural correlates of
sensory awareness, Trends in Cogn. Sci., 5(1), 16-25, 2001.
2. K.-H. Fichtner and L. Fichtner, Bosons and a quantum model of the brain, Jenaer Schriften zur Mathematik und Informatik, Math/Inf/08/05, FSU Jena,
Faculty of Mathematics and Informatics, Jena ,2005, 27 pages.
3. K.-H. Fichtner and L. Fichtner, Quantum models of brain activities I - Recognition of signals, In J.C. Garcia, R. Quezada, and S.B. Sontz, editors, Quantum probability and Related topics, XXIII of QP-PQ: Quantum Probability and White Noise Analysis, 135-144, New Jersey London Singapore, 2008,
World Scientic.
4. K.-H. Fichtner and W. Freudenberg, Point processes and the position distribution of innite boson systems, J. Stat. Phys., 47, 959-978, 1987.
5. K.-H. Fichtner and W. Freudenberg, Characterization of states of innite
boson systems, Comm. Math. Phys., 137, 315-357, 1991.
6. K.-H. Fichtner and W. Freudenberg, The compound Fock space and its application to brain models, In L. Accardi, W. Freudenberg, and M. Ohya,
editors, Quantum Bio-Informatics II, XXIV of QP-PQ: Quantum Probability and White Noise Analysis, 55-67, New Jersey London Singapore, 2009,
World Scientic.
7. K.-H. Fichtner, L. Fichtner, W. Freudenberg, and M. Ohya, On a mathematical model of brain activities, In Quantum Theory, Reconsideration of
Foundations-4, 962 of AIP Conference Proceedings, 85-90, Melville, New
York, 2007. American Institute of Physics.
8. K.-H. Fichtner, L. Fichtner, W. Freudenberg, and M. Ohya, On a quantum
model of the recognition process, In L. Accardi, W. Freudenberg, and M.
Ohya, editors, Quantum Bio-Informatics, XXI of QP-PQ: Quantum Probability and White Noise Analysis, 64-84, New Jersey London Singapore, 2008,
World Scientic.
9. K.-H. Fichtner, L. Fichtner, W. Freudenberg, and M. Ohya, On a quantum
model of the brain activities, In L. Accardi, W. Freudenberg, and M. Ohya,
editors, Quantum Bio-Informatics III, XXVI of QP-PQ: Quantum Probability and White Noise Analysis, 81-92, New Jersey London Singapore, 2010,
World Scientic.
10. K.-H. Fichtner, L. Fichtner, W. Freudenberg, and M. Ohya, Quantum models
of the recognition process — on a convergence theorem, Open Systems and
Information Dynamics, 17(2), 161-187, 2010.
11. K.-H. Fichtner, L. Fichtner, W. Freudenberg, and M. Ohya, In L. Accardi, W.
157
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
Freudenberg, and M. Ohya, editors, to appear in Quantum Bio-Informatics
IV, QP-PQ: Quantum Probability and White Noise Analysis, New Jersey
London Singapore, 2011, World Scientic.
K.-H. Fichtner, L. Fichtner, K. Inoue, and M. Ohya, Internal noise caused
by memory, Preprint.
K.-H. Fichtner, W. Freudenberg and V. Liebscher, Time evolution and invariance of boson systems given by beam splittings, Innite Dimensional
Analysis, Quantum Probability and Related topics, 1(4), 511-531, 1998.
K.-H. Fichtner, V. Liebscher and M. Ohya, A limit theorem for conditionally independent beam splittings, In M. Schürmann and U. Franz, editors,
Quantum Probability and Innite Dimensional Analysis, From Foundations
to Applications, XVIII of QP-PQ: Quantum Probability and White Noise
Analysis, 227-236, New Jersey London Singapore, 2005, World Scientic.
K.-H. Fichtner and M. Ohya, Quantum teleportation with entangled states
given by beam splittings, Comm. Math. Phys., 222, 229-247, 2001.
S. Hamero and R. Penrose, Orchestrated objective reduction of quantum
coherence in brain microtubules: the “Orch OR" model for consciousness,
Mathematics and Computer Simulation, 40, 453-480, 1996.
R. Hari and O.V. Lounasmaa, Neuromagnetism: tracking the dynamics of
the brain, Physical World, 33-38, May 2000.
M. Ohya, Complexity in quantum system and its application to brain function, In T. Hida and K. Saito, editors, Quantum Information , II, 144-160,
Singapore, 2000, World Scientic.
J.W. Philips, R.M. Leahy, and J.C. Mosher, Imaging neural activity using
MEG and EEG, IEEE Engineering in Medicine and Biology, 16(3), 34-42,
1997.
J.M. Schwartz, H.P. Staap, and M. Beauregard, Quantum physics in neuroscience and psychology, Phil. Trans. Royal Soc. Lond., B360(1458), 13091327, 2005.
W. Singer, Consciousness and the structure of neural representations, Phil.
Trans. Royal Soc., B353, 1829-1840, 1998.
W. Singer, Der Beobachter in Gehiru. Essays zur Hirnforschung, Suhrkamp
Verlag, Frankfurt a.M., 2002.
H.P. Staap, Mind, Matter and Quantum Mechanics, Springer, Berlin Heidelberg, 2nd edition, 2003.
H.P. Staap, A model of the quantum-classical and mind-brain connections,
and the role of the quantum zeno eect in the physical implementation of
conscious, arXiv, 0803.1663v1 (physics.gen-ph), 11 March 2008.
J. von Neumann, Mathematische Grundlagen der Quantenmechanik,
Springer, Berlin, 1st edition, 1932.
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Quantum Bio-Informatics V
c 2013 World Scientiﬁc Publishing Co. Pte. Ltd.
pp. 159–170
SKEW INFORMATION AND UNCERTAINTY RELATION
SHIGERU FURUICHI
World Scientic Publishing Co., Inc, 1060 Main Street, River Edge NJ 07661,
USA
E-mail: furuichi@chs.nihon-u.ac.jp
KENJIRO YANAGI
Division of Applied Mathematical Science, Graduate School of Science and
Engineering, Yamaguchi University, 2-16-1, Tokiwadai, Ube City, 755-0811,
Japan
E-mail: yanagi@yamaguchi-u.ac.jp
This article is a short review on our recent results on uncertainty relations with
skew informations. In nal section, we give the problem on uncertainty relations
with skew informations.
Keywords : Uncertainty relation, Wigner-Yanase skew information and
trace inequality
PACS numbers : 03.65.Ta and 03.67.-a
2010 Mathematics Subject Classication : 15A45, 47A63 and 94A17
1. Introduction
In quantum mechanical systems, the expectation value of an observable
(self-adjoint operator) H in a quantum state (density operator) is expressed by T r[H]. Also, the variance for a quantum state and an observ2
able H is dened by V (H) T r[ (H T r[H]I) ] = T r[H 2 ] T r[H]2 .
We start from the famous Heisenberg uncertainty relations 10
1
V (A)V (B) |T r[[A, B]]|2
(1)
4
for a quantum state and two observables A and B. The further strong
result was given by Schrödinger 22 :
1
(2)
V (A)V (B) |Re {Cov (A, B)} |2 |T r[[A, B]]|2 ,
4
159