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Time-evolution during Qubit measurement with a JBA — Physical and Informatics Dynamics —

Time-evolution during Qubit measurement with a JBA — Physical and Informatics Dynamics —

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151

The space L2 (L2 (M + (G))) is called compound Fock space over G. In an

analogous way to (2.14) we can dene 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 aect 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 congurations 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 dene 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 conguration of excited neurons related

to the dierent 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).

152

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 dierences of the operators

Ogˆuk = Oguk  Ogu0 ,

k = 1, . . . , r.

(2.26)

Now, for any function u on G we dene 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

153

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 dene

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 dierent regions are responsible for dierent 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(xzr )2mLr ,

m  0,

x 5 Gr

(3.7)

2

related to the eigen values (2mL1

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)

155

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 dierence of

the values related to the recording electrodes and the reference electrode

(sometimes specialists consider pairs of electrodes). Practice shows that

these dierences 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

Ek = 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

156

We can conclude that concerning EEG-measurement the memory causes a

noise which can be asymptotically identied 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.

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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 Scientic 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

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 Classication : 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 dened 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

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