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On Positive Maps; Finite Dimensional Case Wladyslaw A. Majewski

On Positive Maps; Finite Dimensional Case Wladyslaw A. Majewski

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121

Finally, one introduces the corresponding measure

Z

(1 , 2 ; t)dt .

NBLP () = sup

1 ,2

(18)

>0

It is clear that NBLP () is rather di!cult to compute in practice. It is easy

to prove the following

Proposition 1. If the map is divisible (i.e.

(1 , 2 ; t)  0 (i.e. Markovian II).

Markovian I), then

Interestingly, the converse is not true: if (1 , 2 ; t)  0, then in general

the dynamical map needs not be divisible.

4. Example: qubit dynamics

In this section we illustrate the relation between two concepts of Markovianity (cf. Denitions 2 and 3) on a simple example. Consider the following

dynamics of a qubit

00 (t) = 00 x0 (t) + 11 [1  x1 (t)] ,

11 (t) = 00 [1  x0 (t)] + 11 x1 (t) ,

(19)

01 (t) = 01 (t) ,

where x0 (t), x1 (t) 5 [0, 1], and

|(t)|2  x0 (t)x1 (t) .

(20)

The above conditions for xk (t) and (t) guarantee that the dynamics is

completely positive. One easily nds for the corresponding local generator

1

X



L  = i [ z , ] +

ak Lk  + Lz  ,

2

2

(21)

k=0

where

1

L0  =  +    {   + , } ,

2

1

L1  =    +  { +  , } ,

2

Lz  =  z  z   .

(22)

The time-dependent coe!cients a0 and a1 are dened by

xb 0 (1  x1 ) + xb 1 x0

,

1  x0  x1

xb 0 x1 + xb 1 (1  x0 )

a1 =

,

1  x0  x1

a0 =

(23)

(24)

122

whereas (t) and

(t)

b

a0 (t) + a1 (t)

 Re

,

2

(t)

(t)

b

(t) = Im

.

(t)

(t) = 

(25)

(26)

One has

Proposition 2. The corresponding dynamical map (t, 0) is divisible if

and only if ak (t), (t)  0 for all t  0.

On the other hand one nds the following formula

(1 , 2 ; t) = 

2A(t)200 + [A(t) + 4(t)]|01 |2

p

,

200 + |01 |2

(27)

where A(t) = a0 (t) + a1 (t), and a 2 × 2 matrix  is dened by

 = 1  2 .

(28)

In principle one may have (1 , 2 ; t)  0 even if a0 (t) < 0 or a1 (t) < 0.

Indeed, let us dene

Z t

Z t

f0 ( )d , x1 (t) =

f1 ( )d ,

(29)

x0 (t) =

0

such that 0 

Rt

0

0

fk ( )d  1 , for all t  0. Now, following Ref. 25 let

f0 (t) =  sin t ,

t0,

(30)

and f1 (t) = 0 for t 5 [0, ] together with

f1 (t) =  sin t ,

t ,

(31)

where 0 <  < 1/2. One nds

a0 (t) + a1 (t) =

 sin t

,

1   +  cos t

t 5 [0, ] ,

(32)

and a0 (t) + a1 (t) = 0 for t  . Note, that for t   one has

a0 (t) = a1 (t) =  sin t ,

which proves that (t, 0) is not divisible.

(33)

123

5. Markovianity and the generalized discrimination

problem

Consider now the following generalized discrimination problem: suppose

we are given one of two known states 1 and 2 , with probabilities p and

1  p, respectively. Our goal is to guess which one it is with minimal

error probability. Again, the solution of this problem is well known due to

Helstrom 26 : the p-dependent minimal error probability (MEP(p)) reads as

follows

MEP(p) =

1  Dp [1 , 2 ]

,

2

(34)

where

Dp [1 , 2 ] := ||(1  p)1  p2 ||1 .

(35)

Clearly, for the unbiased problem, that is p = 1/2, one reproduces (11).

Now, following Ref. 25 let us introduce p-dependent information ow

p (1 , 2 ; t) =

d

Dp [1 (t), 2 (t)] .

dt

(36)

e 0) = (t, 0)
1ld of the exFinally, consider an extended dynamics (t,

tended system living in H
H and let

ep (e

1 , e

2 ; t) =

d

Dp [e

1 (t), e

2 (t)] ,

dt

(37)

e 0)e

k (t) = (t,

k . One proves

where e

k are density operators in H
H, and e

(cf. Ref. 25) the following

Theorem 1. A map (t, 0) is divisible (Markovian I) if and only if

ep (e

1 , e

2 ; t)  0 for any density operators e

1 and e

2 , t  0, and any

p 5 (0, 1).

6. Conclusions

In conclusion, we have analyzed two concepts of Markovianity, one based

on the divisibility property of the dynamical map (Markovianity I) and the

other based upon the distinguishability of quantum states (Markovianity

II). We have given very simple example for the dynamics of a single qubit

where these two criteria do not coincide. Furthermore we proposed a way to

make them equivalent, in the sense that Markovianity would be identied by

divisibility, but keeping the interpretation in terms of ows of information.

124

Acknowledgments

It is a pleasure to thank Professor Noboru Watanabe for his warm hospitality during QBIC 2011. Many thanks for his help after the earthquake

on March 11.

References

1. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems

(Oxford Univ. Press, Oxford, 2007).

2. U. Weiss, Quantum Dissipative Systems, (World Scientic, Singapore, 2000).

3. R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applications

(Springer, Berlin, 1987).

4. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge Univ. Press, Cambridge, 2000).

5. R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Rev. Mod.

Phys. 81, 865 (2009).

6. J. Wilkie, Phys. Rev. E 62, 8808 (2000); J. Wilkie and Yin Mei Wong, J.

Phys. A: Math. Theor. 42, 015006 (2009).

7. A. A. Budini, Phys. Rev. A 69, 042107 (2004); ibid. 74, 053815 (2006).

8. H.-P. Breuer, Phys. Rev. A 69 022115 (2004); ibid. 70, 012106 (2004).

9. S. Daer, K. Wódkiewicz, J.D. Cresser, and J.K. McIver, Phys. Rev. A 70,

010304 (2004).

10. A. Shabani and D.A. Lidar, Phys. Rev. A 71, 020101(R) (2005).

11. S. Maniscalco, Phys. Rev. A 72, 024103 (2005); S. Maniscalco and F. Petruccione, Phys. Rev. A 73, 012111 (2006).

12. J. Piilo, K. Härkönen, S. Maniscalco, K.-A. Suominen, Phys. Rev. Lett. 100,

180402 (2008); Phys. Rev. A 79, 062112 (2009).

13. E. Andersson, J. D. Cresser and M. J. W. Hall, J. Mod. Opt. 54, 1695 (2007).

14. A. Kossakowski and R. Rebolledo, Open Syst. Inf. Dyn. 14, 265 (2007); ibid.

15, 135 (2008); ibid. 16, 259 (2009).

15. H.-P. Breuer and B. Vacchini, Phys. Rev. Lett. 101 (2008) 140402; Phys.

Rev. E 79, 041147 (2009).

16. M. Moodley and F. Petruccione, Phys. Rev. A 79, 042103 (2009).

17. B. Vacchini and H.-P. Breuer, Phys. Rev. A 81, 042103 (2010).

18. D. Chru´sci´nski, A. Kossakowski, and S. Pascazio, Phys. Rev. A 81, 032101

(2010)

19. D. Chru´sci´nski and A. Kossakowski, Phys. Rev. Lett. 104, 070406 (2010).

20. D. Chru´sci´nski, A. Kossakowski, P. Aniello, G. Marmo, F. Ventriglia, Open

Systems and Inf. Dynamics 17, 255-278 (2010).

21. D. Chru´sci´nski, A. Kossakowski and S. Pascazio, Phys. Rev. A 81, 032101

(2010).

22. G. Lindblad, Comm. Math. Phys. 48, 119 (1976); V. Gorini, A. Kossakowski,

and E.C.G. Sudarshan, J. Math. Phys. 17, 821 (1976).

23. H.-P. Breuer, E.-M. Laine, J. Piilo, Phys. Rev. Lett. 103, 210401 (2009).

24. Á. Rivas, S.F. Huelga, and M.B. Plenio, Phys. Rev. Lett. 105, 050403 (2010).

125

25. D. Chru´sci´nski, A. Kossakowski and Á. Rivas, Phys. Rev. A 83, 052128

(2011).

26. C. W. Helstrom, Quantum Detection and Estimation Theory (AcademicPress, New York, 1976).

Quantum Bio-Informatics V

c 2013 World Scientiﬁc Publishing Co. Pte. Ltd.

pp. 127–142

HIGH DENSITY LIMIT OF THE DISTRIBUTION OF THE

OUTCOME OF EEG-MEASUREMENTS

K.-H. FICHTNER, L. FICHTNER,

Unversity Jena, Institute of Applied Mathematics, 07743 Jena, Germany

E-mail: chtner@mathematik.uni-jena.de

W. FREUDENBERG

Brandenburg Technical University Cottbus, Department of Mathematics,

03013 Cottbus, Germany. E-mail: freudenberg@math.tu-cottbus.de

M. OHYA

Department of Information Science and Quantum Bio-Informatic Center,

Tokyo University of Science, Noda City, Chiba 278-8510, Japan

E-Mail: ohya@rs.noda.tus.ac.jp

Using EEG measurements one gets information on the densities of excited neurons

located in the regions of the brain. Up to now there exist dierent hypothesises

concerning the distribution of the random outcomes of EEG measurements. Using

classical models for describing brain activities it turned out to be di!cult to explain

the observed properties of these outcomes. We will describe the distribution of

the random outcomes of EEG measurements and certain conditional distributions

in terms of a high density limit. These considerations are based on a quantum

statistical model of the process of recognition that was developed in the last years

(cf. [1] — [10]).

1. Introduction

The procedure of recognition can be described as follows: There is a set of

complex signals stored in the memory. Choosing one of these signals may

be interpreted as generating a hypothesis concerning an ”expected view of

the world”. Then the brain compares a signal arising from our senses with

the signal chosen from the memory leading to a change of the state of both

signals. Measurements of that procedure like EEG or MEG are based on the

fact that recognition of signals causes a certain loss of excited neurons, i.e.

the neurons change their state from excited to non-excited. For that reason

a statistical model of the recognition process should re ect both the change

127

128

of the signals and the loss of excited neurons. Physicists as R. Penrose or

H. P. Stapp (cf. [12, 18, 19]) but also at an increasing rate specialists of

modern brain research (cf. [15, 16, 14, 17]) are convinced that information

processing in the brain cannot be described appropriately by models based

on classical physics or classical stochastics. A rst attempt to explain the

process of recognition in terms of quantum statistics was given in [1]. In

a (still incomplete) series of papers based on a quantum statistical model

of the recognition process the procedures of creation of signals from the

memory, amplication, accumulation and transformation of input signals,

and measurements like EEG and MEG are treated in detail (cf. [1] —

[10]). In the present paper it is not possible to present this approach in

detail. In lieu we will sketch roughly a few of the basic ideas and structures

of the proposed model of the recognition process. Our main purpose is

to describe the distribution of the outcome of EEG-measurements. Using

EEG-measurements one gets information on the densities of excited neurons

located in the regions of the brain. Let us remark that these measurements

are classical ones, and the results can be described by classical statistics

though the underlying statistical model of the recognition process is a nonclassical one. We will describe the distribution of the random outcomes

of EEG-measurements and certain conditional distributions in terms of a

high density limit. The proofs of these limit theorems will be given in a

forthcoming paper.

2. The Space of Signals

In the present section we introduce brie y notions and notations needed

in the sequel. For interpretation and motivation of the introduced notions

we refer to the above mentioned papers. Starting point will be a set G

representing the space where the process of recognition and processing of

the signals takes place. For the mathematical model the concrete structure

can be used as a model for very dierent aspects of the recognition processes

in the brain. So let G be an arbitrary complete separable metric space

and G its -algebra of Borel sets. Further, let  be a xed nite diuse

measure on [G, G]. Especially, we are concerned with the case where G

is a compact subset of Rd and  is the ddimensional Lebesgue measure

restricted to G. The elements of the Hilbert space H = L2 (G, ) can be

interpreted as functions of the excited neurons. From this space of square

integrable complex-valued functions on G we pass over to the Hilbert space

129

L2 (M (G)) = L2 (M (G), F ) where M (G) := {{x1 , . . . , xn } : xj 5 G, n  0}

is the set of all nite symmetric point congurations (including the void

conguration o). To avoid symmetrisation procedures it is convenient to

identify the sets {x1 , . . . , xn } with the corresponding counting measures

* =  x1 +. . .+ xn where  x denotes the Dirac measure in x. So we can make

the identication M (G) = {* : * is a counting measure on G}. Within

this notation |*| = *(G) denotes the number of points in *, and x 5 *

means that * x 5 M (G) (i. e. x belongs to the the conguration *). The

measure F dened on a canonical -algebra M(G) over M (G) restricted to

n-point congurations is (up to symmetrisation) just the product measure

1
n

.

n! 

More precisely,

n

X

X 1 Z

F (Y ) := 1IY (o) +

1IY (

 xj )
n (d[x1 , . . . , xn ]) (Y 5 M(G)).

n!

j=1

n1

Gn

(1)

Hereby, 1IY denotes the indicator function of a set Y . Observe that

F ({o}) = 1, F (M (G)) = exp{(G)} and that F is concentrated on the

set of simple point congurations (without multiple points). One can show

that the space L2 (M (G)) is isomorphic to the usual symmetric (bosonic)

Fock space (L2 (G, )):

L2 (M (G)) 

= (L2 (G, ))

For details we refer for instance to [8]. The space L2 (M (G)) consists of

(complex-valued) functions of congurations of excited neurons, and will

be interpreted as the space of signals. Later on we restrict ourselves to

an essentially smaller subspace. A quantum state on a Hilbert space H is

a positive linear normalized functional \$ on the set L(H) of all bounded

linear operators on H. In our special case a general (normal) state \$ on

L2 (M (G)) can be written in the form

\$(A) =

N

X

n=1

n h n , A n i

(A 5 L(L2 (M (G))))

(2)

from L2 (M (G)) and

where (

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