On Positive Maps; Finite Dimensional Case Wladyslaw A. Majewski
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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. Denitions 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 dened 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) read as follows
(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 dened 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 dene
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 ,
t0,
(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)
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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 p2 ||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 identied 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
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125
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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 dierent 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, amplication, 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
of G is irrelevant. To start with a general space G has the advantage that it
can be used as a model for very dierent 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 diuse
measure on [G, G]. Especially, we are concerned with the case where G
is a compact subset of Rd and is the ddimensional 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 congurations (including the void
conguration 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 identication 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 conguration *). The
measure F dened on a canonical -algebra M(G) over M (G) restricted to
n-point congurations 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
n1
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 congurations (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 congurations 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 hn , An i
(A 5 L(L2 (M (G))))
(2)
from L2 (M (G)) and
where (