{ QUANTUM MEASUREMENTS AND INFORMATION\\ RESTRICTIONS IN ALGEBRAIC QM\\ }
S.N.Mayburov
Lebedev Inst. of Physics
Leninsky Prospect 53
Moscow
Russia
117924
It's argued that Information-Theoretical restrictions for the systems selfdescription are important for Quantum Measurement problem. % In particular G$\ddot{o}$del information incompleteness As follows from Breuer theorem, for the quantum object S measurement by information system $O$ they described by $O$ restricted states $R_O$. $R_O$ ansatz can be introduced phenomenologically from the consistency with Shr$\ddot {o}$dinger dynamics and measurement statistics. % $R_O$ 'preferred' basis $|O_j\rangle$ defined by % $O$ state decoherence via $O$ - E interactions. The analogous restrictions obtained in Algebraic QM considering Segal algebra of S,$O$ observables and the resulting $O$ algebraic states $\{\varphi^O\}$ set defined as its dual space. From Segal theorem for associative (sub)algebras it's shown that $\varphi^O_j$ describes the random 'pointer' outcomes $O_j$ observed by $O$ in the individual events.
quant-ph0402044.tex
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%\begin{titlepage}
 \title{\small{  QUANTUM MEASUREMENTS AND INFORMATION\\ 
RESTRICTIONS IN ALGEBRAIC QM\\ }}
\small\author{S.N.Mayburov \\ 
Lebedev Inst. of Physics\\
Leninsky Prospect 53, Moscow, Russia, 117924\\
E-mail :\quad   mayburov@sci.lpi.msk.su}
\date {}
\maketitle
\begin{abstract}
It's argued that
  Information-Theoretical restrictions for the systems selfdescription
are important for Quantum Measurement problem.
%       In particular G$\it\ddot{o}$del information incompleteness
As follows from Breuer theorem,
for the quantum object S  measurement by information system $O$
they  described by $O$ restricted states $R_O$.  
$R_O$ ansatz  can be introduced phenomenologically from the consistency
with Shr$\ddot {o}$dinger dynamics and measurement statistics.
%           $R_O$  'preferred' basis $|O_j\rangle$ defined by 
%           $O$ state decoherence via $O$ - E interactions.
 The analogous  restrictions obtained in Algebraic QM 
 considering Segal algebra  of S,$O$ observables
and the resulting $O$ algebraic states $\{\varphi^O\}$ set
defined as its dual space. From
 Segal theorem for associative  (sub)algebras  it's shown
 that  $\varphi^O_j$ describes the  random 'pointer'
outcomes $O_j$  observed by $O$ in the individual events.

\end{abstract}
\vspace{6mm}
$ \quad \quad $ {\bf Key Words: \, Quantum Measurements, Information,
  $C^*$ algebras}  
\small { \quad 
%   Talk given on 'Quantum Information and Computations' workshop, \\}
% \small {  \qquad  Triest,  October 2002 }
\vspace{10mm}
%\end{titlepage}
%\begin {sloppypar}
%
%
%

\section  {Introduction}


Despite that Quantum Mechanics (QM) is universally acknowledged
physical theory, there are still several unresolved problems concerned with
its interpretation.
 Of them, the State Collapse or Quantum Measurement
   Problem is the most widely
and long discussed (D'Espagnat,1990; Busch,1996).
In this paper we regard the quantum  measurement process in the
 information-theoretical framework and demonstrate
its importance for the state collapse consideration
(Svozil,1993). Really,   both the quantum and classical measurement
 is, eventually, the  information
acquisition   by the information system $O$ (Observer) 
via the direct or indirect interaction with the studied system S
 (Guilini,1996; Duvenhage,2002).  
Therefore, 
  the possible restrictions on the information
pattern transferred from S to $O$ can be important
in the Measurement Theory (Breuer,1996).
% Such approach to the Measurement Problem
%  corresponds with the general trends of quantum information theory
% extensively developed now.
We  concede in our study that QM description is applicable
both for  a microscopic    and  macroscopic
objects; in particular,
 $O$ state described by  Dirac  vector
$|O\rangle$ or density matrix $\rho$ 
  relative to another  observer $O'$ (Rovelli,1995;Bene,2000).
%  Such approach to QM foundations, assuming universality of QM
%  applicability to micro/macroscopic objects 
%  which called Relational QM leads to some other interesting conclusions.  
 $O$ considered as the information gaining and utilizing system 
(IGUS)  which acquire and memorize the information as the result of
 S interactions   with the measuring system (MS)  which  element is $O$
(below S formally is also regarded as MS element).
In principle, $O$  can be  either a human brain or some automatic device 
processing the information. In all cases it's
the system with some number of  internal degrees of freedom (DF)  which
 interacts during S information acquisition, so
 that $O$ internal state changes after it.
%  The computer information processing or the perception  by human brain
%  supposedly results from some physical objects evolution
%  which on microscopic level obeys to  QM laws.
%  Example  of it are the electron pulses excited in the computer circuits
%   during the information bits memorization.
%   Such correspondence for the mental
%   processes isn't proved,  but there are now the strong
%   experimental evidences that QM can  successfully describe
%   a  complex systems including biological ones (Penrose,1997).   


%  In standard QM framework S quantum state  -  $|S\rangle$
%   defined relative to some object $O_c$ which supposed to be
%  classical.
% In our model MS is the quantum system and due to it  
   S measurement by $O$ described by MS state $|MS\rangle$ evolution
relative to some  $O'$, yet in this model  
 the  acquired S information memorized and processed by $O$, not by $O'$,
 which reflected by $O$ internal state   evolution. Therefore, 
the detailed description of S information recognition    
% internal state perturbed by S,$O$ interaction 
%  and therefore, $O$ internal state
  should be analyzed in the  selfdescription framework (Svozil,1993).
%  is  the new  problem for Quantum theory but 
The information systems selfdescription
was  studied already in the context
of the  selfreference  problem 
 (Finkelstein,1988; Mittelstaedt,1998).
It was shown that the arbitrary system selfdescription
is always incomplete; this result often interpreted as the
analog of G$\it\ddot{o}$del Theorem for Information theory (Svozil,1993).
 In this  framework Breuer developed 
 the  restricted  states formalism
 for the selfdescritpion
 in the measurement process - the  selfmeasurement which is applicable 
 both  in classical and quantum case (Breuer,1996).
 It follows that $O$ internal state $R_O$, which
 is the partial (restricted) MS state,   can differ
 principally from the standard
QM ansatz for $O$ state relative to $O'$ (Mittelstaedt, 1998).
Basing on this  results, we propose here the novel
formalism  which    accounts $O$ selfmeasurement  effects and predicts
 the measured state collapse.
 Its  main feature is the modification of quantum state ansatz which becomes
 the doublet $\Phi= \{ \phi^D, \phi^I \} $, where $\phi^D=\rho$ is  QM
density matrix MS, $\phi^I(n)$ is $O$ restricted state describing $O$
 subjective information 
in the given individual event $n$ 
(Mayburov,2001).
 %  It assumes the corresponding enlargement of quantum states set $L_T$ over
%  standard Hilbert space $\cal H$, so that $L_T=\cal{H}$ $ \otimes L_V$
% where $L_V$ is $V^O$ set.
% which elements describes $O$ information acquired in the individual events. 
$\phi^I$  can be  independent of $\phi^D$ and ,in particular,
  demonstrates the stochastic behavior in S measurements.  
 It will be shown that such formalism
  corresponds to the well-known generalization
of standard QM - algebraic QM based on Jordan,Segal and $C^*$- algebras
applications (Emch,1972). In its framework $\phi^D$ is MS state defined on MS 
observables algebra $\cal{U}$,
 $\phi^I$ corresponds to the 
  state defined on $O$ observables subalgebra $\cal{U}_O$.
%     $\Phi$ states properties  depends on  $O$ decoherence interaction
%     with environment defining $O$ states $\phi^I$ preferred basis.
%  The analogous ideas were formulated  by Kochen  on phenomenological level
%   without  use of selfdescription or algebraic approaches (Kochen,1985); 
%   our first results were published in (Mayburov, 2001).
% Relativistic analysis of EPR-Bohm correlated measurements also indicate
% the subjective character of the state collapse which moment can't be
% defined covariantly for all observers.

%  Here it's necessary to make some  comments on our model premises
%  and review some terminology.

%      The most detailed state description
%      presented by  the ensemble  state ( Gemenge) defined below, but 
%      such description isn't always possible ????
% In our  model we'll suppose that MS  always can
%  be described completely (including Environment E if necessary)
%  by some state vector $|MS\rangle$ relative to 
% $O'$ or by  density matrix for mixed cases.
%  MS can be closed system , like atom in the box or open pure
% system  surrounded by electromagnetic vacuum or  E of other kind.
% We don't assume in our work any special dynamical
%  properties of $O$ internal states beyond standard QM. 
%   In this paper  the brain-computer analogy used without
% discussing its reliability and philosophical 
% implications $\cite {Pen}$.
% We ignore here quantum computer options
%  having in mind only the standard   computers.
%
We must stress that  the observer consciousness 
 never referred directly and doesn't play any role in our theory 
(London,1939).
%  also none of the brain  the special  properties
% beyond standard QM assumed.
 Rather, in our model observer $O$  regarded as the quantum 
reference frame (RF) which interacts with studied object S (Aharonov,1981).
% changing $O$ internal state and via it  storing the information about S.
 S state description 'from the point of view' of the particular $O$
referred by the terms 'S state in $O$ RF' or simply 'S  state for $O$'.
 The terms 'perceptions', 'impressions' used by us to characterize the
   IGUS $O$ description of experimental  results 
and   defined below in strictly physical  terms.
In particular, the perception is  
 the acquisition of some information by IGUS, i.e. the change
of IGUS  state; 
the different $O$ impressions  associated with the different, 
 $O$ physical states.
%  which description sho operationally unambiguous.



\section {Measurements  and Quantum States Restrictions}

Our formalism exploits both the quantum states in the individual events
- i.e. individual states and the 
  statistical states describing the quantum ensembles properties
 (Mittelstaedt,1998).
 Remind that in QM  the individual
states are the pure states which are isomorphic to Dirac vectors 
 $|\Psi\rangle$    
in $\cal H$ ; the statistical states  described by  the 
normalized, positive operators of trace $1$
- density matrixes  $\rho$ on $\cal {H}$.
 If the  $\Psi_l$ composition  is known  for 
the given ensemble, its state can be described in more detail by
 the   ensemble state (Gemenge)    
 presented  by the table $W^e=\{ \Psi_l; P_l\}$ 
where $P_l$ are the corresponding probabilities (Busch,1996).
%  note hence, that
%   such detailed description isn't always physically possible.
%  The  same statistical state 
%  $\rho$,  in general, admits the nonunique
%  decompositions into different $W^e$. 
   Algebraic QM   states  will be considered in chap. 3.



We'll regard 
  the simple  MS measurement model similar 
to  von Neuman model (von Neuman,1932; Busch,1996).
It     includes
the measured binary state  S which interacts with the
 observer $O$  storing the incoming S information.
% $O$ interaction with environment E - decoherence
% will be  accounted at the next stage of the model regarded in chap. 4.
 In our model the detector D omitted in  MS chain,
% because  the only  D effect is
% the amplification of S signal to make it conceivable for O.
the role of $O$ decoherence effects will be discussed below.
%     \psi_s=a_1|s_1\rangle+a_2|s_2\rangle
%
%  The pure S,$O$ states  belongs to MS Hilbert space
%  ${\cal H}_{MS}$ defined in  $O'$ RF.
%  the density matrixes ${\rho}_{MS}$ set defined on
% $\cal{H}_{MS}$ denoted $\Gamma$.
The regarded $O$ has one  internal DF  and  in its Hilbert space $\cal {H}_O$
 the basis  consists of the
  three orthogonal states $|O_{0,1,2}\rangle$
 which are  the eigenstates of $Q_O$  
'internal pointer' observable with eigenvalues $q^O_i$.
We'll  consider    
 the measurement   of the binary
 S observable $\hat{Q}$ on S   state $\psi_s$.
Initial  $O$ state  is  $|O_0\rangle$ and  MS initial state is :
\begin {equation} 
     \Psi^{in}_{MS}=\psi_s |O_0\rangle= 
(a_1|s_1\rangle+a_2|s_2\rangle)|O_o\rangle  \label {AAB}
\end {equation}
 where $|s_{1,2}\rangle$ are $Q$ eigenstates with eigenvalues $q_{1,2}$.
  S-$O$ measuring interaction starts at $t_0$
and finished effectively at some finite $t_1$,  
 by the suitable choice of  S$-O$ interaction Hamiltonian $\hat{H}_I$ 
 Schr$\ddot{o}$dinger equation (SE)  results in
 MS final  state $\rho^p_{MS}$ :
\begin {equation}
   \Psi_{MS}=\sum \Psi^{MS}_j = \sum a_i|s_i\rangle|O_i\rangle
%  a_2|s_2\rangle |O_2\rangle
                                   \label {AA2}
\end {equation} 
%  and thus  $|O_{1,2}\rangle$ are $O$ final states induced by
%   the measurement of eigenstates  $|s_{1,2}\rangle$.
As the result, for any $\psi_s$ one obtains $ \bar{Q}_O=\bar{Q}$
 which means that $O$ performs the unbiased $Q$ measurement.
Meanwhile, for any $O$ observable
 $Q'_O  \ne F(Q_O) ;\, \bar{Q}'_O=0$ independently of 
$\psi_s$. 
% For some situations indicated below, to simplify the calculations,
%  it's convenient to choose $|O_0\rangle=|O_1\rangle$
% and consider the binary subspace $\cal{H}_O$.   
Regarding $O$ as the information system, 
 we'll assume  that $|O_{1,2,0}\rangle$ corresponds to 
 $O$ information pattern - an impressions notified by $q^O_{1,2,0}$
(Guilini,1996).
% 
% It percepted by $O$ as if $Q$ of S has the
 % value $q_{1,2}$ or $q_0$ - absence of input S signal, correspondingly;
 % this assumption isn't final and will be reconsidered   below.
%  Formally Hilbert space ${\cal H}_{MS}$ in $O$ RF can be   obtained
% performing  ${\cal H'}_{MS}$ unitary transformation $\hat{U}'$ to $O$ RF.
%  The set (algebra) of all S, $O$ observables denoted
% $\cal{M}_S,\cal{M}_O$ correspondingly,
%  the set of other (collective) MS observables
% which isn't belong to $\cal{M}_O,\cal{M}_S$   denoted $M_{A}$.
%  $O$ states $|O_i\rangle \in \cal H_O$ Hilbert subspace of ${\cal H}_{MS}$;
%  $\rho_O$ subset on $\cal H_O$ is $L_O\in L_q$.
%  In  our case when we regard  only internal or RF independent states 
%  ${U}'=I$  can be taken and thus  $\Psi=\Psi^O$ for 
%  arbitrary states in $O'$ and $O$ RF correspondingly.
%  In practice  IGUS $O$  have many internal DFs, but
%  their account  doesn't change principally the results obtained below.
%  For the start we omit also $O$-E interaction - decoherence, but later
%  we'll account it and study its influence. 
%
Therefore,  at $t>t_1$  for external  $O'$ 
MS is  in the pure state  $\Psi_{MS}$ of (\ref {AA2})
 which is  the superposition of the states 
corresponding to the different measurement outcomes.
% If one applies standard QM rules for the calculation of MS state
% in $O$ RF $\Psi_{MS}^O$ it  obtained from
% $\Psi_{MS}$ by  the unitary transformation $U'$
% $\Psi^O_{MS}=U'\Psi_{MS}$. If to omit the state component describing
% MS center of mass motion and restricts only to internal MS state,
% it gives $\Psi^O_{MS}=\Psi_{MS}$.
Basing on our  assumptions, 
 from $O$ 'point of view' $\Psi_{MS}$ describes the
simultaneous superposition (coexistence) of two
contradictory impressions : $Q_O=q^O_1$ and $Q_O=q^O_2$ percepted by $O$
 simultaneously.
%      which  as Wigner claimed is nonsense \cite {Wig}.
 Yet it's well known   that experimentally   
the macroscopic   $O$ observes at random one of $Q_O$ values $q^O_{1,2}$.
 From 
that  S final state is $|s_1\rangle$ or $|s_2\rangle$  and
S state collapse occurs. 
In standard QM  with Reduction Postulate S final  state described by
% the statistical ensemble of  individual final states for $O$
  the density matrix  of mixed state: 
\begin {equation}
 \rho_s^m= \sum_i |a_i|^2|s_i \rangle \langle s_i|
                                                              \label {AA33}
\end {equation}
In accordance with it, in our model one
 can  ascribe to MS the corresponding mixed state :
\begin {equation}
 \rho_{MS}^m= \sum_i |a_i|^2|s_i \rangle \langle s_i||O_i \rangle \langle O_i|
                                                              \label {AA3}
\end {equation}
which    differs principally from $\rho^p_{MS}$ of (\ref {AA2}). 
It's quite difficult to doubt both in the correctness of
  MS evolution description by SE
 and in  the state collapse experimental observations.
% which formalism was proposed in \cite {Bre}.
% If $O$ regarded as the quantum object, 
  This  obvious contradiction constitutes famous Wigner 
  'Friend  Paradox' for $O, O'$ (Wigner,1961).
%   which in the form of
% $gedankenexperiment$ was just described above (Wigner,1961).
We attempt here to unite  this alternative  
  systems descriptions 'from outside' by $O'$
and 'from inside' by $O$ in the same  formalism.



% Considering the general measurement formalism, note first, that 
Formally, both the classical and quantum measurement 
 of the arbitrary  system $S'$ is the 
mapping of $S'$ states set $N_S$ on  the given IGUS $O^S$ states set $N_O$ 
(Mittelstaedt,1998).  
%  In the simple cases   some measured $S'$ parameter
%   $A^S$ becomes correlated with with some $O$ parameter $A^O$
%   via  $S',O$  interaction.
% If  $O^S$ interaction with $S'$ can't be neglected
 If the final $O^S$ and $S'$ state can't be factorized,
 then  $O^S$ should be regarded as
 the subsystem of the large system $S_T=S'+O^S$ with the states set $N_T$.
  In this situation - 'measurement from inside'  $N_O$ is $N_T$ subset and 
$O^S$ state  is $S_T$ state projection to $N_O$ -
  the restricted  state $R_O$.
% which describes the information pattern available to $O$.  
From $N_{T}$ mapping properties the principal restrictions on
 $O^S$ restricted states obtained in
 Breuer theorem : if for two arbitrary $S_T$ 
states $\Phi_{S}, \Phi'_{S}$ 
their restricted  states $R_O, R'_O$ coincide, then for $O^S$ this $S_T$ 
states are indistinguishable (Breuer,1996).
% Contrary to  its  seeming triviality,Breuer theorem stipulates the important
%  information systems features   considered below.  
  The origin of this  results  in classical case is easy to 
 understand: $O^S$ has less number
 of DFs then $S_T$ and, therefore,
 can't describe completely $S_T$ state (Svozil,1993).
 In quantum case  
  the  observables noncommutativity and nonlocality introduce
some new features regarded below.
Despite that $R_O$ are incomplete $S_T$ states,
 they are the real physical states
for $O^S$ observer - 'the states in their own right' as Breuer puts it. 

 The described
$S'$,$O^S$,$S_T$ relation corresponds to our MS model which can be
regarded as 'the MS measurement from inside'. 
%   it's natural to expect that $R_O$ state depends only of $O$ observables
%   set $M_O$ and not of other MS observables.
%  Obviously for the
  Breuer results doesn't permit
to derive the restricted states for an  arbitrary  system
directly,
 and as the phenomenological $R_O$ ansatz it was proposed (Breuer,1996)  
to use the partial trace  which for MS final  state  (\ref {AA2}) is equal to:
\begin {equation} 
   R_O=Tr_s  {\rho}^p_{MS}=\sum |a_i|^2|O_i\rangle\langle O_i|
      \label {AA4}
\end {equation}
%
%       Such $R_O$ is in fact $\rho_{MS}$ projection into $\cal H_O$ defined
%       in $O'$ RF and all $R_O$ belong to $\rho_O$  set $\Sigma_O$.
%
%    this was our motivation for application of Algebraic
%   QM for this problem.
%  \cite {Bre}. 
%  $R_O$ can be interpreted as $O$ subjective state which
%  describe his  perception of MS state after measurement, 
%
% $R_O$  components weights are defined in $O$ basis, which
% will be used in the following calculations :
% \begin {equation}
% w_j(t)=Tr (\hat{P}^O_j R_O)   \label {BB00}
% \end {equation}
% where $\hat{P}^O_j$ is $O_j$ projection operator, and $w_j(t_1)=|a_j|^2$
% for MS final state.
in particular, for the incoming  $|s_j\rangle$  
 $R_O=|O_j\rangle \langle O_j|$.
For   MS   state $\rho^{m} _{MS}$
of (\ref {AA3}) appearing in the measurement of the incoming S mixture,
 the  corresponding restricted statistical
 state is the same $R^{mix}_O=R_O$.
This equality doesn't mean  the
 collapse of MS pure state $\Psi_{MS}$
% because it's applied directly only to restricted distributions.  
 because  the collapse appearance
   should be   verified also  for  MS,$O$ individual
  states.
%    as was noticed in the chap.1 for pure MS state
%    it simply coincides with $\rho_{MS}$.
For the pure case MS individual state is always $\Psi_{MS}$, yet
 for  the incoming S statistical mixture (\ref {AA3})
  MS individual state   differs
from event to event:
\begin {equation}
\rho^A (n)=\rho^I_l=|O_l\rangle \langle O_l|| s_l\rangle\langle s_l|
  \label {A44}
\end {equation}
where the random $l(n)$ described by the
 probabilistic distribution $P_l=|a_l|^2$.
% This individual state is  pure and
%  can be initially unknown for $O$ but exists objectively.
 $\rho^A (n)$  differs from
 the  state (\ref{AA2}), correspondingly,
its restricted state 
 $\varsigma^O(n)=|O_l\rangle \langle O_l|$
  also differs in any event  from $R_O$ of (\ref {AA4}). Due to it, 
 the main condition of Breuer  Theorem violated for the individual states
 and $O$ can differentiate pure/mixed states 'from inside'
in the individual events (Breuer,1996). Therefore, the proposed 
formalism doesn't permit to obtain 
 the state collapse  for $O$ selfdescription
 in standard QM framework.
%   It's easy to understand its physical reason : the proposed $R_O$ 
%    describes $Q_O$ uncertainty for MS state $\Psi_{MS}$  and consequently
%   can't result in the definite $Q_O$ in the individual event.
Hence, $R_O$ is the consistent restriction of MS statistical state
$ \rho_{MS}^{p,m}$ to $O$  which coincides  for the
pure and mixed  S states with the same $|a_i|$. 
 $R_O$ ansatz (\ref {AA4})
 regarded also as $O$ individual partial state relative to 
external classical observer  in the standard Quantum Measurement
Theory without selfdescription  (Lahti,1990). 
%    In fact, $R_O$ describes the situation
%    which Wigner defines as $O$ impressions superposition (Wigner,1961).

%   In our model $O$ initial state is $|O_0\rangle$ only, 
%   together with $H_I$ unique S,$O$ interaction, it reduces
%   $\rho_{MS}$ decompositions ambiguity.


% In this framework
% Straightforward $R_O$ interpretation is that it describes
% $O$ internal pointer $Q_O$ 'splitting' percepted by $O$ or $O$ impressions
% superposition in Wigner terms.
Note that even  in Breuer  theory 
%  MS state   'from inside'- $R_O$  is  operationally different
%  from MS state for   $O'$ 'from outside' i.e. $\Psi_{MS}$.
 $O$ can't observe the difference between 
  MS   states with different $D_{12}=a_1^*a_2+a_1 a_2^*$. 
%  the partial  collapse.
 Such difference revealed by 
  MS interference term (IT) observable :
\begin {equation}
   B=|O_1\rangle \langle O_2||s_1\rangle \langle s_2|+j.c.
    \label {AA5}
\end {equation}
In standard QM,
being measured by external $O'$ on S,$O$,
 it gives $\bar{B}=0$ for the mixed MS state (\ref {AA3}),
 but  $\bar{B}\neq 0$  for the pure MS states (\ref{AA2});
 $B$ value principally can't be measured by $O$ 
'from inside';
%    because $O$ performs $Q_O$ measurement and $[Q_O,B] \neq0$. 
%  We'll use below  also $O$ IT observable:
% $$
%    B_O=|O_1\rangle\langle O_2|+|O_2\rangle \langle O_1|
%  $$
 note also that $B,Q_O$  doesn't commute. 
%
%      We don't intend here to analyze Breuer theory in detail;
%       some  its weak points discussed in chap.5 and also in (Mayburov,2002). 
%  
%     Note hence that the formal states difference in QM by itself doesn't mean
%        automatically that  $O$ percepts them as different ones.
%   It should be  at least one observable $A_O \in M_O$ for which $\bar{A}_O$  
%    expectation values are different for such states
%    which isn't proved in Breuer theory.
%
% In relation to it  Algebraic QM  dispatches the additional
% indications for the choice of  alternative $R_O$ ansatz.

Formally, MS individual state  for $O$ can be written  in doublet 
form $\Phi^B(n)=|\phi^D,\phi^I \gg$, where $\phi^D=\rho_{MS}$
is the objective (dynamical)
state component  and the information component
 $\phi^I$ describes $O$ subjective information  in the given event $n$.
% below this ansatz will be  important for us.
 In Breuer theory  
% equal  to $R_O$ for the pure states and $\varsigma^O(n)$ for S mixture
% correspondingly,
%      The resulting states set is
%         $N_T=\Gamma \otimes L_I$ where   $L_I$ the set of restricted states.
 for the pure MS  states $\phi^I$ is just $\phi^D$  projection
but in the alternative formalism described below it will
% but for other theories  
 describe the novel $O$ state features.
%   We don't intend in this paper to analyze Breuer 
%  theory in detail, our main aim is to develop
%   the alternative  selfmeasurement formalism  which 
%   can describe the states collapse appearance.
%    Note only, that $R_O$ states aren't the extremal 
%  points on $\cal{H}_O$ which expected for the individual states;
%  this important feature discussed  at lenght in chap.5.   
%  This assumption doesn't mean that the impression $I^O$
%    corresponds only  to  one of  $q^O_i$ eigenvalues but  that it
%  devoids of any ambiguities which $O$ quantum state can have.
% We can't define this impression - $O$ internal state here
% As the example of the simple $O$
% toy-model  can be regarded the hydrogen-like atom
% for which $O_0$ is  its ground state
% and $O_i$ are the metastable  levels excited by $s_i$, resulting so into the
% final S - $O$ entangled state.
% In our model 'internal pointer' $O_i$
% and $O$ memory  which normally differs supposed to be the same object, but
% it isn't important.
%
%   \section {Doublet Selfmeasurement Formalism}
%
%
In this formalism the state collapse 
appears in MS  'measurement from inside' performed by $O$ and
reflected in its information component $\phi^I$ (Mayburov, 2001).
% Breuer  analysis of quantum measurements
% is quite  valuable because it permits to incorporate IGUS $O$ into
% measurement chain and describe its response to the quantum S signal.
% Yet, it shows that even
%  the account of $O$ selfdescription restrictions doesn't lead to
% state collapse appearance in standard QM.
%  Hence, 
% it prompts  simultaneously  how to  modify standard QM formalism,
% so  that it would describe the state collapse consistently
%  without violation of acknowledged quantum dynamics.
%  The main idea is to exploit the possible  difference of 
%  MS subjective states
% for different observers $O,O'$ permitted  in Selfdescription Theory. 
% Therefore, one can attempt to modify QM formalism so that the state collapse
% would be observed in the 'measurement from inside'.
%
To agree with the
 quantum Schr$\rm\ddot{o}$dinger dynamics  (SD),
 the particular formalism should satisfy to two operational conditions : \\
i) if an arbitrary system $S'$   doesn't interact with IGUS $O^S$,
 then for $O^S$ this  system evolves according to
 Schr$\rm\ddot{o}$dinger-Liouville  equation  (SLE)  \\
%          (for example GRW theory  brokes this condition (Girardi,1987).\\
ii) If $S'$ interacts with $O^S$ and the entangled $S'$,$O^S$ state produced
 i.e. measurement occurs, then  SD can be violated for $O^S$ but   
 for external, stand-by $O'$  the    $S',O^S$  evolution should be
 described by SLE as follows from  condition i).\\
% iii) if input S state approximates to the classical scale.
% than for $O$ and $O'$   the classical limit i.e.
%  objective measurement restored which for $O,O'$ is equivalent. 
% If  both this conditions satisfied
%  it means that in such formalism MS  final states
%  relative to $O$ and $O'$ can be unitarily nonequivalent $\cite {Ume}$.
%  If in addition SD violated  for $O$
%  in such way that for MS and  the pure incoming S state
%  $\psi_s$  of (\ref {AAB}) $O$ percepts random events
%   this phenomena will be called  the weak (subjective) collapse.
% To explain our main idea,
% consider  first  its simplified version in which some evolution
% features omitted. 
%              the  initial state (\ref {AAB}).
     Below it will be argued that
this doublet state formalism (DSF)  corresponds to the measurements
description in Algebraic QM framework.

For the novel MS state:   
 $\Phi=|\phi^D, \phi^I\gg$ 
the dynamical component  $\phi^D$ is also
equal to QM density matrix  $\phi^D=\rho$ and obeys  to   SLE  :
\begin {equation}
    \frac {\partial \phi^D}{\partial  t} =[\phi^D,\hat {H}]  \label {AA8}
\end {equation}
  and the initial $\phi^D$ of (\ref{AAB}) evolves   
 at $t>t_1$ to $\phi^D(t)=\rho^p_{MS}$ of  (\ref{AA2}).
% the initial $\phi^D(t_0)=\rho^p_{MS}(t_0)$ corresponds to $\Psi^{in}_{MS}$
% of (\ref{AAB});
%  SLE solution for MS at $t>t_1$ is $\phi^D=\rho^p_{MS}$ of  (\ref{AA2}).
    $O$ information component  $\phi^I$ differs principally
from Breuer theory because it behaves stochastically
in the individual events. 
% is the initial $O$ subjective information and
  Namely,  for $t \le t_0$ the initial
 $\phi^I=|O_0\rangle \langle O_0|$  - $O$ has
 no information on S at $t_0$. For
    the  final  $\phi^I(t)$ at $t \ge t_1$
   after the measurement
 at $t>t_1$  $\phi^I$ is the stochastic
state $\phi^I(n)=\phi^I_i$, where
$\phi^I_i=|O_i\rangle \langle O_i|$,
 with $i(n)$ described by  the probabilistic distribution with $P_i=|a_i|^2$.
  Therefore, such doublet, individual state $\Phi(n)$ 
can change from event to event  and $\phi^I$
is partly independent of $\phi^D$ being correlated with it only
 statistically.
 DSF $O$ subjective states $\phi^I$  can't  differ
the pure and mixed states with the same $|a_i|^2$. Therefore, Breuer
theorem conditions fulfilled and  the subjective state
collapse observed by $O$. 
  MS ensemble evolution described
 via the doublet statistical states  
$  |\Theta\gg=|\eta_D,\eta_I\gg  $,
where $\eta_D=\phi^D$, $\eta_I(t)$ describes 
the probabilistic distribution $\{P_i(t)\}$  of  $O$  $\phi^I_i$ observations
at given $t$.
% $\eta^I$ can be written as the density matrix; for example for MS at
% $t>t_1$, $\eta_I=\sum |a_i|^2 |O_i\rangle \langle O_i|$.
%  Such simplified theory version  describes MS state $\Phi$
%   at $t<t_0$ and $t>t_1$ ;
%  the detailed $\Phi$ evolution ansatz  given below. 
%   An initial state component
%   $\phi_D=\rho(t_0)$  defined also by standard QM rules.
%   Thus before the measurement starts
%   $O$ state vector is $|O_0\rangle$ (no information on S) and
%  the  doublet state is $\Phi=|\rho(t_0), V_0^O \gg $ where
%   $V_0^O=|O_0\rangle \langle O_0|$ is  the initial $O$ information.
%
Thereon, $\eta_I(t)$ defined by $\eta_D(t)$ which  obeys to SLE.
 Due to it,
 $\Theta$ evolution is reversible and the acquired $O$ information
can be erased completely. 
Naturally, 
the quantum  states for external $O'$ (and other observers)  also has
the same doublet form $\Phi'$. In the regarded situation
 $O'$ doesn't interact with MS  and so  $O'$ information
doesn't change after S measurement by $O$, eventually,
 for $O'$ MS evolution described by SLE only.
% because $O'$ doesn't acquire any new information during it.
% Due to it, after S measurement
%  MS,$O'$  state for $O'$ is
% $\phi^{D'}=|\Psi_{MS}\rangle |O'_0\rangle$;
% $\phi^{I'}=|O'_0\rangle \langle O'_0|$. 
%  and this situation corresponds to the weak 
%  (subjective) collapse phenomenon as it defined above.
%

%  proposed formalism satisfies to all demands for
%  the consistent  selfdescription theory.


  Witnessing Interpretation proposed by Kochen (Kochen,1985) 
  is quite close to DSF but doesn't exploits the selfdescription
effects. It
 phenomenologically supposed that for apparatus $A$ ($O$ in our
notations) some S
measured value $Q$ in pure state always has random definite value
$q_j$ relative to $A$, yet 
  no new mathematical formalism different
from standard QM  wasn't constructed for its proof (Lahti,1990).



% \section {Decoherence and Preferred Basis(PB)}

Plainly, in DSF $|O_i\rangle $ constitutes the  
 preferred basis (PB) in $\cal H_O$ and its appearance should be explained
in the consistent theory,
this problem   is well-known
 in standard QM with the Reduction Postulate (Busch,1995). 
In DSF  PB problem acquires the additional aspects related
to the information recognition by $O$. 
The plausible explanation prompts $O$ decoherence - i.e.
$O$ interaction with environment E (Zurek,1982; Guilini,1996). 
In this case the produced,  entangled S,$O$,E state  admits
 the unique,  orthogonal decomposition  which extracts $O$
 PB for the final MS states (Elby,1994).
 Tuning the interaction parameters,  PB can be made equivalent to 
$|O_i\rangle$ basis which resolves formally PB problem
(Mayburov,2002).  Despite its importance,
 for the simplicity
   the decoherence consideration and its influence on  
$O$ selfdescription omitted here. 
We plan to present this results in the forcoming paper,
 the preliminary calculations shows that the decoherence account
doesn't changes our principal conclusions
(Mayburov,2002). 


\section {   Selfmeasurement in Algebraic QM}

Now   the quantum  measurements and $O$ selfdescription
will be regarded
in  Algebraic QM framework (Bratelli,1981).
%  and derive MS restricted states
%  $C^*$ algebras  formalism \cite {Bra}.
% We'll argue that algebraic QM 
% applied to  the systems selfdescription  in fact our
% phenomenological   theory.
  Besides the standard quantum effects,
Algebraic QM    describes successfully
the phase transitions and other nonperturbative phenomena 
which  standard QM fails to incorporate (Emch,1972). 
Consequently, there are the serious premises to regard Algebraic QM
as the consistent generalization of standard QM.
%  The Operators Algebras formalism 
% for the quantum measurements with decoherence effects
%  studied  in  (Primas,1990).
Algebraic QM was applied extensively
 to the superselection model of quantum measurements
when the detector D or environment E  regarded as the infinite systems
(Pimas,1990; Guilini,1996).
The algebraic formalism of nonperturbative QFT  
 was applied also to  the study of measurement 
dynamics in some realistic systems (Mayburov,1998).
%
%   For the simplicity  we don't consider $O$,E decoherence in this section 
%   in detail but use  obtained $O$ long-living states basis as being settled  
%   $a \; priory$.
%
In standard QM the fundamental structure is the fixed states set 
- Hilbert space $\cal H$ on which an observables - Hermitian operators
defined. Yet for some systems 
 the states set structure  principally differs
 from the arbitrary $\cal{H}$ and the standard QM
axiomatics becomes preposterous. 
In distinction, in Algebraic QM   the fundamental structure 
 is the  Segal algebra $\cal{U}$ of observables $A,B,...$
which incorporate the main properties of the studied system $S_f$
and eventually, defines
$S_f$ state set $\Omega$  (Emch,1972).
 Technically, it's more  convenient to  consider 
 $C^*$-algebra $\cal{C}$  for which $\cal{U}$ is the subset
and calculate $\cal{U}$ properties afterward.
%   Roughly speaking, $\cal{C}$ is algebra of
% complex elements for which $\cal{U}$ is the subset 
% of its real elements.
 For our problems $\cal{C}$, $\cal{U}$ are in the unambiguous correspondence 
$\cal{C} \leftrightarrow \cal{U}$
and below their use is equivalent in this sense. 
%
%  This elements in Algebraic QM
%  are the linear operators for  which the sum $A+B$ and
%   product $A \cdot B$  defined.
   $S_f$   states set $\Omega$ defined by
$S_f$ $\cal{U}$    via the notorious
 GNS  construction; it demonstrates that
  $\Omega$ is the  vector space dual to
 the corresponding $S_f$ $\cal C$ (Bratelli,1981).
 Such states
called here the algebraic states $\varphi \in \Omega$ and
are defined as the normalized, positive, 
linear functionals on $\cal {U} $:
 $\forall A \in {\cal U}$;$ \; \forall \varphi \in \Omega$ it gives
$\bar{A}=\langle \varphi;A\rangle$.
%  The pure algebraic  states $\varphi$  
%  are the  extremal $\Omega$ points.
%  Consequently, the states set $\Omega$
% has the flexible structure, due its dependence on $\cal{U}$ 
%  which  can be varied significantly depending on $S_f$ physical properties.
%   In particular, below it used for the restricted states set calculations
%   related to the incomplete $S_f$ observables set - $\cal U$ subalgebra. 

Here only unitarily equivalent $S_f$ will be regarded;
 for  them $S_f$ $\varphi$ formally
corresponds   to QM density matrixes $\rho$ (Segal,1947).
% ; yet for such $S_f$
%  the regarded Algebraic formalism leads to some novel effects
%  in comparison the standard QM (Segal,1947).
%   and  by the slight abuse
%  of mathematical precision one can take  $\varphi=\rho$.
% It doesn't mean
% that $\varphi$ always are equivalent to $S_f$ statistical states $\rho$;
%     they are  equivalent when $\varphi$ is pure state;
%  for the discussion see (Primas,1983).
%   For us their possible difference isn't
% essential  because  for our measurement formalism we need only
% $\bar{A}$ values which in any case are the same.
%       Due to it, throught our discussion
%       we don't discriminate   the algebraic states $\varphi$
%       and the statistical states, keeping in mind their possible difference. 
%
% For the  measurement theory 
%  $S_f$ individual  states  are of primary importance;  
The algebraic pure states are    $\Omega$  extremal points and they
  regarded as the algebraic individual states
 (AIS) $\xi$; their set denoted $\Omega^p$  (Emch,1972; Primas,1990).
%
%   Each  AIS possess the arbitrary objective property (eigenvalue)
%   $q^A_j$  associated with the projector   $P^A_j$.
%    AIS $\xi$  aren't $a\; priory$ equivalent to QM individual states.  
The arbitrary $\varphi$ doesn't admit the  unambiguous
decomposition into AIS $\xi_i$ ensemble,
except the  situation when  $\varphi$ is pure; in this
case   $\xi=\varphi$.
%   AIS are equivalent to
%   QM individual states for the finite dimensional systems,
%  n particular, for MS $\xi^{MS} corresponds to \Psi_{MS}$.
 The algebraic mixed states $\varphi_{mix}$
can be constructed as $\xi_i$ ensembles;
  the   ensemble states $W_A$
  defined analogously to  the  described  QM ansatz.

 

% To simplify our formalism
% we neglect E presence and decoherence
%  in this model assuming that $O$  states basis defined phenomenologically.

  In many practical situations 
 only some restricted 
   linear subspace $\cal M_R$  or  subalgebra
 $\cal{U}_R$ of $S_f$ observables  algebra $\cal U$
is available for the observation.
%         It was shown that for  $\cal {U}_R$
 For such subsystems the restricted  algebraic states $\varphi_R$
 can be defined consistently
%   \cite {Em}. Alike   $S_f$ complete states the restricted  $\varphi_R$
 via $A_R \in \cal{U}_R$ expectation values :
\begin {equation}
    \bar{A}_R=\langle \varphi;A_R\rangle=\langle \varphi_R;A_R\rangle
  \label {CC12}
\end {equation}
%                   which coincide for $\Omega, \Omega_R$ states.
defining $\varphi \rightarrow \varphi_R$ restrictions;
their set denoted $\Omega_R$.
 $\varphi_R$ doesn't depend on any $A' \notin \cal{U}_R$,
therefore, $\forall \varphi_R,\,\langle \varphi_R;A'\rangle=0$ (Emch,1972).
For our MS only $O$ observables supposedly are available
for the observation (perception) and that makes the subalgebras
studies important for us.
%
%
Remind that  any classical system $S^c$  can be
described by some associative Segal algebra $\cal {U}^C$
 of $S^c$ observables $\{A\}$ (Emch,1972);
 in algebraic QM $\cal U$ associativity
corresponds to QM observables commutativity.
The theorem by Segal   proves that 
%    for all $A_j\in \cal{U}'$
%   commute and unit operator $I\in \cal {U}'$, i.e.
any  associative Segal  (sub)algebra $\cal{U}'$
is isomorphic to some algebra $\cal {U}^C$ of classical  
observables (Segal,1947); thereon, its  $\varphi^a$ states set
  $\Omega^a$ is isomorphic to
  the set $\Omega ^c$ of the  classical statistical states $\varphi^c$.
%   The corresponding AIS properties stipulated by  Segal theorem
%   can be formulated in our terms as follows :
% for the associative  (sub)algebra $\cal{U}'$
% its states set $\Omega'$ has the extremal points
The corresponding AIS   - i.e. the pure states   
corresponds to the classical,
individual  states $\xi^c_i$ - points in $S^c$ parameters space.
%  with the  observables eigenvalues $q^A_i$ (Emch,1972).
For us the most important is the  case when
  $\cal{U}'$ includes only $I$ and single $A \ne I$; 
   there $\xi^c_i=\delta(q^A-q^A_i)$,  corresponding to 
 $A$ eigenvalues $q^A_i$ spectra.
 Consequently, even if quantum $S_f$ described by nonassociative $\cal{U}$,
 it contains the subalgebra $\cal{U}'\in U$
 ( and may be not unique)  for which
 the restricted AIS $\xi^c_i$ are classical with  the objective properties
$q^A_i$.
%  Therefore, Algebraic QM admits the multiplet quantum states
% structure analogous to the regarded DSF $\Phi$; each member of multiplet is 
% the algebraic state $\varphi^j_R$ corresponding to the particular subalgebra
% $\cal{U}^J_{R}$ $ \in \cal {U}$.
%
%     responding to $P^A_i$ projectors.
%  The importance of this results for our problem is obvious; it 
% explains how classicality can appear in the restricted quantum system
% arising in the measurements.
% Restricted subalgebras $\cal{U}_R$  were applied already
% for the description of the information restrictions
%  stipulated by  the practical impossibility of 
%  arising  in the large, collective  systems measurements (Emch,1972); 
%  here their application  extended to the restrictions induced 
%   by IGUS $O$ selfdescription incompleteness.


%   are AIS analog in most aspects; each of them possess some definite
%    property $P^A_i \in \cal {U}_R$, and is $\Omega_R$ extremal point. 


%  The advantage of Operators Algebra is that  in the same formalism
%  the arbitrary system $S_f$ can be  described
% both as  classical and quantum one (Emch, 1972).
%  Transfering to quantum $S_f$ properties   $\cal{U}^C$ mapped
%  to the corresponding   algebra $\cal{U}$ of $S_f$ quantum observables.
% In this case,  the observables number is larger, due to the appearance of
%  interference terms and other specifically quantum observables.

For the classical observing system $S^c_T$ described by some $\cal{U}^C$
its  selfmeasurement  $O$ restrictions are easy to find -
$O$ state depends only on $S^c_T$ coordinates $\{x^O_j\}$
 which are $O$ internal
coordinates (Breuer,1996). 
They constitute $\cal{U}^C_R$ subagebra of $\cal{U}^C$ but the realistic
 $O$ effective subalgebra $\cal{U}^C_O\in \cal {U}^C_R $  
 can be even smaller because
some $x^O_j$ can be uninvolved directly into the measurement process.  
%  Breuer theorem leaves the quantum $O$ restricted states 
%  ansatz  partly ambiguous, if one applies 
QM  Correspondence principle  prompts that for the transition to
 the quantum case  $S^c_T \rightarrow S_T$
 $O$ restricted subalgebra $\cal{U}_R$ also
includes only $O$ internal observables. In quantum case
any effective subalgebra
 $\cal{U}_O \in \cal{U}_R$ stipulating
   the restricted states  sets $\Omega_R,\Omega_O$, correspondingly.   
 Our main hypothesis is
 that in any individual event to the arbitrary  $S_T$ AIS $\xi$ 
  responds  some  restricted $O$ AIS $\xi^O_j$. It 
 advocated below for $\cal U_O$,
for $\cal U_R$ it accepted $ad\;hoc$.
% the analogous assumption  used in Breuer theory.
%  but in Algebraic QM
%   it leads to some new features discussed below in detail.
%  It means that, if $S_T$ state described by AIS for some external $O'$,
%  it described also by AIS for $O$ , i.e. 'from inside'. 

%   From $\cal{C}\rightarrow \cal{U}$ relation it results in
%   $\cal{U}_R$ $\cal{U}_O$  $O$ observables subalgebras.

% Alike to Classical Mechanics  each such state $\varphi^R$
% corresponds to one of $A^1,..,A^n \in \cal{U}_R$
%  eigenvalues set $q^1_{j},...,q^n_l$.
%  Statistical state for systems ensemble $\varphi^E_R$ defined
%  according to Algebraic QM rules \cite {Em}.

  MS  described by
 $ \cal{U}$ Segal algebra  for MS  observables  which defines
  $\varphi^{MS} \in\Omega$ properties.
 $O$  subalgebra is $\cal{U}_R$ which includes all $O$ 
internal observables. Then, $\varphi^R \in\Omega_R$
 is equivalent to $O$ QM statistical states $\rho$ set.
 Consequently, 
$O$ AIS set $\Omega^p_R$ is equivalent to $\cal{H}_O$  and
 any $O$  AIS $\xi^R_i $ corresponds to some  $O$
 state vector $ |O^r_i \rangle \in \cal H_O$.
%  which seems quite natural MS$\rightarrow O$ restriction.
%       It differs from Breuer $R_O$ of (\ref {AA4})
%      which isn't the individual state on $\cal{H}_O$.
We don't study here $\Omega_R$ states further,
%   because it's not our primary aim,
note only that Breuer $O$ state (\ref {AA4}  )  $R_O \notin  \Omega^p _R$
and can't be AIS  on $\cal {U}_R$ for $a_{1,2} \ne 0$. 
%  any $\Omega_O \in \Omega_R$,
%  so any $\xi^O \in \Omega_O$ is one of $|O^r\rangle$.  
%  because the effective $\cal{U}_O$ which is practically important for us
%  leads to the  different solutions.  
%  it should have the correspondence with the effective $\cal U_O$.
% Thereon, for MS algebraic states
% for the restricted states $\varphi^O$
%
To define   $\cal U_O$,  let's consider
  $\varphi^{MS} \rightarrow \varphi^O$ restriction properties.
 Remind that
  for the regarded MS dynamics of  (\ref {AA2}) $O$ 
 can measure only the  observable $Q_O$, 
%   illustrated by $\Psi_{MS}$ of.
for any other $Q'_O \neq F(Q_O)$  the final $\bar{Q}'_O=0$;
 it means
 $ \forall \varphi^O; \, \langle  Q'_O ; \varphi^O\rangle=0$
for $O$  restricted, algebraic states.
%  As was shown,
%  $O$,E decoherence dynamics  in MS+E model also results in the 
%  same  $\bar {Q'}_O=0$ values for the arbitrary $\rho_{MS}$.
%   For all $Q'_O$ aren't involved directly in 
%   S measurement   from (\ref {CC12}) it follows that
%   $  \forall \varphi^{MS}; \varphi^{MS} \rightarrow
%
%  which means both for MS and MS+E models that
%  $Q'_O \notin \cal{U}_O$  (Segal,1947).
%
%  Formally, if $Q'_O \in \cal{U}_O$ and
%  $\forall \varphi^O \in \Omega_O;\, \langle Q'_O, \varphi^O \rangle=0$
%  it means $Q'_O=0$, so $Q'_O \notin \cal{U}_O$ (Segal,1947).
%
From that follows that 
$\cal{U}_O \in \cal {U}_R$  effective  $O$  subalgebra
 includes only  $Q_O$ and $I$.
Really, only in this case 
 $ \forall \varphi' \in \Omega_O; \, \langle  Q'_O ; \varphi'\rangle=0$;
 each $\varphi^O$ corresponds to $\varphi'$ with the same $\bar{Q}_O$
and vice versa.
 Therefore,  $\varphi^O$ set $\Delta_O$ is isomorphic to $\Omega_O$.
 There is no other $\cal U_R$ subalgebras with such properties and
  that settles $\cal U_O $ finally. Therefore,
obtained $\varphi^O$ are  equivalent to $R_O$,  in agreement with 
  MS the statistical states $\rho_{MS}$ restriction
to $O$ which are equal to $R_O$ of (\ref{AA4}) as was shown above.
% Therefore, we can conclude that both MS and MS+E models leads to the same
% effective $\cal {U}_O$, yet the decoherence MS+E dynamics 
%  results in unambigous PB definition which makes the theory better founded.
%
From Segal theorem  for  $\cal{}U_O$  
 the restricted algebraic $O$ 
 states $\varphi^O\in \Omega_O$   are isomorphic to classical,
probabilistic $q^O_i$ distributions,
%  To establish MS, $O$ AIS relations, it's necessary to analyze  the
 $O$ AIS $\xi^O$
 are  isomorphic to the classical, pointlike  states:
$$
            \xi^O_i= \delta(q^O-q^O_i)
$$
for $Q_O$ eigenvalues.  
%  can be derived  directly from Segal Theorem  but $\cal{U}_O$ 
% is quite simple and  to illustrate its properties, it's instructive to
% start   from its semiqualitative consideration. 
%
  For  the incoming S state
$\psi_s=|s_i\rangle$  results in 
 $\Psi^{MS}_i=|s_i\rangle|O_i\rangle$ which are $\Omega$ extremal points,
% algebraic MS state 
% $\varphi =|S_i\rangle \langle S_i| |O_i\rangle\langle O_i|$.
% then as was proved above
%  such MS  restricted state  associated with our 
%  $O$  state $\varphi^O_i \sim |O_i\rangle\langle O_i|$
 $O$ restricted states $\varphi^O_i =  |O_i\rangle\langle O_i|$
  are $\Omega_O$ extremal point  and   AIS $\xi^O_i=\varphi^O_i$.
%  From (\ref {CC12}) it defines  $Q_O$   spectra $q^O_i$ on $\cal U$.
% Alike QM individual states $|\psi_l\rangle$, $\{\xi^O_i\}$ can be interpreted
% as $O$ subjective states in the individual events. 
In any $Q$  eigenstate $|s_i\rangle$   measurement 
 the final MS restricted
 state  from  $O$ 'point of view' describes the definite 
$Q_O$ value   $q^O_i$ which establishes
operationally $\xi^O_{i,j}$ distinction in the individual events.
% established operationally by the detection (perception)
% of $q^O_{i,j}$ difference.
% Therefore, we can interpret $\xi^O_i$ as $O$ individual states.  
%   the individual events.
% to which corresponds $\phi^I$ of our phenomenological theory.

  For the incoming S mixture with the $|s_i\rangle$ probabilities $|a_i|^2$
 MS algebraic final state is
  $\varphi_{mix}=\rho^m_{MS}$ of  (\ref {AA3}); 
%
%     the incoming S mixture  of $|s_i\rangle$ eigenstates
%  the final statistical MS state described for $O$ as the mixture
%        $\rho^m_O=\sum |a_i|^2 \varphi^O_i$
%    which defined by the single $\cal{U}_O$ independent observable
%
 the corresponding $O$ restricted state  $\varphi^O_{mix}$
 defined from the relation for $\bar{Q}_O$ :
$$
    \bar{Q}_O=\langle\varphi^O_{mix} ;Q_O\rangle=
    \langle\varphi_{mix};Q_O\rangle=\sum |a_i|^2 q^O_i
$$
which results in the solution  $\varphi^O_{mix}=\sum  |a_i|^2\varphi^O_i$.
From the regarded correspondence
of MS $\xi^{MS}$ and $O$ AIS $\varphi^O_{mix}$
 represents the stochastic mixture of AIS $\xi^O_i$ 
 described by $O$  ensemble state 
  $W^O_{mix}=\{ \xi^O_i;\,P_i=|a_i|^2;\,i=1,2\}$. If  the incoming  S state  
 is $Q$ eigenstates superposition
$\Psi_{MS}^{in}$ of (\ref{AAB}),  MS final algebraic state
$\varphi^{MS}$ with the same $|a_i|^2$ results in  the same $\bar{Q}_O$ value.
Therefore, its  restricted algebraic state
coincides with the mixed one $\varphi^O=\varphi^O_{mix}$.
%    Note that, $\Psi_{MS}$ is MS individual QM state to which correponds
%     AIS $\xi^{MS}$.
%  From that, we can expect that for  $O$ ensemble states
%  $W^O=W^O_{mix}$ also, i.e. $W^O$ is also $\xi^O_i$ probabilistic mixture. 
% which describes the maximal $O$ information.
% and restricted statistical MS state is :$R^s_O=\sum |a_i|^2\varphi_i^O$.
%  that no $\varphi^O_i$ state corresponds to such $\bar{Q}_O$.
%  Their equality follows  
From   Segal theorem 
%   Really,  from (\ref CC{12}) it follows that
%   $  \forall \varphi^{MS}; \varphi^{MS} \rightarrow \varphi^O; \,
%    \langle  Q'_O ; \varphi^O\langle=0$ 
 in $\Omega^p_O$ all $O$  individual states are AIS $\xi^O_i$,  
possesing the definite properties $q^O_i$.
%   with the projectors $P^O_i$.
 There is no
%   other projectors  in $\cal{U}_O$ and  no
other individual states $\xi^O_a \ne \xi^O_{i}$, consequently,
 MS restricted AIS  in each event can be only one of $\xi^O_i$.
 Eventually,
 if MS state is $\Psi^{MS}_j$ superposition of (\ref {AA2}),
 to dispatch the correct $\bar{Q}_O$ for $\xi^O$ ensemble,
$\xi^O_i$ should appear at random  with the probabilistic distribution $P'_i$
defined by $\bar{Q}^O$.
Yet for such $W^O$ content the only solution which results in the
necessary $\bar{Q}_O$ value is  $P'_i=|a_i|^2$ and so
 $W^O=W^O_{mix}$.
 It demonstrates
  that $\xi^{MS}\rightarrow \xi^O$ restriction map is stochastic. 

% To advocate the obtained picture,
%remind that
In general, any two physically different states operationally
discriminated by the particular
 observation procedure which reveals this states
difference via the  difference of some observables values
distributions.
For the statistical states it demonstrated by their probabilistic
distributions parameters, for the individual states $\xi_{a,b}$ 
such difference can be extracted from some observables $A,B$
 eigenvalues $q^{A,B}_{i,j}$ for which this states are the eigenstates.
 In that case   this values 
 can be obtained and compared in the single event per each state
(Mittelstaedt,1998). 
%      which means  that  $\xi^O_i$ covers all $Q_O$ spectra. 
In our case the only $\cal{U}_O$ observable is $Q_O=\sum q^O_i P^O_i$ and
$\xi^O_{i,j}$ difference reflected by $q^O_{i,j}$ difference. 
 If to assume that some other $\xi^O_a \ne \xi^O_i$ exists,
it needs also some other observable $Q^e \ne F(Q^O);\, Q^e$ should
belong to  $ \cal{U}_O$ to differ it from $\xi^O_i$, 
but it's inconsistent with the obtained $\cal{U}_O$ structure.
 In particular, Breuer restricted state $R_O$ of (\ref {AA4}) analog
$\xi^O_R=\sum |a_i|^2 \xi^O_i$  for $a_i \ne 0$
can't be $O$ individual state on $\cal{U}_O$ because it
  isn't $\Omega_O$ extremal point.
%   The  physical reasons of it explained in chap. 2 :
%   $R_O$ is $O$ partial state for external $O'$ but not for $O$ himself. 
%       Note also that $R_O=\varphi^O_{mix}$ for AIS  at statistical level. 
 Moreover, the arbitrary $\varphi^O$ admits the unique
decomposition into $\xi^O_i$ set and  can be interpreted as $\xi^O_i$ 
ensemble  with the given probabilities.
%
%  The evolution eq. (\ref {CC}) are applicable also
%  for $\{\varphi ,\varphi^O\}$ doublet.
  Since $\xi^O_i \in \Omega _R$, 
it can be taken  also as the  possible ansatz for MS  states  restriction on 
   $\cal {U}_R$.
DSF doublet state $\Phi$ components $\phi^D,\phi^I$
 are equivalent to $\xi^{MS}$, $\xi^O$ correspondingly.
%  $O$ impressions $I^O(n)=q^O_i$ are unambiguous in each event. 

If to analyze this results  from the Information-Theoretical premises,
 note that
 the difference between the pure and
mixed MS states reflected by $B$ IT of (\ref {AA5}) expectation values.
Therefore, $O$ possible observation of S pure/mixed $W^O$
 states difference means
that $O$ can acquire the information on   $B$  expectation value.
 But $B \notin \cal {U}_R$ and isn't correlated with $Q_O$ via
S,$O$ interaction alike $Q$ of S; so this assumption  is prepostereous.
%  Briefly, for MS state with $\bar{B}=0$
%  $W^O$ was defined as function of  $\bar{Q}_O$.  
%    $W_c^O=\{ \xi^O_{1,2};\quad P_{1,2}=|a_{1,2}|^2\}$.
%   Because $B\notin \cal{U}_O$,  it follows that the same AIS distribution
%  $W^O(\bar{Q}_O)=W^O_{mix}$ persists for arbitrary $\bar{B}$.
%
%  
% Summing up all the given arguments, if the restricted $O$ exists
% and percpted by $O$ in each event, then
% for the pure incoming  $\psi_s$
% $O$ observes stochastic $q^O_i$ distribution with 
% probabilities $|a_i|^2$ described by $W^O$ statistical mixture
%  $\varphi^O=\sum |a_i|^2\xi_i^O$ and  can't be discriminated
% from incoming S mixture with the same $P_i=|a_i|^2$.
% This randomness of the observed $O$ states can be related to the fact
 Note that MS individual states $\xi^{MS}$ symmetry
 is larger than the symmetry   of the restricted $O$ states.
%  In particular, the
% states $\xi_i^O$ doesn't possess the symmetry of the incoming measured state
% $\Psi^{in}_{MS}$.
In Algebraic QM such symmetry reduction results in
 the phenomena of Spontaneous Symmetry Breaking, by the
analogy the  discussed  randomness apearence can be
called Information Symmetry Breaking.
%  Note that Selfrefence problem
% avoided in this case by use of the natural 
% assumption that all observers are similar in relation to
% their information acquisition properties (Svozil,1993).
%  In fact, Algebraic QM supports
%  quite reasonable measurement picture : any $O$
%  can observe only the object parameters for which it possess the suitable
%  'internal' instrument. For example, $O$ can't observe $|O_i \rangle$  
%  superpositions because $O$ detecting structure doesn't permit
%  IT observable $B_O$ to be measured simultaneously with $Q_O$.
%   because first of all they doesn't commute.
%   Such considerations
%   were already regarded above in DSF phenomenolgical picture,
%   but in Algebraic QM they acquire additional support.
%   In our model it supposed that
% the only available for $O$ observable is $Q_O$ leading to the 
% simplest associative MS subalgebra $\cal{U}_O$.
%  The algebraic calculations performed here for the very simple
%  IGUS $O$ model which  is more primitive then any realistic
%   IGUS structure.
 In practice
it's possible that $O$ effective subalgebra is larger then $\cal U_O$
 but this case
will demand more complicated calculations which we plan to present
in the forcoming papers.
 In Algebraic QM 
the only important condition for the classicality appearance
is $\cal{U}_O$ observables commutativity and it's reasonable to expect
 it to be feasible also for complex IGUS structures.

%  Yet because for any $Q'_O \ne F(Q_O)$ 
% $\bar{Q}'_O$ doesn't depend on MS state, we can't expect any
%  principally new features.



Despite  of the acknowledged  Algebraic QM achievements,
 its foundations are still
discussed and aren't finally established. In particular, 
it's still unclear whether all the algebraic states corresponds to the
physical states (Primas,1983). This questions are  important by themself 
and  are essential for our formalism feasibility. By our choice
of the initial MS states we avoid it in the regarded model.
In particular, we admitted  without proof
 that for MS AIS $\xi^{MS}$  some   $O$ restricted AIS responds
in any event. It agrees with the restricted states consideration 
 as the real physical states, 
% 'the states in their own right' defined on $O$ subalgebra $\cal{U}_O$
 but on the whole, this assumption needs further clarification.



% \section { Discussion}


% The present report  argues    that the account of the information

%In this paper the
% system quantum properties permit to explain the quantum state collapse
% as the result of  the principal 
For the conclusion, the
 information-theoretical restrictions on the quantum measurements
were  studied  on the simple selfdescription  model of IGUS $O$.
Breuer selmeasurement study shows that
by itself  the $O$ inclusion  as the quantum object
into the measurement scheme doesn't result in the
 state collapse appearance (Breuer,1996).
Our considerations indicates that to describe the
 state collapse and in the same time to conserve
Schr$\it\ddot{o}$dinger linear evolution, it's necessary 
 to extend the quantum states set over standard QM Hilbert space.
Such modification proposed in DSF involving the doublet states $\Phi$,
where one of its componenets $\phi^I$ corresponds to $O$
 subjective information - i.e. $O$ selfdescription.
  Algebraic QM  
presents the additional arguments in favour of this approach,
 in its   formalism $O$ structure
  described by $O$ observables algebra $\cal {U}_O$
which defines the multiplet states set analogous to $\Phi$. 
In Algebraic formalism the stochastic events appearance
 stipulated by  MS individual states restriction to $O$.
% We present here the simple measurement theory and we don't regard
% it as the final solution of measurement problem. 
In our opinion  the obtained results evidence
 that it's impossible to solve  the Measurement Problem
 without accounting of the information system 
$O$ interactions at quantum level and its information
acquisition restrictions (Zurek,1998).
%    This approach to the measurement problem  has much i
%   common with Quantum reference frames theory  by Aharonov  $\cite {Aha3}$
\\
\\
\qquad \qquad \qquad {\Large { References}}\\
\\
%\end {sloppypar}
%\begin {footnotesize} 
%\begin {thebibliography}{99}
% \bibitem {Aha2}
 (1981) Y.Aharonov, D.Z. Albert Phys. Rev. D24, 359 
%\\
\\
%\bibitem {Be}
 (2000) G.Bene, quant-ph 0008128
%\\
\\
% \bibitem {Bra}
 (1979) O.Bratteli, D.Robinson 'Operators Algebra and
Quantum Statistical Mechanics' (Springer-Verlag, Berlin)
%\\
\\
% \bibitem {Bre}
 (1996) T.Breuer, Phyl. of Science 62, 197 (1995),
 Synthese 107, 1 (1996)
%\\
\\
%\bibitem {Busch}
 (1996) P.Busch, P.Lahti, P.Mittelstaedt,
'Quantum Theory of Measurements' (Springer-Verlag, Berlin,1996)
%\\
\\
%\bibitem {Desp}
 (1990) W. D'Espagnat, Found Phys. 20,1157,(1990)
%\\
\\
% \bibitem {De}
%  (1985) D.Deutsch, Int J. Theor. Phys. 24, 1 
%
% \bibitem {Du}
%\\
% \\
 (2002) R.Duvenhage, Found. Phys. 32, 1799  
%\\
\\
%\bibitem {Elb}
 (1994) A.Elby, J.Bub Phys. Rev. A49, 4213 
%\\
\\
%\bibitem {Em}
 (1972) G.Emch, 'Algebraic Methods in Statistical Physics and
Quantim Mechanics',\\
 (Wiley,N-Y) 
%\\
\\
%  \\bibitem {Fin}
 (1988) D.Finkelstein, 'The Univrsal Turing Machine:
 A Half Century Survey', (ed. R.Herken, University Press, Oxford) 
%\\
\\
% \bibitem {Gir}
%  (1986) GC. Girardi, A.Rimini, T.Weber Phys.Rev. D34, 470  
% \\
% \\
% \bibitem {Gui}
 (1996) D.Guilini et al., 'Decoherence and Appearance of
Classical World', (Springer-Verlag,Berlin) 
%\\
\\
%  \bibitem {Alb} D.Z.Albert, Phyl. of Science 54, 577 (1986)
%  \bibitem {Koch}
 (1985) S.Kochen 'Symposium on Foundations of Modern Physics'
  , (World scientific, Singapour)
%\\
\\
%  \bibitem {Lah}
 (1990) P. Lahti Int. J. Theor. Phys. 29, 339 
%\\
\\
% \bibitem {Lon}
 (1939)  London F., Bauer E. La theorie de l'Observation
 (Hermann, Paris)   
%\\
\\
% \bibitem {May2}
 (1998) S.Mayburov, Int. Journ. Theor. Phys. 37, 401 
%\\
\\
%  \bibitem {mm0}
 (2001) S.Mayburov  Proc. V  QMCC Conference, Capri, 2000,
(Kluwer, N-Y); $ \setminus$quant-ph 0103161
%\\
\\
%  \bibitem {mm}
 (2002) S.Mayburov Proc. of Vth Quantum Structures conference,
Cesenatico, 2002; Quant-ph 0205024; Quant-ph 0212099
%\\
\\
% \bibitem {Mitt}
 (1998) P.Mittelstaedt 'Interpretation of
Quantum Mechanics and Quantum Measurement\\ Problem',
(Oxford Press, Oxford)
%\\
%\\ 
% \bibitem {Nam}
%  (1992) M.Namiki, S.Pascazio, Found. Phys. 22, 451
%  \\
\\
% \bibitem {Ne}
 (1932) J. von Neuman 'Matematische Grunlanden
 der Quantenmechaniks' , (Berlin)
%\\
\\
% \bibitem {Pen} (1994) R.Penrose, 'Shadows of Mind' (Oxford Press) 
% \bibitem {Pri2}
 (1983) H.Primas,  'Quantum Mechanics,
 Chemistry and Reductionism' (Springer, Berlin)
%\\
%\\
%  \bibitem {Pri}
 (1990) H.Primas,  in  'Sixty two years of uncertainty'
,ed. E.Muller, (Plenum, N-Y)
%\\
\\
%  \bibitem {Rov}
 (1995) C. Rovelli, Int. Journ. Theor. Phys. 35, 1637; 
quant-ph 9609002  
%\\
\\
% \bibitem {Seg}
 (1947) I.Segal, Ann. Math., 48, 930    
\\
%\\
%  \bibitem {Svo}
 (1993) K.Svozil 'Randomness and undecidability in Physics',
(World Scientific, Singapour)
%   \bibitem {Ume} H.Umezawa,H.Matsumoto, M.Tachiki, 'Thermofield 
%  Dynamics and Condensed States' (North-Holland,Amsterdam,1982)
% \bibitem {Fuk} R. Fukuda, Phys. Rev. A ,35,8 (1987)
%\bibitem {Mayb5} S.Mayburov Quant-ph/0103161
% \bibitem {Jan} Jansson B. 'Random Number Generators' (Stokholm, 1966)
% \bibitem {May3} S.Mayburov Quant-ph/9911105 
%\bibitem {Don} M.Donald, Found. Phys 25, 529 (1995)
% \bibitem {Scu}
%\\
\\
%  (1982) M.Scully, K.Druhl Phys. Rev. A25, 2208  
%
%\bibitem {Shum} B. Schumaher , Phys. Rev. A51, 2738 (1995)
% \bibitem {Vaid} (1996) L.Vaidman, quant-ph 9609006 
% \bibitem {Hep} K.Hepp, Helv. Phys. Acta 45 , 237 (1972)
%\bibitem {Sna} S.Snauder Found., Phys. 23, 1553 (1993) 
% \bibitem {Wig}
 (1961) E.Wigner,  'Scientist speculates',(Heinemann, London)
\\
% \bibitem {Zur}
 (1982) W.Zurek, Phys Rev, D26,1862 
\\
%  \bibitem {Zur2}
 (1998) W.Zurek Phys. Scripta , T76 , 186 
% \bibitem {Aha3} Y.Aharonov, T.Kaufherr Phys. Rev. D30, 368 (1984)
 % \bibitem {Whi} A.Whitaker, J. Phys., A18 , 253 (1985)
\\
\\
%\bibitem {Bel} J.S.Bell, Helv. Phys. Acta 48, 93 (1975)
% \end {thebibliography}
\end {document}


quant-ph0402044.extracted_bib
\begin {thebibliography}{99}
% \bibitem {Aha2}
 (1981) Y.Aharonov, D.Z. Albert Phys. Rev. D24, 359 
%\\
\\
%\bibitem {Be}
 (2000) G.Bene, quant-ph 0008128
%\\
\\
% \bibitem {Bra}
 (1979) O.Bratteli, D.Robinson 'Operators Algebra and
Quantum Statistical Mechanics' (Springer-Verlag, Berlin)
%\\
\\
% \bibitem {Bre}
 (1996) T.Breuer, Phyl. of Science 62, 197 (1995),
 Synthese 107, 1 (1996)
%\\
\\
%\bibitem {Busch}
 (1996) P.Busch, P.Lahti, P.Mittelstaedt,
'Quantum Theory of Measurements' (Springer-Verlag, Berlin,1996)
%\\
\\
%\bibitem {Desp}
 (1990) W. D'Espagnat, Found Phys. 20,1157,(1990)
%\\
\\
% \bibitem {De}
%  (1985) D.Deutsch, Int J. Theor. Phys. 24, 1 
%
% \bibitem {Du}
%\\
% \\
 (2002) R.Duvenhage, Found. Phys. 32, 1799  
%\\
\\
%\bibitem {Elb}
 (1994) A.Elby, J.Bub Phys. Rev. A49, 4213 
%\\
\\
%\bibitem {Em}
 (1972) G.Emch, 'Algebraic Methods in Statistical Physics and
Quantim Mechanics',\\
 (Wiley,N-Y) 
%\\
\\
%  \\bibitem {Fin}
 (1988) D.Finkelstein, 'The Univrsal Turing Machine:
 A Half Century Survey', (ed. R.Herken, University Press, Oxford) 
%\\
\\
% \bibitem {Gir}
%  (1986) GC. Girardi, A.Rimini, T.Weber Phys.Rev. D34, 470  
% \\
% \\
% \bibitem {Gui}
 (1996) D.Guilini et al., 'Decoherence and Appearance of
Classical World', (Springer-Verlag,Berlin) 
%\\
\\
%  \bibitem {Alb} D.Z.Albert, Phyl. of Science 54, 577 (1986)
%  \bibitem {Koch}
 (1985) S.Kochen 'Symposium on Foundations of Modern Physics'
  , (World scientific, Singapour)
%\\
\\
%  \bibitem {Lah}
 (1990) P. Lahti Int. J. Theor. Phys. 29, 339 
%\\
\\
% \bibitem {Lon}
 (1939)  London F., Bauer E. La theorie de l'Observation
 (Hermann, Paris)   
%\\
\\
% \bibitem {May2}
 (1998) S.Mayburov, Int. Journ. Theor. Phys. 37, 401 
%\\
\\
%  \bibitem {mm0}
 (2001) S.Mayburov  Proc. V  QMCC Conference, Capri, 2000,
(Kluwer, N-Y); $ \setminus$quant-ph 0103161
%\\
\\
%  \bibitem {mm}
 (2002) S.Mayburov Proc. of Vth Quantum Structures conference,
Cesenatico, 2002; Quant-ph 0205024; Quant-ph 0212099
%\\
\\
% \bibitem {Mitt}
 (1998) P.Mittelstaedt 'Interpretation of
Quantum Mechanics and Quantum Measurement\\ Problem',
(Oxford Press, Oxford)
%\\
%\\ 
% \bibitem {Nam}
%  (1992) M.Namiki, S.Pascazio, Found. Phys. 22, 451
%  \\
\\
% \bibitem {Ne}
 (1932) J. von Neuman 'Matematische Grunlanden
 der Quantenmechaniks' , (Berlin)
%\\
\\
% \bibitem {Pen} (1994) R.Penrose, 'Shadows of Mind' (Oxford Press) 
% \bibitem {Pri2}
 (1983) H.Primas,  'Quantum Mechanics,
 Chemistry and Reductionism' (Springer, Berlin)
%\\
%\\
%  \bibitem {Pri}
 (1990) H.Primas,  in  'Sixty two years of uncertainty'
,ed. E.Muller, (Plenum, N-Y)
%\\
\\
%  \bibitem {Rov}
 (1995) C. Rovelli, Int. Journ. Theor. Phys. 35, 1637; 
quant-ph 9609002  
%\\
\\
% \bibitem {Seg}
 (1947) I.Segal, Ann. Math., 48, 930    
\\
%\\
%  \bibitem {Svo}
 (1993) K.Svozil 'Randomness and undecidability in Physics',
(World Scientific, Singapour)
%   \bibitem {Ume} H.Umezawa,H.Matsumoto, M.Tachiki, 'Thermofield 
%  Dynamics and Condensed States' (North-Holland,Amsterdam,1982)
% \bibitem {Fuk} R. Fukuda, Phys. Rev. A ,35,8 (1987)
%\bibitem {Mayb5} S.Mayburov Quant-ph/0103161
% \bibitem {Jan} Jansson B. 'Random Number Generators' (Stokholm, 1966)
% \bibitem {May3} S.Mayburov Quant-ph/9911105 
%\bibitem {Don} M.Donald, Found. Phys 25, 529 (1995)
% \bibitem {Scu}
%\\
\\
%  (1982) M.Scully, K.Druhl Phys. Rev. A25, 2208  
%
%\bibitem {Shum} B. Schumaher , Phys. Rev. A51, 2738 (1995)
% \bibitem {Vaid} (1996) L.Vaidman, quant-ph 9609006 
% \bibitem {Hep} K.Hepp, Helv. Phys. Acta 45 , 237 (1972)
%\bibitem {Sna} S.Snauder Found., Phys. 23, 1553 (1993) 
% \bibitem {Wig}
 (1961) E.Wigner,  'Scientist speculates',(Heinemann, London)
\\
% \bibitem {Zur}
 (1982) W.Zurek, Phys Rev, D26,1862 
\\
%  \bibitem {Zur2}
 (1998) W.Zurek Phys. Scripta , T76 , 186 
% \bibitem {Aha3} Y.Aharonov, T.Kaufherr Phys. Rev. D30, 368 (1984)
 % \bibitem {Whi} A.Whitaker, J. Phys., A18 , 253 (1985)
\\
\\
%\bibitem {Bel} J.S.Bell, Helv. Phys. Acta 48, 93 (1975)
% \end {thebibliography}