{ Quantum Measurements and Information \\ % Restrictions in Algebraic QM \\ }
S.N.Mayburov
Lebedev Inst. of Physics
Leninsky Prospect 53
Moscow
Russia
117924
Information-Theoretical restrictions on the systems self-description and the information acquisition are applied to Quantum Measurements Theory. % In particular G$\ddot{o}$del information incompleteness For the measurement of quantum object S by the information system $O$ such restrictions are described by restricted states $R_O$ formalism. $R_O$ ansatz can be introduced phenomenologically from the agreement 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 from the consideration of Segal algebra $\cal U_O$ of $O$ observables; the resulting $O$ restricted states $\{\xi^O_i\}$ set is defined as $ \cal U_O$ dual space. From Segal theorem for associative (sub)algebras it's shown that $\xi^O_j$ describes the random 'pointer' outcomes $q_j$ observed by $O$ in the individual events.
quant-ph0404153.tex
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%\begin{titlepage}
% \title{{ Quantum Measurements and Information \\
% Restrictions in Algebraic QM   \\ }}
%
 \title{{ Information Systems Self-description \\
 and Quantum Measurement Problem   \\ }}
\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}

  Information-Theoretical restrictions on the systems self-description
and the information acquisition are  applied to Quantum Measurements Theory.
%       In particular G$\it\ddot{o}$del information incompleteness
For the measurement of quantum object S  by  the information system $O$
such restrictions  are  described by restricted states $R_O$
formalism.  
$R_O$ ansatz  can be introduced phenomenologically from 
 the agreement
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 
 from the consideration of Segal algebra $\cal U_O$  of $O$ observables;
the resulting $O$  restricted states $\{\xi^O_i\}$ set
is defined as $ \cal U_O$  dual space. From
 Segal theorem for associative  (sub)algebras  it's shown
 that  $\xi^O_j$ describes the  random 'pointer'
outcomes $q_j$  observed by $O$ in the individual events.

\end{abstract}
\vspace{6mm}
% $ \quad \quad $ {\bf Key Words: \, Quantum Measurements, Information,
%   $C^*$ algebras}  
\small { \quad
To appear in  Int. J. Theor. Phys. (2004)  }  
%   Talk given on 'Quantum Information and Computations' workshop, \\}
% \small {  \qquad  Triest,  October 2002 }
\vspace{10mm}
%\end{titlepage}
%\begin {sloppypar}
%
%
%

\section  {Introduction}

There are several fundamental problems concerning the interpretation of 
 Quantum Mechanics (QM), mainly involving the measurement process
(Jauch,1968; Aharonov,1981). 
% is a universally acknowledged
% theory, yet there are still several unsolved questions
% concerning its interpretation.
 The oldest and most prominent of them is probably
  the State Collapse or Quantum Measurement
   Problem  (D'Espagnat,1990; Busch,1996).
In this paper we shall analyze the quantum  measurements  within the
 Information-Theoretical framework and demonstrate
the importance of such consideration. Indeed,   the
  measurement of any kind results in   the  reception of data about
 the observed system S parameters   by another  system $O$ (Observer) 
% via the direct or indirect interaction with the studied system S
 (Guilini,1996; Duvenhage,2002).  
Therefore   the studies of
  Information-Theoretical restrictions on the information which can be
 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.
%
%  Such approach to QM foundations, assuming universality of QM
%  applicability to micro/macroscopic objects 
%  which called Relational QM leads to some other interesting conclusions.  
In the model regarded here
 $O$ is   the information gaining and utilizing system 
(IGUS);  it processes and memorizes the  information acquired as the result of
 S interactions   with the measuring system (MS) which element is $O$.
% and below S  is also regarded formally as MS element.
%As often assumed  considering the quantum measurements,
 We  assume  that QM description is applicable
both for  a microscopic    and  macroscopic
objects, this is the standard approach in Quantum Measurements Theory
(Busch, 1996). In particular,
 $O$ state supposedly is described 
by the  quantum statistical state  $\rho$ 
  relative to another  observer $O'$ (Rovelli,1995;Bene,2000).
In principle,in our approach $O$  can be  either a human brain
 or some automatic device,  in all cases it's
the system  
  which final  state correlates with the input data.
%  The computer information processing or the perception  by human brain
%  supposedly results from some physical objects evolution
%  which on microscopic level obeys   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 $O$ can be described by MS state $\rho _{MS}$ evolution
relative to external $O'$, yet  in our approach we regard    
  S information recognition and memorization  performed by $O$
itself, not by $O'$.
%  which reflected by $O$ internal state   evolution.
 Therefore the reception of information by $O$ during the measurement    
% internal state perturbed by S,$O$ interaction 
%  and therefore, $O$ internal state
  should be analyzed within the  self-description framework (Svozil,1993).
%  is  the new  problem for Quantum theory but 
The  systems self-description
was  studied extensively in 
  Self-reference  Problem context 
 (Finkelstein,1988; Mittelstaedt,1998).
It was shown that the self-description of an arbitrary system 
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  formalism of restricted  states 
 The self-description
 in the measurement process that is called also the  measurement from inside
 was considered in the  formalism of inference maps 
between MS and $O$ (Breuer,1996). It
 leads to some general results for the measurement properties
 but doesn't permit to derive the internal (restricted) $O$
 states $R_O$ from the  first  principles of  the theory.
% is the partial (restricted) MS state can differ principally from the standard
% QM ansatz for $O$ states  (Mittelstaedt, 1998).
Following this approach, we propose here the novel
formalism  which under simple assumptions
describes MS states mapping to $O$ states.
%   accounts $O$ self-description  effects and predicts
% Its  main feature is the modification of
In this formalism MS quantum state is represented by 
 the doublet $\Phi= \{ \phi^D, \phi^I \} $, where $\phi^D=\rho_{MS}$ is  MS
density matrix, $\phi^I(n)$ is $O$ restricted state which describes $O$
 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,
 It follows that $\phi^I$ is 
  the stochastic state which describes the random outcomes
 in the measurement of S states superpositions and this effect
can be interpreted as the subjective state collapse observed by $O$.  
 It will be shown that such formalism
  corresponds to the well-known generalization of standard QM
 - algebraic QM based on Segal and $C^*$- algebras of observables.
 (Emch,1972). In its framework $\phi^D$ is 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 should stress that  the observer's consciousness 
doesn't play any role in our theory and isn't referred to directly anywhere
(London,1939).
%  also none of the brain  the special  properties
% beyond standard QM assumed.
% In our approach the observer $O$  regarded as the quantum 
% reference frame (RF) which can interacts with the 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 $O$
%
 The terms 'perceptions', 'impressions' used here
%   IGUS $O$ description of experimental  results 
 are  defined  in strictly physical  terms.
In our model the perception is  
 the acquisition of some information by $O$, i.e. the change
of $O$  state;   a different
 $O$ impressions are associated with a different  $O$ states.
 The systems states are defined in  $O$ reference frame (RF)
 (or other  $O'$ RF)
 and are referred to also as 'S  state for $O$'.

%  which description sho operationally unambiguous.



\section {Measurements  and Quantum States Restrictions}

Our formalism exploits both the quantum states in the individual events
- i.e. the  individual states and the 
  statistical states which describe the quantum ensembles properties
 (Mittelstaedt,1998).
 In QM  the individual
states are the pure states  represented by Dirac vectors 
 $|\Psi_l \rangle$    
in $\cal H$ ; the statistical states are  described by  the 
normalized, positive operators of trace $1$
- density matrixes  $\rho$ on $\cal {H}$.
 If   $|\Psi_l \rangle$ composition  is known  for 
the given ensemble, its state can be described in more detail by
 the   ensemble state (Gemenge)    
 represented 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 shall start from the analysis of
   simple  MS  model
% which similar 
% to  von Neuman model of measurement (von Neuman,1932; Busch,1996).
which    includes
the measured state  S and
 IGUS  $O$   storing the incoming S information.
S represented by two-dimensional state vector $\psi_s$,
whereas $O$ is described by three-dimensional 
% $O$ interaction with environment E - decoherence
% will be  accounted at the next stage of the model regarded in chap. 4.
%     \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
  Hilbert space $\cal {H}_O$. Its
  basis  consists of 
  the orthogonal states $|O_{0,1,2}\rangle$,
 which are  the eigenstates of $Q_O$  
'internal pointer' observable with eigenvalues $q^O_{0,1,2}$.
 In our model the detector D is 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.
Let us   consider    
 the measurement   of  S observable $\hat{Q}$
%  on S   state $\psi_s$;
%  the initial  $O$ state  is  $|O_0\rangle$ and  thus
for MS initial state:
\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,$O$  interaction $\hat{H}_I$ starts at $t_0$
and finished effectively at some finite $t_1$, in that case    
 Schr$\ddot{o}$dinger equation for $\Psi^{in}_{MS}$ 
%  the suitable   S$-O$ interaction Hamiltonian $\hat{H}_I$
supposedly 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  $|s_{1,2}\rangle$.
%  for the corresponding .
It turns out that $ \bar{Q}_O=\bar{Q}$, so
 $O$ performs the unbiased $Q$ measurement (Von Neuman,1932).
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 $\cal{H}_O$.   
Concerning  the information recognition by IGUS $O$, 
 we  assume  that $|O_{1,2,0}\rangle$ correspond to 
the certain information patterns - an impressions percepted by $O$
 as $Q_O$ values $q^O_{1,2,0}$ (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. 
%
For the external observer $O'$  at $t>t_1$, 
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 $\Psi^O_{MS}=\Psi_{MS}$. 
whereas 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}.
However, it's well known   that experimentally   
the macroscopic   $O$ observes at random one of $Q_O$ values $q^O_{1,2}$
in any individual event. It means 
that  S final state is $|s_1\rangle$ or $|s_2\rangle$  and
S state collapse occurs. 
   S final  state  is 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 one
 can  ascribe to MS the 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}). 
% which formalism was proposed in \cite {Bre}.
% If $O$ regarded as the quantum object, 
  This   discrepancy constitutes the basis of  Wigner 
  'Friend  Paradox' for $O, O'$ (Wigner,1961).
%   which in the form of
% $gedankenexperiment$ was just described above (Wigner,1961).
% It's quite difficult to doubt both in the correctness of
%  MS evolution description by SE
%  and in  the state collapse experimental observations;
% In our opinion this paradox  reproduces, in fact, the true
%  descriptions of quantum measurement for different observers.
  We shall propose here   the formalism which incorporate consistently
this two   
  MS descriptions 'from outside' by $O'$
and 'from inside' by $O$.


% Considering the general measurement formalism, note first, that 
Generally,  the  measurement 
 of an arbitrary  system $S'$ is the 
mapping of $S'$ states set $N_S$ to  the given IGUS $O^S$ states set $N_O$.   
%  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 $S'$ states can't be factorized,
 then  $O^S$ should be considered as
 the subsystem of the large system $S_T=S'+O^S$ with the states set $N_T$
(Mittelstaedt,1998).
  In this situation - 'measurement from inside',  $N_O$ is $N_T$ subset and 
  the inference map of $S_T$ state  to $N_O$ defines
$O^S$ state  called also  the restricted  state $R_O$.
% which describes the information pattern available to $O$.  
% From $N_{T}$ mapping properties
 The important property of
 $S^T \rightarrow O^S$  inference map is formulated by
 Breuer theorem: if for two arbitrary $S_T$ 
states $\Lambda_{S},\Lambda'_{S}$ 
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.  
%  In classical case the origin of this  effect   is obvious:
  $O^S$ has less degrees of freedom
  than $S_T$ and 
 can't discriminate all possible $S_T$ states,  due to it
some number of $S_T$ indistinguishable states should exist for any $S_T$
 and $O^s$ (Svozil, 1993).
 In quantum case  
  the  observables noncommutativity and nonlocality introduce
some novel 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 
characterizes them. 

 The obtained
$S'$,$O^S$,$S_T$ relations are applicable to our MS model which can be
treated as  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's results leave the considerable freedom for the choice of
the inference map and
 don't permit
to derive the restricted states ansatz 
directly.
 It was proposed phenomenologically (Breuer,1996)  
that $R_O$ coincides with MS  partial trace 
 which for MS final  state  (\ref {AA2}) is equal:
\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,  $R_O=|O_j\rangle \langle O_j|$ for the initial
%   $\psi_s=|s_j\rangle$.
For   MS mixed  state $\rho^{m} _{MS}$
of (\ref {AA3}) 
 the  corresponding restricted 
 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 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  $\Psi_{MS}$ of (\ref{AA2}), yet
 for  the incoming S statistical mixture  - Gemenge (\ref {AA33})
 it  differs
from event to event:
\begin {equation}
\varsigma(n)=\rho^I_l=|O_l\rangle \langle O_l|| s_l\rangle\langle s_l|
  \label {A44}
\end {equation}
where the random $l(n)$ frequency is stipulated by the
 probabilistic distribution $P_l=|a_l|^2$.
% This individual state is  pure and
%  can be initially unknown for $O$ but exists objectively.
In any event $\varsigma(n)$  differs from
 MS  state (\ref{AA2})  and 
its restricted state 
 $\varsigma^O(n)=|O_l\rangle \langle O_l|$
  also differs   from $R_O$ of (\ref {AA4}) in any event. Because of it 
 for the
restricted individual states the main condition of Breuer  Theorem
is violated
 and $O$ can differentiate pure/mixed states 'from inside'
in the individual events (Breuer,1996). Therefore the proposed 
formalism doesn't result in 
 the state collapse  for the measurement from inside
 in  standard QM.
% In any case, $R_O$ is the correct ansatz for
% $O$ partial state in $O'$ RF (Lahti,1990),
% but, as will be argued below, isn't unambigous ansatz for $O$ 
% restricted state.
%   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.
Note also that the formulae (\ref {AA4}) 
describes also   the
  restricted $O$   statistical state
$R^{st}_O=R_O$ for the  pure and mixed MS  state $ \rho_{MS}^{p,m}$ 
and this ansatz is correct  both in $O$ and $O'$ RFs.

% 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   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}
which characterizes $O$,S quantum correlations.
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}).
Yet  $B$ value  can't be measured by $O$ 
'from inside', so $O$,S correlations are 
unavailable for $O$ directly. 
In standard QM $R'_O=R_O$
 is regarded  as $O$ individual (partial) state in $O'$
RF (Lahti,1990). Such $R'_O$ choice is advocated first of all by the correct
$\bar{Q}_O,\bar{Q}'_O$ values obtained for that state.
Let's consider as the example MS state of (\ref {AA2})
 with $a_{1,2}=\frac{1}{\sqrt{2}}$, in this state
$B$ has  the eigenvalue $\tilde {b}=1$.
 It means that in $R'_O$ state  $Q_O$ is prncipally uncertain for $O'$:
 $q_1^O \le \tilde {q}^O \le q^O_2$. In particular,  such $B$ value excludes
the ignorance interpretation of $Q_O$ uncertainty for $O'$,
which assumes that in that state $Q_O$ in any event is sharp and
is equal either to $q^O_1$ or $q^O_2$ (Busch,1996).
However, such reasoning fails for $O$ observer because
$B$ value is unobservable for him, and therefore it's impossible
to conclude in that theory whether $Q_O$ is uncertain or sharp in $O$ RF.   
 Hence $R_O$ of (\ref {AA4}) is just
the phenomenological choice which should be advocated additionally,
and other solutions for $O$ individual  state, as we argue
below, are possible. In particular, in this case the 'subjective'
ignorance interpretation, in which $Q_O$ value is sharp for $O$
but is  simultaneously uncertain for $O'$, can't be excluded.

%    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.

MS individual state  for $O$ can be rewritten  in doublet 
form $\Phi^B(n)=|\phi^D,\phi^I \gg$ where $\phi^D=\rho_{MS}$
is the dynamical
state component,   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.
 $\phi^I$ is just $\phi^D$  trace, however,
 in the alternative self-description formalism described below it 
% but for other theories  
 represents 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 case    
the state collapse 
appears in MS  measurement from inside  and
is described by the 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),
 this doublet state formalism (DSF) 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
 the measurement of some $S'$
observable occurs, then  SD can be violated for $O^S$ but,
  as follows from  condition i),   
    $S',O^S$ evolution
 for the external   observer  $O'$ should obey SLE.\\
  Below it will be shown that
DSF  corresponds to the measurements
description in Algebraic QM.
% 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}).
% Regarding DSF for our MS model it turns out that
For DSF state   
 $\Phi=|\phi^D, \phi^I\gg$ 
the dynamical component  is also
equal to QM density matrix  $\phi^D=\rho$ and obeys     SLE  :
\begin {equation}
  \frac {\partial \phi^D}{\partial  t} =[\phi^D,\hat {H}]    \label {AA8}
\end {equation}
  For our MS model 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  $\phi^I$  can
%  result in the stochastic outcomes   in the individual events. 
% is the initial $O$ subjective information and
   For $t \le t_0$,  the initial
 $\phi^I=|O_0\rangle \langle O_0|$; 
%  - $O$ has  no information on S. 
  %  the  final  $\phi^I(t)$ at $t \ge t_1$
   after S measurement finished
 at $t>t_1$,   its $\phi^I$ outcome  is supposed to be stochastic:
 $\phi^I(n)=\phi^I_i$,  where
$\phi^I_i=|O_i\rangle \langle O_i|$;
 here $i(n)$ 
 described by  the probabilistic distribution with $P_i=|a_i|^2$.
 It turns out  that such doublet individual state $\Phi(n)$ 
can change from event to event,  and $\phi^I(n)$
is partly independent of $\phi^D$ being correlated with it only
 statistically.
  $O$ subjective states $\phi^I$ ensembles coincide for
the pure and mixed states with the same $|a_i|^2$. Therefore the conditions of
 Breuer theorem are fulfilled and  the subjective state
collapse is observed by $O$. 
%   MS ensemble evolution can be described
%   by means of the doublet statistical states ansatz  
%  $  |\Theta\gg=|\eta_D,\eta_I\gg  $,
%  where $\eta_D=\phi^D$, $\eta_I(t)$ describes 
%  the probabilistic distribution $\{P_i(t)\}$  of   $\phi^I_i$ observations
% by $O$ 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.
%
%    $\eta_I(t)$ is defined by $\eta_D(t)$ which  obeys  SLE, and
% because of it
%  $\Theta$ evolution is reversible.
%  and S information acquired by $O$ can be erased completely . 

 An arbitrary ensemble evolution can be 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 $\{P^f_i(t)\}$  of  $\phi^I_i$ observations
 by $O$ at given $t$:
\begin {equation}
% \frac{\partial{\eta_D}}{\partial{t}}=[\eta_D,\hat{H}_c]    \nonumber \\  
P^f_j(t)=tr(\hat{P}^O_j \eta_D(t)) \label {CC} 
% \eta_I=\sum P^f_l |O^A_l \rangle \langle O^A_l| \nonumber
\end {equation}
where $\hat{P}^O_j$ - the projector on the given $\phi^I_j$.
For MS in particular,
 if S don't interact with $O$ (no measurement), 
 then $\eta_I$ is time invariant and MS obeys  the
standard QM SLE evolution for the dynamical component $\eta_D$.
Therefore our doublet states are important only for the
 measurement-like  processes
with direct  $ S-O$ interactions.
% $\eta^I$ 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.
%
 $\eta_I(t)$ is defined by $\eta_D(t)$ which  obeys  SLE, 
 because of it
 $\Theta$ evolution is reversible and the acquired $O$ information
can be erased completely (Mayburov,2002). 


Plainly, in this theory 
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 $\phi^I$
doesn't change during S measurement. Consequently,
 MS  state evolution  for $O'$ described by
$\phi^D$, which obeys SLE. Because of it  MS state
collapse isn't observed by  $O'$ in agreement with  the conditions i, ii.
 As the example of the simple $O$
 toy-model  can be regarded the hydrogen-like atom $A_H$
 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 differ supposed to be the same object.
% it isn't important.
  Witnessing QM Interpretation proposed by Kochen  
  is quite close to DSF but doesn't exploits the self-description approach
(Kochen,1985; Lahti,1990).   

% 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.



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


 
%  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)}





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 called also the basis degeneracy   is well-known
 in  Quantum Measurement Theory  (Lahti,1990; Elby,1994). 
% Plainly, in DSF $\phi^I_i=|O_i\rangle $ constitutes the  
%  preferred basis (PB) in $\cal H_O$;
%  and its appearance should be explained
%  in the consistent theory,
% this problem, called also basis degeneracy,  is well-known
%  in Quantum Measurement Theory (Busch,1995).
In its essence, the theory consistency demands that the final
 MS states decomposition should be unique,
 but this isn't the case for $\Psi_{MS}$ of (\ref {AA2}).
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).
Such interactionresults in the final entangled S,$O$,E state
which decomposed on some orthogonal $O$ basis  $|O^E_i\rangle$
 (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   $H_{O,E}$ interaction
parameters,   $|O^E_i\rangle$ basis can be made equivalent to 
$|O_i\rangle$ basis. 
% 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). 
For the initial MS state $\Psi^{in}_{MS}$ of
 (\ref {AAB}) it results in the final MS-E state :
\begin {equation}
    \Psi_{MS+E}=\sum a_i|s_i\rangle|O_i\rangle|E_i\rangle
                                             \label {DD1}
\end {equation}
where  $|E_i\rangle$ are final E  states.
% and the analogous states obtained for the general S-$O$ or S-D interaction. 
It was proved that such triple decomposition is unique, even
if $|E_i\rangle$ aren't orthogonal (Elby,1994).
%  Thus, PB problem formally resolved, if the decoherence accounted and
% this is essential   for our selfdescription theory.
 It isn't necessary
in our case to use $ N\rightarrow \infty$ limit, E can be also  a finite
 system.


 In addition, the  decoherence results in the
important consequences for  the   information storage by
$\phi^I$.
Really, for the practical IGUS $O$
the memorized states $|O^C_i\rangle$ excited by $|s_i\rangle$ signals
 must be stable or at least long-living. But as follows from eq. (\ref{DD1}),
 any  state $|O^C_j\rangle$ different from one of $|O_i\rangle$
in the short time would split into $|O_i\rangle$ combinations - entangled
$O$,E  states superpositions, so that 
 $|O^C_j\rangle \ne|O_i\rangle$ are in fact a virtual states, 
%  that's why $|O_i\rangle$ is true PB.
%   But  our calibration condition  demands that at least
%   $Q^O$ eigenstates should be conserved copiously and not transferred
%   to any combinations ??????
%
%%%% $$ Tuning specially $H_{int}$ we can get $|O_i\rangle=|O^E_i\rangle$.
Consequently, in this model  $O$-E decoherence  selects 
 the basis of long-living $O$ eigenstates which supposedly
describes $O$ events perception and
memorization. 
It means that, if our S signal is $Q$ eigenstate which
 induces $Q_O$ eigenstate $|O_i\rangle$, then it's memorized
by $O$ for the long time. It makes also $Q_O$  the preferable observable
for $O$ observables set, since for any
$Q'_O \ne F(Q_O)$ it gives $\bar {Q}'_O=0$ for  $\rho_{MS+E}$
resulting from the arbitrary $\rho_{MS}$ 
 (even if it's not the entangled state of
(\ref {AA2})).  This measurement 
scheme denoted as MS+E model will be considered below,
together with  MS model in which PB settled $a\, priory$.
%    As the result, the only $O$ observable
%   which can be memorized by $O$ is $Q_O$ 'pointer' ;
%  for any other $O$ observable
%  $Q'_O$ and any $\Psi_{MS}$ one obtains that in MS,E final state
%   $\bar{Q}'_O=0$. It corresponds to our calibration condition for $I^O$.
% Decoherence influence should be accounted in any 
%  $O$ selfdescription formalism.
%  In Breuer ansatz $O$ interaction with E states can be  accounted
% analogously to  S states, so that \\
% $R_O=Tr_{S,E}\rho_{MS,E}$ derived taking trace both on $S$ and $E$ states.
%  For our dual formalism
%  an analogous approach permits to derive $| \Theta \gg$, thus defines 
%  $\phi^I$ distribution. In the individual events $\phi^I$
% correspons to $|O_i\rangle|S_i\rangle|E^O_i\rangle$ branch.
% and corresponding $\Psi_{MS,E}$ branch in each event.
 In other aspects
 decoherence doesn't change our selfmeasurement model;  its
most important role is the unambigous definition of $O$ PB.
% describing its information recognition. 
In fact, $\cal {H}_O$ symmetry broken dynamically 
by $H_{O,E}$ interaction which makes majority of $O$ states unstable.
Despite that $O$,E decoherence clearly indicates PB existence,
 it doesn't mean that our DSF follows from first QM principles.
Our theory is still phenomenological and decoherent PB in MS+E
model only reveals
how $\phi^I$ basis should be chosen if such theory is correct.



%  Witnessing QM Interpretation proposed by Kochen  
%  is quite close to DSF but doesn't exploits the self-description
% (Kochen,1985; Lahti,1990).
% Considering the measurement by the apparatus $A$ ($O$ in our
% notations), it
% supposes phenomenologically that for  the measured  object $S$
%in pure state the
%measured value $Q$  always has the random  value
%$q_j$ relative to $A$. However,
%  no new mathematical formalism different
%from standard QM  wasn't proposed for the proof
%of this hypothesis (Lahti,1990).



\section {   Quantum Measurements in Algebraic QM}

Now   the quantum  measurements and $O$ selfdescription
 for the finite quantum systems will be regarded
in  Algebraic QM formalism (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 reasons to regard Algebraic QM
as the consistent generalization of standard QM with the fixed Hilbert space. 
%  The Operators Algebras formalism 
% for the quantum measurements with decoherence effects
%  studied  in  (Primas,1990).
Algebraic QM was applied extensively
 to the superselection models of quantum measurements,
in which the detector D or environment E  are regarded as the infinite systems
with $ m,V \rightarrow \infty $ (Primas,1990; Guilini,1996).
The algebraic formalism of nonperturbative QFT  
was used also in  the study of measurement 
dynamics in some realistic  systems (Mayburov,1998).
% Generally, even  not  accounting the algebraic QM,
%  the  methods of operator algebras 
%  are applied extensively in Quantum Physics (Jauch,1968; Primas,1983).
%
%   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  states set 
- Hilbert space $\cal H$ on which an observables - Hermitian operators
are defined. It can be shown that for some nonperturbative
systems  the structure of states set   differs principally
 from  $\cal{H}$, and the standard QM
axiomatics becomes preposterous. In distinction, 
   the fundamental structure  of Algebraic QM
 is   Segal algebra $\cal{U}$ of observables
which incorporates the  main properties of  regarded system $S_f$ (Emch, 1972).
%  and  defines $S_f$ state set $\Omega$  (Emch,1972).
 It's more   convenient technically   to  deal with 
 $C^*$-algebra $\cal{C}$  for which $\cal{U}$ is the subset.
% and calculate $\cal{U}$ properties afterward.
   Roughly speaking, $\cal{C}$ is an algebra of
 complex elements for which $\cal{U}$ is the subset 
 of its real (Hermitian) elements. 
%
  This elements in Algebraic QM
  are the linear operators for  which the sum $A+B$ and
   product $A \cdot B$  defined.
 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.
   $S_f$   states set $\Omega$ defined by
 $\cal{U}$    via the notorious
 GNS  construction; it proves that
  $\Omega$ is the  vector space dual to
 the corresponding  $\cal C$ (Bratelli,1981).
 Such states are
called here the algebraic states $\varphi $ and
are  the normalized, positive, 
linear functionals on $\cal {U} $: for any observable $A \in \cal{U}$,
  $ \, \forall \varphi \in \Omega;$, 
$\,\bar{A}=\langle \varphi;A\rangle$.
%  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. 
%
% ; 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  pure states -  i.e.   $\Omega$  extremal points are 
  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 $W^A$ are
  defined analogously to  their  QM ansatz.
Here only  a finite-dimensional $S_f$ will be considered,
 for  them  $\varphi$ states set is isomorphic to some
 QM density matrixes $\rho$ set. In that case 
we regard the corresponding $\varphi$ and $\rho$ as equivalent,
despite that, strictly speaking, they are  the different 
 objects  (Segal,1947).
 

% 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 $S_f$   algebra $\cal U$
is available for the observation.
% plainly, regarded MS measurement from inside 
% has much in common with this case.
%         It was shown that for  $\cal {U}_R$
 For such subsystems the restricted  algebraic states $\varphi_R$
 and their set $\Omega_R$ can be defined consistently
%   \cite {Em}. Alike   $S_f$ complete states the restricted  $\varphi_R$
 via the expectation values of $A_R \in \cal{U}_R$:
\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;
 $\varphi_R$ doesn't depend on any $A' \notin \cal{U}_R$,
thereby $\forall \varphi_R,\,\langle \varphi_R;A'\rangle=0$.
% $\varphi_R$ set denoted $\Omega_R$. (Emch,1972).
% For our MS (or any other IGUS) only $O$ observables  are available
% for the observation by $O$, 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 $\{A\}$;
 in algebraic QM $\cal U$ associativity
corresponds to its observables commutativity (Emch,1972).
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, its  $\varphi^a$ states set
  $\Omega^a$ is isomorphic to
  the set $\Omega ^c$ of the  classical statistical states $\varphi^c$
(Segal,1947).
%   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   
correspond to the classical
individual  states $\xi^c_i$ - points in $S^c$ phase space.
%  with the  observables eigenvalues $q^A_i$ (Emch,1972).
For the systems self-description the most important is the  case when
  $\cal{U}'$ is elementary, i.e. includes only $I$ - unit operator
 and  the single $A \ne I$, 
   then $\xi^c_i=\delta(q^A-q^A_i)$ corresponds to    
 $A$ eigenvalues $q^A_i$ spectra.
 Consequently, even if quantum $S_f$ is described by nonassociative $\cal{U}$,
 it should  contain the subalgebras $\cal{U}'\in U$
   for which                 
 the restricted states 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 $S^c_T=S'+O^c$ 
 described by some $\cal{U}^C$,
IGUS  $O^c$  self-description  restrictions are simple and straightforward -
the restricted $S^c_T$ states depend only on those
 $S^c_T$ coordinates $\{x^O_j\}$,
 which are $O$ internal degrees of freedom (Breuer,1996). 
They constitute the subalgebra $\cal{U}^C_R \in \cal{U}^C$.  In practice
 $O^c$ effective subalgebra $\cal{U}^C_O \in \cal {U}^C_R $ which
really defines the measurements 
 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 quantum IGUS $O$
%   observing system  $O^c \rightarrow O$
the  restricted $O$ subalgebra $\cal{U}_R$ also should
include only $O$ internal observables. In this case
any effective self-description subalgebra 
 $\cal{U}_O$ belongs to $\cal{U}_R$,
%   $\cal{U}_O, \cal{U}_R$ stipulates
   their  states  sets are denoted $\Omega_O, \Omega_R$ correspondingly.   
 The  main assumption of our theory is as follows:
 given  the subalgebra $\cal{U}_A \in \cal{U}$, 
  in any individual event $n$ an arbitrary  MS AIS $\xi^{MS}(n)$ 
 induces  some    restricted AIS $\xi^A(n)$ spanned on $\cal {U}_A$.
% which, as was defined above,is $\Omega_R$ extremal point.
%   ($\cal{U}_R$ in particular case).
This hypothesis seems to us quite natural  and  
 below the additional arguments in its favor will be presented
for the particular $\cal{U}_A$.
For illustration we regard it first 
for $\cal U_R$, and  accept $ad\;hoc$ that, according to the
formal  definition of the individual states,
such states are $\Omega_R$ extremal points.
% 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 is described by
 $ \cal{U}$ Segal algebra  of MS  observables  which defines
  $\varphi^{MS} \in\Omega$ properties,
  $MS+E$ model involves $\cal {U}_{MS,E}$ algebra correspondingly.
 $O$  subalgebra  $\cal{U}_R$  includes $I$ and all $O$ 
internal observables, so it means that
  $\Omega_R$
 is isomorphic to $O$  statistical states $\rho^O$ set $\Gamma_O$.
 Consequently, 
$O$ AIS set $\Omega^p_R$ is isomorphic to $\cal{H}_O$ and 
 any $O$  AIS $\xi^R_i $ corresponds to some  
 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 $\cal{H}_O$.
% The propsed MS
%  states restriction to $O$ seems  to us quite natural because 
% in such ansatz  $O$  states set  defined always  by $O$ observables
%  algebra only. 
We don't study here $\xi^R$  states in detail because they
 are unimportant for our aims,
%   because it's not our primary aim,
note only that Breuer  state (\ref {AA4}  ) for $O$ $R_O \notin  \Omega^p _R$,
and can't be $O$ AIS  on $\cal {U}_R$ for  an arbitrary $a_{1,2} $. 
%  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   $\cal U_O$,
%   $\varphi^{MS} \rightarrow \varphi^O$ restriction properties.
 remind that
  for the regarded MS dynamics  $O$ 
 can measure only the  observable $Q_O$. 
%   illustrated by $\Psi_{MS}$ of.
For any other $Q'_O \neq F(Q_O)$   one obtains  $\bar{Q}'_O=0$
for any $\Psi^{in}_{MS}$,
%In MS $\rightarrow O$  restriction framework
 therefore for  any  $\varphi^O \in  \Omega_O$ it  follows that  
 $ \langle  \varphi^O; Q'_O \rangle=0$.
%  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).
%
It means that  
  $O$ effective    subalgebra $\cal{U}_O$ is equal to
the elementary $\cal U^I_R$, which
 includes only  $Q_O$ and $I$.
Really, only in this case  
$ \langle   \varphi'; Q'_O \rangle=0$
 for all $ \varphi' \in \Omega^I_R$ defined on  $\cal U^I_R$;
 each $\varphi'$ corresponds to $\varphi^O$ with the same $\bar{Q}_O$
and vice versa;
 so  $\varphi^O$ set $\Omega_O$  is isomorphic to $\Omega^I_R$.
 There is no other $\cal U_R$ subalgebras with such properties and
  that settles $\cal U_O $ finally. It turns out that
 the obtained $\varphi^O$ are  equivalent to $R^{st}_O$ of  (\ref{AA4}) which
 are    MS  statistical states $\rho_{MS}$ restriction to $O$.

% to $O$ which, as was shown above, is equal to $R_O$ of.
% 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     the restricted algebraic $O$ 
 states $\varphi^O\in \Omega_O$   are isomorphic to the classical
 $q^O_i$ distributions;
%  To establish MS, $O$ AIS relations, it's necessary to analyze  the
meanwhile $O$ AIS $\xi^O$ -  $\Omega_O$ extremal points
 are the positive states:
\begin {eqnarray}
            \xi^O_i= \delta(q^O-q^O_i) \label {AA66}
\end {eqnarray}
 which correspond to the classical pointlike  states.
% Their norm is $\int \xi^O dq^O=1$.
% which are the map of   $R^O_i =  |O_i\rangle\langle O_i|$ to $\Omega_O$.
%  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. 
%
  In particular, such $\xi^O_i$ can appear in 
%  $\xi^{MS}_i$ restriction to $O$ which is MS final state in
$\psi_s=|s_i\rangle$  measurement  by $O$ as the restriction of MS final
state $\xi^{MS}_i=|s_i\rangle|O_i\rangle$.
% 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.
%
As was supposed above,  the formal individual state $\xi^O_i$
  percepted by  $O$  as the definite 
$Q_O$ value   $q^O_i$. Such information pattern permits to reveal
 $\xi^O_{i,j}$ distinction in the single event per each state
which is important for our theory interpretation.
Note also that Algebraic formalism, in fact,  extracts PB
 $|O_i\rangle \sim \xi^O$ in $\cal {H}_O$ 
even without  the account of E decoherent interaction. The 
 decoherence, as argued below, only duplicates this effect. 

% 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.
 
 The incoming S mixture - $|s_i\rangle$ Gemenge results in
 MS algebraic final state 
  $\varphi_{mix}$ which is equivalent to $ \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
%
  $O$ restricted state  $\varphi^O_{mix}$ 
is  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  $\varphi^O_{mix}=\sum  |a_i|^2\varphi^O_i$,
where $\varphi^O_i=\xi^O_i$ of (\ref {AA66}).
From the  correspondence
of MS state $\xi^{MS}_i$ and $O$ state $\xi^O_i$ in the individual events,
the restricted algebraic
 state $\varphi^O_{mix}$
 represents the statistical 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}) resulting in
%
 If MS final state is  the pure state $\Psi_{MS}$ of (\ref {AA2}) and
 corresponding MS  algebraic state is
$\varphi^{MS}=\xi^{MS}$,  it results 
 in  the same $\bar{Q}_O$ value.
Therefore its $O$ restricted algebraic state
coincides with the mixed one: $\varphi^O=\varphi^O_{mix}$.
%    Note that, $\Psi_{MS}$ is MS individual QM state to which corresponds
%     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 $\Omega_O$ all $O$  individual states are AIS $\xi^O_i$ of (\ref{AA66}),  
% 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,
From that one can define what are $O$ individual states induced by
MS pure states $\xi^{MS}$, in particular, whether they  differ from
the obtained $\Omega_O$ extremal points $\xi^O_i$ -
the formal individual states.
% To find them  the following method will be used:
%first,  analogously to $\cal {U}_R$ case, we assume that
% MS restricted AIS  in each event can be only one of $\xi^O_i$,
% then we shall argue that other solutions are excluded.
% Eventually,
%   if MS state is $\Psi^{MS}_j$ superposition of (\ref {AA2}),
Remind that any  physically different states can be  operationally
discriminated by the particular observation procedure,
which puts in correspondence  to this states some parameters values.
%   which reveals this states
%   difference via the  difference of some observables values distributions.
For the statistical states it can be their probabilistic
distributions parameters. The individual states $\psi_{a,b}$
are   the eigenstates of some $A,B$ observables  
and their difference  is revealed by their
 eigenvalues $q^{A,B}_{i,j}$ which exist objectively.  In principle,
   this values 
 can be extracted  from the single event per each state
(Mittelstaedt,1998). 
%      which means  that  $\xi^O_i$ covers all $Q_O$ spectra.
If to suppose  that such correspondence maintained also
for the restricted states, then 
in our case the only $\cal{U}_O$ observable is $Q_O$  and
$\xi^O_{i,j}$ difference reflected by $q^O_{i,j}$ values. 
 If to suppose that some other $\xi^O_a \ne \xi^O_i$ exists,
 then some other observable 
 $Q^e \in \cal{U}_O$, for which $\xi^O_a$ is the eigenstate,
 should exist, such that $\, Q^e \ne F(Q^O)$.
%  shouldbelong to  $ \cal{U}_O$ to differ it from $\xi^O_i$, 
However this assumption is inconsistent
 with the obtained $\cal{U}_O$ structure.
 In particular, Breuer  state $R_O$  analog
$\xi^O_R=\sum |a_i|^2 \xi^O_i$ 
can't be $O$ individual state on $\cal{U}_O$, because it
  isn't $Q_O$ eigenstate and can't be $\Omega_O$ 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. 

To illustrate this reasoning,  consider
the superposition $\xi^{MS}_s=\Psi_{MS}$
 of ({\ref{AA2})  with $a_{1,2}=\frac{1}{\sqrt{2}}$; we denote the
(fuzzy) values of $Q^O$ and $B$ of (\ref {AA5}) as $\tilde{q}^O,\tilde{b}$.
Let's define the conditions to which
 the  restricted $\xi^{MS}_s$ state - $\xi^O_s$ should satisfy.  
In $O'$ RF $\xi^{MS}_s$ is $B$ eigenstate with the eigenvalue $\tilde {b}=1$ 
which represents  IT condition,
and $q^O_1 \le \tilde {q}^O \le q^O_2$ is the spectral condition.
Taken together this conditions indicate that $\tilde{q}^O$ is 
 located within the interval $[q^O_1,q^O_2]$ and is
principally uncertain inside it,
as $\tilde b$ value evidences.
 $\xi^{MS}$ restricted state $\xi^O$ is defined on
 $\cal {U}_O$ $=\{I,Q_O\}$,
therefore the spectral condition
%    $q^O_1 \le \tilde {q}^O \le q^O_2$
 also holds for $O$. Now because $B \notin \cal{U}_O$
and unavailable for $O$,
IT condition can be dropped.  Without it the spectral condition alone
  means  that $\tilde{q}^O$ is   localized in $[q^O_1,q^O_2]$ interval,
 but can be either uncertain or sharp.
Hence $\xi^O_{1,2}$ states with  the sharp $\tilde{q}^O=q^O_{1,2}$
 satisfy to that only necessary  condition for the restricted states.  
 Meanwhile there are no state $\xi^O$ in $\Omega_O$
with $\tilde{q}^O$ uncertain.
The reason of it, as was explained above,
is that  such $\xi^O$ should be an eigenstate of some IT 
observable   which value, alike $\tilde b$,
 can  demonstrate   $\tilde{q}^O$  uncertainty for $O$.
Yet $\cal{U}_O$ doesn't include any such observable.
 Consequently, $\xi^O_{1,2}$ are the only suitable candidates and 
 the formal
 solution for $\xi^{MS} \rightarrow \xi^O_s$ restriction in the 
individual event $n$   is:
 $$
          \xi^O_s(n)=\xi^O_1.or.\xi^O_2
$$
 and it can be shown to be
the same for any $a_{1,2} \neq 0$. It means that $MS\rightarrow O$
 map is ambigous and
  breaks  $Q_O$ symmetry of $\xi^{MS}$. 
 Note also that an arbitrary $\varphi^O$ admits the unique
decomposition into $\xi^O_i$ set and so can be interpreted as $\xi^O_i$ 
ensemble  with the given probabilities $P'_i$.
Thus, in each event $\xi^O_{1}$ or $\xi^O_2$ can appear at random.
For the pure MS ensemble regarded above 
% $\xi^O_i$ should appear at random  with some probabilities $P'_i$.
%  defined by $\bar{Q}^O$.
% It turns out that
% for such $\varphi^O$ content
 the only solution which gives  the correct $\bar{Q}_O$  is  $P'_i=|a_i|^2$ 
, and so $W^O=W^O_{mix}$.
 The obtained results demonstrate
  that for MS ensemble
$\xi^{MS}\rightarrow \xi^O$ restriction map is stochastic 
and results in  the  subjective state collapse  observed by $O$ 
analogously  to  DSF state collapse  described above.
% To advocate the obtained picture, remind that
%
%  The evolution eq. (\ref {CC}) are applicable also
%  for $\{\varphi ,\varphi^O\}$ doublet.
  Since $\xi^O_i \in \Omega _R$, 
$\xi^O_s$ can be taken  also as the  possible ansatz for
individual MS  states  restriction on 
   $\cal {U}_R$.
DSF doublet state $\Phi$ components $\phi^D,\phi^I$
 correspond to $\xi^{MS}$, $\xi^O$;
the  algebraic states $\varphi(t),\varphi^O(t)$ describe 
$q^O_i$ statistical distributions analogously to DSF $\eta_D,\eta_I$.
%  $O$ impressions $I^O(n)=q^O_i$ are unambiguous in each event. 
%



 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 which leads to 
the randomness of outcomes for the models of  measurements in the infinite
systems (Guilini,1996; Mayburov,1998).
The algebraic self-description  permits to extend
such randomness mechanism on the finite systems.
From  the mathematical point of view the
 Algebraic QM formalism contains the generic structure -
the restricted AIS set $\Omega^I_R$ defined on  the
elementary subalgebra
 $\cal {U}^I_R \in \cal U$ ($ \cal {U}^I_R=\cal{U}_O$ in our model).
  $\Omega^I_R$ extremal points, being treated as the individual
 states,  describe the  collapse of the pure states defined on $\cal U$.  
Hence  in this theory
the quantum state reduction results from the reduction of  a system 
algebra to its associative subalgebra.



If to analyze this results  in the Information-Theoretical framework,
 remind that
 the difference between the pure and
mixed MS states reflected by $B$  of (\ref {AA5}) expectation values
(and other IT observables).
Therefore $O$ possible observation of S pure/mixed
 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 S,$O$ correlations
supposedly are unobservable from inside by $O$ (Mittelstaedt,1998).
It corresponds to Wigner conclusion that the perception by $O$
of the superposition
of two contradictive impressions is nonsense
and should be excluded in the consistent theory (Wigner, 1961).
Consequently, it's possible that Information Theory rules
by itself can lead
 to the subjective state collapse in the quantum measurements.


% one demands that the selfdescritpion formalism
% should  consistently describe the information 
% percepted by $O$, it seems that  the only plausible
% solution is the proposed algebraic  $\xi^O(n)$ ansatz. 
%  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 Self-reference 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 phenomenological 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 than $\cal U_O$,
 but this case
 demands more complicated calculations which we plan to present
in the forcoming paper. In Algebraic QM 
 the only important condition for the classicality appearance
 is $\cal{U}_O$ associativity but it 
  feasible, in principle, also  for the complex IGUS structures.
 If to consider MS+E system 
and its $\cal {U}_{MS,E}$ algebra, the effective $O$ information subalgebra
will be the same $\cal {U}_O$ considered above. 
Therefore $O$ subalgebra and its states set
properties can't depend directly on the surrounding E properties.
%
%  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  settled finally. In particular, 
it's still unclear whether all the algebraic states $\varphi$ 
correspond to the
physical states (Primas,1983). This question is  important by itself 
and  can be essential for our formalism feasibility.
% By our choice
% of the initial MS states it avoided in the regarded model.
 We admitted  also
 that for  MS arbitrary  $\xi^{MS}$  some   $O$ restricted AIS responds
in any event.
 It agrees with the consideration of the restricted states  
 as the real physical states, 
% 'the states in their own right' defined on $O$ subalgebra $\cal{U}_O$
 however, this assumption needs further clarification.
For the regarded simple MS model Algebraic QM formalism in many aspects
is analogous to Orthomodular Algebra of propositions or Quantum Logics
(Jauch, 1968; Emch, 1972). The possibility of
 its application to the systems self-description and the measurement
from inside demands the additional investigation.   
% \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
%    This approach to the measurement problem  has much i
%   common with Quantum reference frames theory  by Aharonov  $\cite {Aha3}$

 \section { Discussion}

in this report the
 information-theoretical restrictions on the quantum measurements
were  studied  in the simple selfdescription  model of IGUS $O$.
Breuer self-description theory shows that
by itself   $O$ 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 leads to the doublet states $\Phi$ ansatz,
where one of its components $\phi^I$ corresponds to $O$
 subjective information - i.e. $O$ selfdescription.
  Algebraic QM  
presents the additional arguments in favor 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 $O$.
 Algebraic QM formalism
-  Segal and $C^*$-algebras of operators is acknowledged
generalization of standard QM (Emch, 1972).
In addition, from the mathematical point of view 
the duality of the operators algebra and the states set
is better founded, than the states set priority
postulated in standard QM (Bratelli, 1981).
 In this paper it was applied for 
the simplest measurement model but if this formalism universality
will be proved, it would mean that the proposed measurement
 theory follows from the established Quantum Physics realm (Emch,1972).
 Algebraic formalism  permits to calculate
  the restricted individual $O$ states $\xi^O$  ansatz
 without any phenomenological assumptions.
From the formal point of view the only novel feature of our approach 
is the use of Segal algebra for the individual restricted
 states $R_O(n)$ calculations.
We don't see any compelling mathematical arguments why  
 for the individual states $R_O$ 
 ansatz should be chosen  the same QM formulae (\ref {AA4})   
  which used for  the statistical restricted states
$R^{st}_O$ (Lahti,1990).  


%   Hence we believe EPR-Bohm correlations
%   deserve the detailed study in this dual framework. 



%  From this considerations
%   the natural question arise : does the observation of random
%  outcomes $\phi_I=V^O_j$ means that before the measurement starts 
%   S state can be characterized by some objective  'hidden parameter'$j_S$ ?

Our  theory demonstrates that the probabilistic realization
 is generic and unavoidable for QM and without it QM supposedly can't
acquire any operational meaning. Wave-particle dualism
was always regarded as characteristic QM  feature but in our theory
it has straightforward  correspondence in our  DSF.
 Our  approach stresses also the
 dual character of
quantum measurement : this is the interaction 
of studied S with IGUS $O$ and in the same
time the information acquisition and recognition by IGUS. 
  Note that Self-reference 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).
%  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).

 
%  In this  approach different  observers  becomes nonequivalent
% in a sense that the  physical reality description becomes principally different
% for each of them (Rovelli,1995). This nonequivalence
% reflected by the presence of subjective component $\xi^O$ available directly
% only for particular $O$.
%   Eventually, the consistent selfmeaurement formalism 
% constructed alternative to Breuer  formalism.  


Under the realistic conditions, the rate of E atoms interactions with 
macroscopic detector D is very high, and due to it in a very short time $t_d$
S,D partial state $\rho_p=Tr_E \rho_{SDE}$ becomes approximately equal
to the mixed one, as $\rho_p$ nondiagonal elements become
very small. This fact induced the claim
that the objective state collapse   can be completely explained by detector  
state decoherence without the observer's inclusion, but it
was proved to be incorrect (D'Espagnat,1990).
% The interesting approach to the state collapse 
The development of decoherence approach proposed
  in Zurek 'Existential interpretation (Zurek,1998).
% Analogously to our MS model,
 IGUS   $O$  regarded as the  quantum object and
 included in the measurement chain; 
%  $O$ state decoherence
%  via interaction with E 'atoms' is quite important in this case.
%  The  simple  IGUS model regarded
% in which special attention given to
%  $O$ information memorization on quantum level.
%  because $O$ perception  reflected by its memory.In this model 
  memorization of input S signal  
occurs in several binary memory cells $|m^j_{1,2}\rangle$ (chain)
which are the analog of  the brain neurons. 
 $O$ memory state suffers  the decoherence from surrounding E 'atoms'
which results in the system state analogous to (\ref {DD1}).  
Under  practical  conditions, the decoherence time $t_d$ 
is also  small and for $t \gg t_d$
 S,$O$ partial state $\rho_p$ differs from the mixture
very little.  From that  Zurek concludes that $O$ 
percepts input pure S signal as the random measurement outcomes.
%  But such theory conserves all the faults of discussed
%  above DC theory. Really, analogously as it  was argued  for DC theory  
%  for external $O'$,
Yet the system S,$O$,E 
 is still in the pure state even at $t\gg t_d$ and  IT observable
$B$ analogous to (\ref {AA5})
  which proves it exists.  Therefore in standard QM framework
it's incorrect to claim that IGUS percepts random events. 
The regarded IGUS model doesn't differs principally
from our MS scheme;
%   Due to it,   DSF  can be applied  
%  to it without significant modifications resulting in our IM analog.
% if to apply DSF to it,
 in Algebraic formalism IGUS  subjective perception  is described
by  $O$ restricted  state $\xi^O$ defined on $\cal {U}_O$
 which describes the random
outcomes for the input pure S state.  Consequently, 
Algebraic QM  application 
  to Zurek IGUS model supports Existential Interpretation.


It's important to stress that all the   experiments in Physics  at the
final stage include  the human subjective perception
which simulated by $O$ state in our model.
The possible importance of observer in the measurement
process was discussed first by London and Bauer (London,1939). They supposed
that  Observer Consciousness (OC) due to 'introspection
action' violates in fact Schrodinger equation  
and results in the state reduction.
%         This idea was seriously  criticized  by Wigner \cite {Wig}.
 In distinction, in our  theory $O$ perception     
 doesn't violate MS Schrodinger evolution from $O'$ point of view.
%   But the   subjective perception of measurement results
%   in it also performed by OC
%    and its results partly independent of dynamics, due to
%   its dependence on the stochastic, subjective component $\phi^I$.
%    This effect deserves further
%   discussion, but we believe that  such probabilistic behavior
%   is general IGUS property not related to OC only.
%  Basing on our approach, we formulate here the simple
%  human perception model - IM in which
%   $O$ impression $I^O$ in the individual
%  events  described by the random $Q_O$ eigenvalues $q^O_i$
%  with the probabilities $|a_i|^2$.
In principle,
Self-description Theory  permits  to regard the relation between
MS, IGUS $O$ states and $O$ subjective information (impression), and
 one can try to extend it on the human perception.
%   von Neuman Psychophisical parallelism hypothesis  (von Neuman,1932).
 This is the separate, important problem which is beyond
our scope and here
we consider briefly only some its principal points which 
formulated here as the simple Impression Model (IM). 
% For realistic IGUS $|O_{1,2}\rangle$ can correspond to some
% excitations of $O$ internal collective DFs
%  like phonons, etc., which memorize this $Q$ information,
%  but we don't consider its possible physical mechanisms here.
Following our approach, $O$ perception affected only by 
$O$ internal states defined on $O$ observables subalgebra $\cal {U}_O$. 
For  $O$ perception   the following, calibration assumption
introduced:  for any Q eigenstate $|s_i\rangle$ 
 after S measurement finished at $t>t_1$ 
and $O$ 'internal pointer'state is $|O_i\rangle$ observer $O$
 have the definite impression $I^O$ 
corresponding  to $I^O=q^O_i$ eigenvalue  - the information 
pattern percepted by $O$.
%  which interpreted as if  the input S state is $|s_i\rangle$.
% This calibration assumption seems speculative, due to Hilbert space 
% symmetry relative to the basis choice but  it advocated here
%by the decoherence effects influence.
%  The calibration condition  for $O$ state  is  nontrivial  because 
It  settles the hypothetical correspondence between MS quantum
dynamics model and human perception. 
Impression $I^O$ is $O$ subjective information which isn't
dynamical parameter and  its introduction can't have
any influence on  the theory dynamics.
%  Note that Breuer restricted state $R_O$ agrees with such assumption. 
%   It also   related to 'Preferred basis' problem discussed below 
%   (Lahti,1990; Elby, 1994).
 Furthermore, we assume that
  if S state is the superposition $\psi_s$  
 then its measurement by $O$ also  results in appearance
for each individual event $n$ of some 
 definite  and unambiguous $O$ impression  $I^O=q^{sup}(n)$. 
 In Algebraic formalism the corresponding
  $O$ subjective information - impression in the individual
event  is equal to  $I^O(n)=q^O_i$ and can be consistently defined
in this ansatz for our IM as the stochastic state 
appearing  with probability $|a_i|^2$.
%  In DSF the corresponding
%   $O$ subjective information - impression in the individual
% event  is equal to  $I^O(n)=q^O_i$.
In algebraic QM  framework they correspond to the restricted $O$ AIS $\xi^O_i$
defined on $\cal{U}_O$.
% is the suitable candidate for  the human perception description.
The obtained picture is the analog
 of Von Neuman psychophisical parallelism hypothesis.
%  without violation of Schr$\it\ddot{o}$dinger dynamics. 
%  Supposed computer-brain perception analogy,
% in fact, means that the human signal perception also defined by $O$
% observables values.
% Despite that this analogy looks quite reasonable, we can't
% give any proof of it.
% In our model
%  in fact  the state collapse have  the subjective character and 
%  occurs initially only for single observer $O$ $\cite {Rov}$.
Yet we must  stress that, on the whole,  our  theory doesn't need any
addressing to  human OC. Rather, in this model IGUS
$O$ is active RF which internal state excited by the interaction
with the studied object.

 
To conclude, the quantum measurements were studied within the
 Information-Theoretical framework and  self-description
 restrictions  on the information acquisition
are shown to be important in the Measurement Theory.
% were  studied  on the simple selfdescription  model of IGUS $O$.
% Breuer selmeasurement study shows that
% by itself   $O$ 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.
%  We proposed the  DSF   doublet state $\Phi$ formalism,
%in which one of its components $\phi^I$ corresponds to $O$
%  subjective information and describes the subjective state collapse.
% The same $\Phi$ ansatz follows from 
Algebraic QM represents
the appropriate formalism of systems self-description,  
% Algebraic QM,
 % presents the additional arguments in favor of this approach,
 in particular, 
 IGUS  $O$ observable algebra $\cal {U}_O$ defines $O$ restricted states
$\xi^O$ set $\Omega_O$.
% which defines the multiplet states ansatz  analogous to $\Phi$. 
The appearance of stochastic events 
 stipulated by  MS individual states restriction to  $\xi^O$ states
and results in the  state collapse observation by $O$. 
The regarded IGUS model is quite simple and on the whole doesn't
permit us to make any final conclusions at this stage.
% We present here the simple measurement theory and we don't regard
% it as the final solution of measurement problem. 
Yet  the obtained results evidence
% that it's impossible to solve  the Measurement Problem
 that the  IGUS information
 restrictions and its interactions with the observed system 
should be accounted in Quantum  Measurement Problem analysis  (Zurek,1998).
\\
\\
\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
Quantum 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)
\\
(1968) J.M.Jauch 'Foundations of Quantum Mechanics' 
(Adison-Wesly, Reading)
\\
%  \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-ph0404153.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
Quantum 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)
\\
(1968) J.M.Jauch 'Foundations of Quantum Mechanics' 
(Adison-Wesly, Reading)
\\
%  \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}