{ 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 measurements 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 the formalism of restricted states which obtained 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 derived in Algebraic QM formalism 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-ph0506065.tex
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%\begin{titlepage}
 \title{{ Quantum Measurements and Information \\
 Restrictions in Algebraic QM   \\ }}
\small\author{S.N.Mayburov \\ 
Lebedev Inst. of Physics\\
Leninsky Prospect 53, Moscow, Russia, 117924\\
E-mail :\quad   mayburov@sci.lpi.msk.su}
\date {}
\maketitle
\begin{abstract}

  Information-Theoretical restrictions on the systems measurements
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 the formalism of
 restricted states which obtained 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 derived in Algebraic QM  formalism
 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,
  Observable algebras}  
\small { \quad 
%   Talk given on 'Quantum Information and Computations' workshop, \\}
% \small {  \qquad  Triest,  October 2002 }
\vspace{10mm}
%\end{titlepage}
%\begin {sloppypar}
%
%
%

\section  {Introduction}

There are still  several unsettled problems
 concerning the interpretation of 
 Quantum Mechanics (QM), and the majority of them
 involves to some extent the quantum measurement processes
(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 Objectification
   Problem  (D'Espagnat,1990; Busch,1996).
In this paper we shall analyze the quantum  measurements  within the
 Information-Theoretical framework which  considers some  of their important
aspects.
 % which in principle can lead to   the important conclusions. 
Indeed,   the
  measurement of any kind is   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  possible 
  Information-Theoretical restrictions on the information 
 transfer from  S to $O$  can influence
on the observed measurement effects  (Svozil,1993).
% 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
%(QMT),despite that it's not quite well founded
%(Busch, 1996).
this is the standard assumption of  Quantum Measurement theory, 
despite that it's not quite well founded (Busch,1996).
 In particular, in our approach  $O$ state  is described 
by the  quantum state  $\rho$ 
  relative to another  observer $O'$ (Rovelli,1995;Bene,2000).
In this 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  
Even if 
  S measurement by $O$ is described by the evolution
of  MS state $\rho _{MS}$ 
relative to some external $O'$,  our aim is
 to calculate    
  S information  acquired  by $O$ itself.  In general  $\rho_{MS}$
ansatz isn't sufficient for that,
%  which reflected by $O$ internal state   evolution.
% The reception of information by $O$ during the measurement    
% internal state perturbed by S,$O$ interaction 
%  and therefore, $O$ internal state
 and the description of MS state  relative to $O$  
 should be regarded in the  self-description framework
of Information Theory (Svozil,1993).
%  is  the new  problem for Quantum theory but 
Earlier the  systems self-description
was investigated in the context of
 the general problem of mathematical  Self-reference 
 (Finkelstein,1988; Mittelstaedt,1998).
It was argued 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 
 Recently the self-description
 in the measurement process,  called also the  measurement from inside,
 was considered in the  formalism of inference maps $M_O$  
 connecting MS and $O$ states $\rho_{MS}, R_O$ (Breuer,1996). It
 leads to some general results for the quantum measurements, 
 but by itself doesn't permit to derive $M_O$  
%  the resulting restricted $O$
  from the  first  QM principles.
% is the partial (restricted) MS state can differ principally from the standard
% QM ansatz for $O$ states  (Mittelstaedt, 1998).
% In this vein the phenomenological 
% formalism  was proposed
% which under simple assumptions
% describes M_O$ ansatz  and  the set of $O$ states (Mayburov,2001,2004).
%   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. 
% $\phi^I$ 
In this paper we argue that the  inference map 
or arbitrary systems can be  
%   corresponds to the well-known generalization of standard QM
calculated in Algebraic QM framework which exploits
    Segal  algebras of observables - i.e. $C^*$- algebras formalism.
 (Emch,1972). In this formalism MS  state $\xi^{MS}$ is defined on MS 
observables algebra $\cal{U}$,
$O$ state $\xi^O$  
  is defined on $O$ observables subalgebra $\cal{U}_O$ which is 
the restricted set of MS observables.
It will be shown that for  MS in the pure state,  $\xi^O$ 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 $O$.  
Earlier the analogous inference map was obtained phenomenologically
in  the formalism of doublet states (Mayburov,2001,2004).
Alternatively  this 
 results can be interpreted also as the consequences of principal 
restrictions on
the information transfered from S to $O$. 

%     $\Phi$ 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;
 their properties  are discussed  also in the final chapter.
 The  states  of objects defined in  $O$ reference frame (RF)
 (or other  $O'$ RF)
   are referred to also as 'S  state for $O$'.

%  which description sho operationally unambiguous.



\section {Measurements  and Quantum States Restrictions}

The essence of the state collapse problem is the 
  possibility of discrimination between 
pure and mixed states in quantum measurements, so
% In our theory several sorts of quantum states are used, and 
% to avoid confusion it's 
it's instructive to remind the classification of quantum states.
 First, there are the 
individual states - i.e.
 the quantum states in the individual events, in 
standard QM  they are 
the pure states - Dirac vectors $|\psi_l\rangle$ in Hilbert space $\cal H$;
see also chap.4  (Primas, 1981).
The   statistical states  describe the  properties
of quantum ensembles, and are represented 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, it 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 $W^e$. 
Here and below under standard QM we understand Schr$\ddot{o}$dinger
 formalism  which postulates that the quantum states
are defined on complex Hilbert space
 introduced $ad\quad hoc$ (Jauch, 1968).     
The properties of states in
   Algebraic QM     will be considered in chap. 3.


We shall use the  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.
For the simplicity the detector D is dropped in MS chain,
   the role of $O$ decoherence effects will be discussed below.   
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 $q^O_{0,1,2}$, which 
for the convinience is taken to be equal to $0,1,2$ correspondingly.  
For the simplicity of calculations it assumed that $|O_{1,2}\rangle$ 
constitutes the basis of two dimensional susbspace $\cal{H}'_O$. 
% 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 $\hat{S}_z$
%  on S   state $\psi_s$;
%  the initial  $O$ state  is  $|O_0\rangle$ and  thus
for MS initial  pure 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 $q_{1,2}$.
% To extract the  studied effect 
% it will be compared with result for $|s_i\rangle$ mixture $W^s$
% with the same $s_z$ probabilities $P_i=|a_i|^2$.
%
In our model  S,$O$  interaction $\hat{H}_I$ starts at $t_0$
and finishes effectively at some finite $t_1$;  
for the suitable $H_I$    
ansatz Schr$\ddot{o}$dinger equation for $\Psi^{in}_{MS}$ 
%  the suitable   S$-O$ interaction Hamiltonian $\hat{H}_I$
would result in
 MS final  state $\rho^p_{MS}$ responding to the state vector:
\begin {equation}
   \Psi_{MS} = \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 
$$
 \bar{Q}_O=\bar{S}_z=|a_1|^2-|a_2|^2
$$
, so $O$ performs the unbiased $Q$ measurement (Von Neumann,1932).
To reveal  the studied effect, the measurement of  incoming
  S  mixture with the same $\bar{S}_z$ should be regarded also,
  its composition is described by the gemenge 
$W^s=\{|s_i\rangle,|a_i|^2\}$ to which correspond the density matrix:  
\begin {equation}
 \rho_s^m= \sum_i |a_i|^2|s_i \rangle \langle s_i|
                                                              \label {AA33}
\end {equation}
%
% Meanwhile for any $O$ observable
% $Q'_O  \ne F(Q_O) ;\, \bar{Q}'_O=0$ independently of 
% $\psi_s$ (except $Q'_O=I$, but it's   trivial). 
% 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 $O$,
 at this stage 
 we  assume arbitrarily  that $|O_{1,2,0}\rangle$ eigenstates
corresponds to 
the certain information patterns  $J^O_l$
percepted by $O$ as $Q_O$ values $q^O_{l}$.
Below  more arguments in favor of this assumption
will be presented, however it's worth to notice here that
in this case $q^O_i$ are $O$ real properties,
so this is quite natural assumption  (Busch, 1996).
 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 $|s_i\rangle$,
for $|s_i\rangle $ superposition it results
  into the final S - $A_H$ entangled state. 


% 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 $t>t_1$ and   the incoming $\psi_s$

The pure state  $\Psi_{MS}$ of (\ref {AA2}) 
   describes MS state relative to external observer $O'$,
%  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 information patterns (impressions):
 $Q_O=q^O_1$ and $Q_O=q^O_2$ percepted by $O$
 simultaneously (Wigner,1961).
%      which  as Wigner claimed is nonsense \cite {Wig}.
However, it's well known that experimentally   
the macroscopic   $O$ would observe 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 $|s_2\rangle$ for $O$,  and
S state collapse occurs. 
   S final  state  is described by $\rho^m_s$
% the statistical ensemble of  individual final states for $O$
which stipulated by the gemenge 
$W^{MS}_{mix}=\{ |s_i\rangle|O_i\rangle, |a_i|^2 \}$.
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}
relative to $O'$;
% to it responds the genenge 
% $W_{MS}=\{|s_i\rangle |O_i\rangle\,|a_i|^2\}$,
it    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  describes the phenomenon of state collapse,
     it illustrates also  the famous  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 regard here   the formalism which temptatively can  unify 
this two    MS descriptions 'from outside' by $O'$
and 'from inside' by $O$. The first clue prompts the comparison
of $O$ average response to S pure and mixed states. Let's
consider for the simplicity only the incoming  $\psi_s$
with $r_s=\frac{a_1}{a_2}$ real, and denote them as $\psi_s^r$, 
 we introduce also  the z-symmetric state $\psi^s_s$
 for which $r_s=1$.
%  , and the corresponding z-symmetric $\Psi_{MS}^s$.
 For incoming  $\psi^r_s$, one obtains 
 $\bar {S}_x=2a_1a_2,\,\bar{S}_y=0$; 
 for $s_i$ mixture $W^s$ of (\ref {AA33}) $\bar{S}_x=0, \bar{S}_y=0$, so the 
  rate of S state purity $\beta_p$  
 is characterized by $|\bar {S}_x|$.
Meanwhile, beside $\bar{Q}_O=\bar{S}_z$, for any other $O$ observable
 $Q'_O  \ne F(Q_O) $ the final $\bar{Q}'_O=0$ independently of the initial
S state. Hence 
 the information about the rate of state purity characterized by $\beta_p$
 doesn't transferred to $O$ at all, at least via $O$ expectation values.
%  (except $Q'_O=I$, but it's   trivial). 
That's not quite surprising, because $S_z, S_x$ doesn't commute,
and so from the uncertainty relations follows that the precise $S_z$
meausrement would spoil $S_x$ measurement.
The analogous results 
 can be obtained for an arbitrary $a_{1,2}$,
but in place of $S_x$ some linear combination of $S_x, S_y$
should be used.
In Information Theory framework  MS and any other
  measurement set-up can be regarded formally as the information
channel transferring the information from S to $O$.
   Despite that the conclusive picture can be obtained only
from the analysis of individual  events, 
this calculations  indicates that
%  if to consider MS as the information channel
%  transferring the information from S to $O$, then  
 the information transfer restrictions
 can be important  in    Quantum Measurement
theory. 



% Considering the general measurement formalism, note first, that 
From the formal point of view  the  measurement 
 of an arbitrary  system $S'$ is the 
mapping of $S'$ states set $N_S$ to  the given IGUS $O^I$ states set $N_O$
(Svozil,1993).   
%  In the simple cases   some measured $S'$ parameter
%   $A^S$ becomes correlated with with some $O$ parameter $A^O$
%   via  $S',O$  interaction.
% If  $O^I$ interaction with $S'$ can't be neglected
% If the final $O^I$ and $S'$ states can't be factorized,
   $O^I$ can be considered  formally as
 the subsystem of the large system $S_T=S'+O^I$ with the states set $N_T$
(Mittelstaedt,1998).
  In this approach - 'measurement from inside', $N_O$ is $N_T$ subset and 
  the inference map $M_T$ of $S_T$ states   to  $N_O$  defines
$O^I$ state  $R_O$ called   $S_T$ restricted  state.
% which describes the information pattern available to $O$.  
% From $N_{T}$ mapping properties
 The important property of
 $S_T \rightarrow O^I$  inference map is formulated by
 Breuer theorem: if for two arbitrary $S_T$ 
states $\Gamma_{S},\Gamma'_{S}$ 
their restricted  states $R_O, R'_O$ coincides, then for $O^I$ 
observer this $S_T$ 
states are indistinguishable (Breuer,1996).
Under simple assumptions about $S_T,O^I$  at least several such  $S_T$  
states should exist.
% 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:
  $O^I$ has less degrees of freedom  than $S_T$ and 
 can't discriminate all possible $S_T$ states, and because of it
some number of $S_T$ indistinguishable states 
 always should exist for any classical $S_T$
 and $O^I$ (Svozil, 1993).
 In quantum case  
  the  observables noncommutativity and nonlocality introduces
some novel features regarded below.
% The main advantage of Breuer theory is  the proof of 
% existence of the complete set of restricted states
% for an arbitrary  $S_T$ structure and correspondingly
% for  any consistent inference map $M_T$. 
Despite that $R_O$ are incomplete $S_T$ states,
 they are the real physical states
for $O^I$ observer - 'the states in their own right', as Breuer 
characterizes them. 

 The obtained
$S'$,$O^I$,$S_T$ relations are applicable to our MS model which also 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 leaves the considerable freedom for the choice of
the inference map $M_T(S_T \to O^I)$, and
 doesn't permit
to derive the restriction ansatz for $S_T$ individual states  
directly.
%  It was proposed phenomenologically (Breuer,1996)  
It can be shown   that
 $R^{st}_O$ -  the restricted statistical state
 coincides with the  partial trace of $S_T$ state over $S'$
 degrees of freedom; 
  for MS   state  (\ref {AA2}) it is equal to:
\begin {equation} 
   R^{st}_O=Tr_s  {\rho}^p_{MS}=\sum |a_i|^2|O_i\rangle\langle O_i|
      \label {AA39}
\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
%   $\psi_s=|s_j\rangle$.
From that for   MS mixture $\rho^{m} _{MS}$
of (\ref {AA3}) 
 the  corresponding restricted 
 state is the same $R^{mix,st}_O=R^{st}_O$.
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}), 
 for  the incoming S  mixture  - Gemenge $W^s$ of (\ref {AA33})
 MS state   also differs from event to event:
\begin {equation}
\varsigma^{MS}(n)=|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 $P_l=|a_l|^2$, so 
% This individual state is  pure and
%  can be initially unknown for $O$ but exists objectively.
in any event $\varsigma^{MS}(n)$  differs from
 MS  state (\ref{AA2}).
It was proposed phenomenologically that for an arbitrary system $S_T$
its restricted individual state   is  also equal to the
partial trace of $S_T$ individual state over $S'$ (Breuer,1996), 
 so for MS pure state:  
%   $R_O=R^{st}_O$ of (\ref {AA4}) (Breuer,1996). Therefore
\begin {equation} 
   R_O=Tr_s  {\rho}^p_{MS}=\sum |a_i|^2|O_i\rangle\langle O_i|
      \label {AA4}
\end {equation}
Obviously such ansatz excludes beforehand any  
kind of probabilistic $R_O$ behavior.
Thereon  in  any event $n$ for MS Gemenge of (\ref {AA33})
its restricted individual  state  
 $\varsigma^O(n)=|O_l\rangle \langle O_l|$
   differs   from $R_O$  
(below  also $\varsigma^O_i =|O_i\rangle \langle O_i|$ notation is  used).
 It follows then, that   for the
restricted individual states the main condition of Breuer  Theorem
is violated. From that Breuer concluded
that  $O$ can differ pure/mixed states 'from inside'
in the individual events, therefore  this  
formalism doesn't result in 
 the state collapse  
 (Breuer, 1996). However,  the formal
difference of two  restricted states doesn't mean automatically
that this states are physically different.
That's the necessary but not sufficient condition,
in general  it's necessary also
that  some MS parameters measurable  by   $O$  
 are different for that states.
 Otherwise it can turn out that this  states are equivalent,
%  i.e. the expectation values of all observables available for $O$
and $O$ can't detect their difference
(Mittelstaedt,1998). For $\varsigma^O_i$ 
the only parameter i.e. observable value
  which characterizes it in an individual events
is $q^O_i$. In Breuer paper $R_O$ parameters available for $O$
      weren't  calculated, without it
  the conclusion about $R_O,\varsigma^O_i$
 discrimination  seems preposterous;
the  discussion of this point will be continued in   chap. 4.
 Below by the slight abuse of terminology,
  MS state restriction to O is called $O$ restricted state. 
% 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 unambiguous 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.
As was shown above the difference between 
 the initial S pure state and
$|s_i\rangle $ mixture with the same $\bar{S}_z$ reflected by 
$\bar{S}_{x,y}$ values. As the  example, for  $\psi^s_s$ with
 $a_{1,2}=\frac{1}{\sqrt{2}}$ for pure and mixed state one obtains
 $\bar {S}_x=1,0$   correspondingly.    
%  MS state   'from inside'- $R_O$  is  operationally different
%  from MS state for   $O'$ 'from outside' i.e. $\Psi_{MS}$.
% $O$ can't measure 'from inside' the difference between 
%  MS   states with different $D_{12}=a_1^*a_2+a_1 a_2^*$; 
%  the partial  collapse;
For MS   states  the same difference reflected 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  of (\ref {AA3}),
 but  $\bar{B}\neq 0$  for the pure MS states (\ref{AA2}).
For example, for the incoming $\psi^s_s$ one obtains $\bar{B}=1$. 
% Qualitatively  for  the initial state  $\psi^s_s$,
% as the result of S,$O$ interaction,
% $S_z$ observable  mapped to $Q_O$, whereas
% $S_x$ to $B$  observable.
% On the whole IT observables are nonlocal
% and  constitutes the large class  $\{B^{MS}\}$ which will be used below.
 However,  $B$ value  can't be measured by $O$ 
 'from inside', so $O$,S correlations are 
 unavailable for $O$ directly. The 
principal possibility for $O'$ to  measure $B$
and send the information to $O$ doesn't change 
the situation, because such $O'$ measurement is incompatible
with $S_z$  measurement by $O$.
 On the whole IT observables are nonlocal
and constitutes the special class of observables $\{B^{MS}\}$.
%which members will be used below.
% 
%In standard QM $R'_O=R_O$ Beside the restricted states
% QM formalism also exploits $O$ partial state $R'_p$ 
% defined in $O'$ RF,  for the individual states $R'_p=R_O$
% (Busch,1996). 
%  choice is advocated first of all by the correct
%  $\bar{Q}_O,\bar{Q}'_O$ values obtained for that state.
% Yet they are the different objects and 
% To illustrate $B$ importance for MS restrictions 
Note that $\{B^{MS}\}$  observabilty excludes
the Ignorance Interpretation (II) of pure states,
which assumes, in particular, that  MS  pure state
  $\Psi_{MS}$ is equivalent to $|s_i\rangle|O_i\rangle$
gemenge (Busch,1996).
% $Q_O$ uncertainty for $O'$,
As the example, for the symmetric $\Psi_{MS}$ of (\ref{AA2}) 
 with $a_{1,2}=\frac{1}{\sqrt{2}}$ II  
 claims that  $Q_O$ value in any event is sharp and
is equal either to $q^O_1$ or $q^O_2$ with the same probability.
Meanwhile  in this state
$B$ has  the eigenvalue $\tilde {b}=1$
  which means that   $Q_O$ value $\tilde{q}^O$
 is principally uncertain for $O'$:
 $q_1^O \le \tilde {q}^O \le q^O_2$, and so II is incorrect for $O'$.  
However such reasoning fails for $O$ observer, because
$B$ value is unobservable for him together with  other
 $B^{MS}$ observables.
%  Rather for the restricted set of MS observables
% available for $O$, it's impossible
% to conclude  at this stage 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 and deserve detailed considerations.
   Hence the 'subjective' II,
 in which $Q_O$ value can be sharp for $O$,
being  simultaneously uncertain for $O'$, can't be excluded
beforehand  in self-description approach. 
% Note also that for the gemenge (\ref {AA3}) $Q_O$ values $q^o_i$
% in the separate event $n$ are objective values (Busch, 1995), 
% it excludes the possibility of $\varsigma^O(n)$ recoherence 
%  admitted by Breuer. 
% Also Breuer conceded that the properties of 
% restricted states for given $S_T$
% are practically independent of the choice of
% the inference map $M_O$, but  the sufficient arguments 
%  of that weren't demonstrated.    
% Hence from this review  it seems that
% the self-description  theory deserves
%  further investigation beyond the described phenomenology. 


%    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.

In the regarded approach
MS individual state  can be rewritten formally in doublet 
form $\Phi^B(n)=|\phi^D,\phi^I \gg$ where $\phi^D=\rho_{MS}$
is the dynamical
state component i.e. MS state for $O'$,   the information component
 $\phi^I$ describes $O$ subjective information, i.e.
$O$ restricted state  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.
 $\phi^I=R_O$ is just $\phi^D$  trace, however
in the alternative theory regarded here it  describes the independent 
$O$ degrees of freedom.
% in the alternative doublet state formalism (DSF) it
% but for other theories  
%  represents the independent $O$ state components (Mayburov,2001,2004);
%  its main  properties conserved also in Algebraic QM self-description, 
% therefore it cited here briefly.
%   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, 2004).
% 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,
 any self-description theory should satisfy
 to the following operational conditions : \\
i) if an arbitrary system $S'$   doesn't interact with IGUS $O^I$,
 then for $O^I$ 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 $O^I$ and
% the entangled $S'$,$O^I$ state produced
 the measurement of some $S'$
observable occurs, then the  quantum dynamics can be violated for $O^I$, but
  as follows from  condition i),   
  in this case  $S',O^I$ evolution 
 for the external   observer  $O'$ should obey SLE.\\
To reconcile this conditions with the state collapse
at the phenomenological level we proposed recently 
 doublet state formalism (DSF) 
% but for other theories  
%  represents the independent $O$ state components
 (Mayburov,2001,2004). It's reviewed here briefly, because
 it has much in common with the
 description of measurements  in Algebraic  QM. 
%  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}
  therefore  for MS
the initial $\phi^D=\rho^{in}_{MS}$ 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
$\phi^I$ describes $O$ restricted state, so
   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 $t>t_1$,  in DSF 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 $i(n)$ outcome frequencies are
 described by  the probabilistic distribution with $P_i=|a_i|^2$.
 Hence 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. Consequently the ensembles of
  $O$ subjective states $\phi^I$  coincides for
the pure and mixed states with the same $|a_i|^2$, the conditions of
 Breuer theorem are fulfilled and  the subjective state
collapse can be observed by $O$.
% Its appearance in DSF can be explained,
%if to concede that $O$ state $\phi^I$ defined on the set of $O$ internal
%observables, but IT $B$ which values discriminates the pure/mixed states
%doesn't belong to it; see more details below.
%  which corresponds to 'subjective ignorance' . 
%   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;
eventually this theory responds to the subjective II
regarded above.
% 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 of PB appearance prompts $O$ decoherence - i.e.
$O$ interaction with environment E,
which is practically unavoidable in lab. conditions (Zurek,1982).
Such interaction  results 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  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 the
 decoherence doesn't change our measurement model;  its
most important role is the unambiguous definition of $O$ PB.
% describing its information recognition. 
In fact $\cal {H}_O$ symmetry is broken dynamically 
by $H_{O,E}$ interaction which makes majority of $O$ states unstable.
As will be shown   in Algebraic QM PB is defined by S,$O$ interactions
only and is independent of $O$ decoherence,
 however the  account of decoherence effects is necessary in any consistent   
measurement theory.

%  Despite that $O$,E decoherence clearly indicates PB existence,
%   it doesn't mean that  DSF follows from first QM principles,
%  it is only  the phenomenological formalism and  PB appearance in MS+E
%  scheme 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 $O$ self-description
 for the finite quantum systems will be regarded
in  Algebraic QM formalism; analogously to DSF
it should also satisfy to the conditions i) - ii) formulated above.
%  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; Blanchard,2003).
% 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  
%   $a \; priory$.
%

What is the fundamental QM entity -
the Hilbert space of states or the algebra of observables
is  the problem dusputed since  the time of QM formalism construction 
 (Von Neuman, 1932; Emch,1972).
In standard QM the fundamental structure is the 
 Hilbert space  of states $\cal H$
 on which an observables - Hermitian operators
( or POV) are defined (Busch,1996).
 However, it was found that for some nonperturbative 
systems  the structure of states set   differs principally
 from  $\cal{H}$, and the axiomatics of standard QM
 becomes preposterous (Bratelli,1981). In distinction, 
   the fundamental structure  of Algebraic QM
 is   Segal algebra $\cal{U}$ of observables
which incorporates the  main properties of any  regarded system $S_f$
 (Emch, 1972).
%  and  defines $S_f$ state set $\Omega$  (Emch,1972).
From the mathematical point of view 
the duality between the operators algebra and the states set
is more natural, than the states set priority
postulated in standard QM (Bratelli, 1981).
In its framework  it's more   convenient technically   to  deal with 
 $C^*$-algebra $\cal{C}$  for which $\cal{U}$ 
constitute  the subset.
% and calculate $\cal{U}$ properties afterward.
 $\cal C$ and $\cal U$ elements in Algebraic QM
  are the linear operators for  which the sum $A+B$ and
   product $A \cdot B$  defined.
  Roughly speaking, $\cal{C}$ is the complex algebra 
  for which $\cal{U}$ is the subset 
 of its real (Hermitian) elements.
 For any system $\cal{C}$, $\cal{U}$ 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; in its framework
  $\Omega$ 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;$, its expectation value
$\bar{A}=\langle \varphi;A\rangle$. 
Eventually $\varphi$ states are the analog of QM density matrixes $\rho$.
%  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  which is a convex shell
denoted $\Omega^p$;
 the further consideration of AIS properties
is given below in chap.4 (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 $\xi^{MS} corresponds to \Psi_{MS}$.
All  other   elements $\varphi \in \Omega$ are treated
as the algebraic mixed states, despite that their physical
meaning isn't settled finally (Primas, 1981).    
Independently of it any algebraic  state $\varphi$
can be constructed operationally as $\xi_i$ ensemble,
  the   ensemble states $W^A$ are
  defined analogously to  QM ansatz given in chap.2.
Here only  a finite-dimensional $S_f$ will be considered,
in this case both regarded QM formalisms are principally equivalent, 
but Algebraic QM is more convinient for our problems,
because in its framework  the correspondence between states and 
observables is more straightforward.
In this case $\varphi$ states set $\Omega$ is isomorphic to $S_f$ set of
 QM density matrixes $\rho$.



%  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.


% From the arguments regarded in the previous chapter
Concerning  the application of Algebraic QM to the measurements,
 in particular  for our MS scheme, 
 $O$ self-descripiton restrictions can be formulated 
 as the restrictions
on the set of MS observables  available for  $O$. 
The  situations 
in which for some system $S_f$ only some restricted 
   linear subspace $\cal M_R$  or  subalgebra
 $\cal{U}_R$ of   algebra $\cal U$
is available for the observation
were  extensively studied  in (Emch, 1972).
% plainly, regarded MS measurement from inside 
% has much in common with this case.
%         It was shown that for  $\cal {U}_R$
 In this case the restricted  algebraic states $\varphi_R$
% and their set $\Omega_R$ can be defined
%   \cite {Em}. Alike   $S_f$ complete states the restricted  $\varphi_R$
can be constructed
 starting from  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 $\forall \varphi_R,\,\langle \varphi_R;A'\rangle=0$.
% the consistency  of $\cal {U}_R$ and the corresponding
% set of states $\Omega_R$  is proved in (Emch, 1972).
% $\varphi_R$ 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.
%
%
In Algebraic QM  any classical system $S^c$  
described by the commutative system of observables $\{A^c_i\}$
which constitute the associative Segal algebra $\cal {U}^C$.
% of $S^c$ observables $\{A\}$; $\cal U$ associativity
%  corresponds to its observables commutativity (Segal,1947).
% The theorem by Segal   proves that 
%    for all $A_j\in \cal{U}'$
%   commute and unit operator $I\in \cal {U}'$, i.e.
Conversely any  associative Segal  (sub)algebra $\cal{U}'$
is isomorphic to the algebra $\cal {U}^C$    which
 describes some $S^c$, its  $\varphi^a$ states set
  $\Omega^a$ is isomorphic to
  the set $\Omega ^c$ of the  classical statistical
$S^c$ 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   
corresponds 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  self-description the most important is the  case when
  $\cal{U}'$ is elementary, i.e. includes only $I$ - unit operator
 and  the single $A \ne I$ (it's also called A-subsystem). 
   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 includes the subalgebras $\cal{U}'\in U$
   for which       the restricted states are classical.
Hence  in Algebraic QM $S_f$  state can be described
formally as the multiplet, each member of
which is the state defined on the particular (sub)algebra
 analogously to DSF state regarded above.
%  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  system $S^c_T=S'+O^c$ 
 described by some $\cal{U}^C$, the self-description restrictions for
IGUS  $O^c$   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 
From the analogous reasons
   for  the quantum localized  IGUS $O$ 
%   observing system  $O^c \rightarrow O$
 its subalgebra $\cal{U}_R$ of MS algebra $\cal U$ also should
include only $O$ internal (local) observables. 
It corresponds   to the locality principle  acknowledged
in Quantum Physics. Really $O$ perception of
 any other MS observable $A_{MS} \notin \cal U_R$
 involves  the instant  measurement of S which can be miles away
from $O$ at that time, the example is IT $B$ of (\ref {AA5}).
Any effective $O$ subalgebra  $\cal{U}_O \in \cal{U}_R$,
%   $\cal{U}_O, \cal{U}_R$ stipulates
   their  states  sets are denoted $\Omega_O, \Omega_R$ correspondingly.   
Concerning with $O$ individual states 
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 state $\xi^{MS}(n)$ 
 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 $\cal{U}_A$.
% From that  and QM locality, because IT observable $B$ of (\ref {AA5}) 
% $B \notin \cal{U}_R$, then  
% $O$ states shouldn't depend on $\bar B$ value. 

% 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
  $\varphi^{MS}$ set  $\Omega$. In case of $O$ decoherence
  $MS+E$ 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 $\cal V_O$. 
 Then $O$ AIS set $\Omega^p_R$ is isomorphic to $\cal{H}_O$, and 
 any $O$  AIS $\xi^R_i $ should correspond 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 the effective
$O$ subalgebra  $ \cal U_O $ differs from $\cal U_R$,
and our aim is to calculate $O$ states defined on it.
% 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
   in the used MS dynamics of S measurement   for any final
$\Psi_{MS}$ 
  only  $\bar{Q}_O \ne 0$ in MS final state, $\bar{Q}_O=\bar Q$; 
%   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 corresponding AIS $\xi^O \in  \Omega^p_O$
 it  follows that  
 $ \langle  \xi^O; Q'_O \rangle=0$.
Any $\varphi^O \in \Omega_O$ is the convex state of $\{\xi^O \}$, hence
$\langle \varphi^O,Q'_O\rangle=0$.
%  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 $\Omega^I_R$ 
(Segal, 1947).
 There is no other $\cal U_R$ subalgebras with such properties, and
  that settles $\cal U_O $ finally as the classical algebra of
single 'pointer' observable $Q_O$. Consequently 
such $\cal U_O$ defines also PB for $O$ states unambiguosly,
and so $O$-E decoherence only can only duplicate it;
  more detailed consideration of the decoherence effects  is given below.
%  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 unambiguous PB definition which makes the theory better founded.
%

 \section {Algebraic QM Restrictions and Self-description} 

%  The use of $w^*$ algebras ????
% Dependence of results of O model ??

Now   the relation between  the  pure  and individual states
 is regarded in more detail, because   it will be used
 below in the construction
 of  restricted $O$ states in Algebraic QM -  AIS.
 Despite that mathematically
this question is quite plain, it's 
  considered here in most detailed way,  because in the discussion
of QM foundations it became 
the source of many confusions   (Bene,2000; Breuer, 1996).
In this consideration of individual  quantum states
 i.e. the states distinguishable in the individual events
   we shall follow the approach  of (Primas, 1990).
As was mentioned in chap. 2,
the  physically different states of any kind can be  operationally
discriminated by the particular experiment
which puts in correspondence  to this states some
measured parameters values.
%   which reveals this states
%   difference via the  difference of some observables values distributions.
For QM statistical states  it can be the parameters of  
 experimental distributions  which can be expressed via
the expectation values of some observables. 
 For the finite-dimensional nondegenerate $S_f$ any pure state $\psi_{a}$
is   the eigenstate of some  observable $A$  
%and their difference  is revealed by their
with the eigenvalue $q^{A}_{a}$; in this case
 $q^{A}_a$ corresponds to some real $S_f$ property (Busch, 1996).
 It permites $\psi_a$ discrimination from 
any  other pure state $\psi_b$ in a single event
by the objective (non-collapse) $A$ measurement,
  hence  any $\psi_a$ is the individual state. 
% In principle,   this values 
% can be extracted  from the single event of measurement
% per each state, that's  just one can expect for the 
% practical difference of individual states (Mittelstaedt,1998). 
There is no other classes of states which satisfies to this conditions, 
in particular, the convex states can't be an individual states.
To illustrate it, let's regard as
 the  example  S state $\psi_s$ of (\ref {AAB}),
 for an arbitrary $a_i$ it is the eigenstate of some operator
$\vec{S}\vec{n}$, where $\vec{n}$ direction defined by $a_i$.
Now let's  consider   the convex state
$ \phi_z=\sum w_i|s_i\rangle \langle s_i|$ with $w_i \ge 0$, in this
     framework it has no real properties. Hence it's
impossible to prove that $\phi_z$ is  the superposition
of states with $s_z= \pm 1$, alike the real property 
 $s_x=1$ demonstrates it for $S_x$ eigenstate $\psi_s^s$ with
 $a_{1,2}=\frac{1}{\sqrt{2}}$.
$\phi_z$ is smeared on $s_z$ axe, i.e. $-1 \le s_z \le 1$, but the
 same inequality is true for $|s_{1,2}\rangle$ states also. Consequiently
 by no means
 $\phi$ can be discriminated from $|s_{1,2}\rangle$ 
in the individual events, and because of it
 $\phi$ can't be the feasible individual state. 
In this framework the ansatz for 
the individual states  is defined by the operational conditions only,  
and hence will be  principally the same in Algebraic QM -
i.e. AIS should be also  the pure states. For the restricted  algebraic
states this  ansatz doesn't change principally,  as shown below
    for $\cal U_O$ its derivation is even more simple and straightforward. 

As follows from Segal theorem,
for the obtained $\cal U_O$ subalgebra
     the  algebraic $O$ 
 states $\varphi^O\in \Omega_O$   are isomorphic to the classical
 $q^O_i$ probabilistic  distributions;
%  To establish MS, $O$ AIS relations, it's necessary to analyze  the
meanwhile  $\xi^O$ -  $\Omega_O$ extremal points
 are the positive states:
\begin {eqnarray}
            \xi^O_i= \delta(q^O-q^O_i) \label {AA66}
\end {eqnarray}
 which corresponds 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 $\xi^O_i$  appears in 
%  $\xi^{MS}_i$ restriction to $O$ which is MS final state in
$|s_i\rangle$  measurement  by $O$
 as the restriction of MS final
state $\xi^{MS}_i \sim |s_i\rangle|O_i\rangle$.
% 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.
%
% Such information pattern permits to reveal
 $\xi^O_{i}$  are $O$ individual states, their 
distinction  can be revealed in the single event from the
the difference of $Q_O$ eigenvalues $q^O_i$. The
existence of such restricted $O$ states at least for some MS states
 restriction is  important for our formalism, because it  
permits to analyze  any other  restricted states (if they exist)
 by the comparison with $\xi^O_i$. Following the   
consideration  of chap. 2, we concede that $O$ percepts this states as
the information pattern $J^O_i=q^O_i$.


% Note also that the formalism of Algebraic QM, in fact,  extracts PB
% $|O_i\rangle \sim \xi^O$ in $\cal {H}_O$ 
% even without  the account of E decoherent interaction,
% but only from the selection of the unique $O$ subalgebra $\cal {U}_O$
% in S, $O$ interation. 
%  The $O$-E 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  $W^s$ 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 $\bar{Q}^n_O$, in particular :
$$
    \bar{Q}_O=\langle\varphi^O_{mix} ;Q_O\rangle=
    \langle\varphi_{mix};Q_O\rangle=\sum |a_i|^2 q^O_i
$$
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 in this case
the statistical mixture of AIS $\xi^O_i$ 
 described by $O$  ensemble state 
\begin {equation}
  W^O_{mix}=\{ \xi^O_i;\,P_i=|a_i|^2;\,i=1,2\} \label {AA55}
\end {equation}
%  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 $\xi^{MS}$
which corresponds to $\Psi_{MS}$ of (\ref {AA2}),
 then MS  algebraic state is
$\varphi^{MS}=\xi^{MS}$, and  it results 
 in  the same $\bar{Q}^n_O$ value as the regarded mixture.
Therefore its $O$ restricted algebraic state
coincides with the mixed one: $\varphi^O=\varphi^O_{mix}$ 
for the same $|a_i|$.
%    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
% As will follow it defines  the restriction $\xi^O_f$ of arbitrary
% MS pure states $\xi^{MS}$  to $O$ 
% , in particular, its  possible discrimination 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}),

%      which means  that  $\xi^O_i$ covers all $Q_O$ spectra.
%  It plausible to assume that the correspondence of AIS and the
% eigenvalues, described above, maintained  also
% for the restricted individual states, but supposedly concerns only with 
%  the observables  of the regarded subsystem. 
% In our case for MS$\to O$ restriction,
%  the only $\cal{U}_O$ observable is $Q_O$ ( or $ F(Q_O)$) and
% $\xi^O_{i,j}$ difference reflected by $q^O_{i,j}$ values,
% thereon no other individual states $\xi^O$ can appear
% as $\xi^{MS}$ restriction.  
%  Note  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 $P'_i$,
% the solution which gives  the correct $\bar{Q}^n_O$  is  $P'_i=|a_i|^2$.

% 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.
%
%   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 the  derivation  of MS restriction to $O$ 
for  individual states,
let's consider it for $\xi^{MS}_s$ which is equivalent to $\Psi_{MS}$
 of ({\ref{AA2}) for  z-symmetric initial state $\psi^s_s$ with
 $a_{1,2}=\frac{1}{\sqrt{2}}$.
 Let's express   the properties  which
MS  restricted  state - $\xi^O_s$ should possess via  
 its relations with  the  observables
  $Q_O$ and $B$ of (\ref {AA5}), their values (in general uncertain)  
 are denoted as $\tilde{q}^O,\tilde{b}$.
In $O'$ RF $\xi^{MS}_s$ is $B$ eigenstate with the eigenvalue $\tilde {b}=1$ 
which is called   IT property;
 $Q_O$ obeys the inequality:
 $q^O_1 \le \tilde {q}^O \le q^O_2$, it is called the spectral property.
Taken together this properties indicates that for $O'$ $\tilde{q}^O$ is 
 located within the interval $[q^O_1,q^O_2]$, and  so is
principally uncertain inside it,
as $\tilde b=1$ value evidences.
  $\xi^O_s$ is defined on
 $\cal {U}_O$ $=\{I,Q_O\}$,
therefore the spectral property
%    $q^O_1 \le \tilde {q}^O \le q^O_2$
 also holds for $O$. Now because $B \notin \cal{U}_O$
and so is unavailable for $O$, for $\xi^O_s$
IT property can be dropped.
  Without it the spectral property alone
  means  that $\tilde{q}^O$ is   localized in $[q^O_1,q^O_2]$ interval,
 but $\tilde{q}^O$  can be either uncertain or sharp  in its limits.
 Meanwhile $\xi^O_{1,2}$ states with  the sharp $\tilde{q}^O=q^O_{1,2}$
  possess   that only  property which MS restricted state should have.
Therefore the
 solution for $\xi^{MS}_s \rightarrow \xi^O_s$ restriction in the 
individual event $n$  can be formally written as:
 \begin {equation}
          \xi^O_s(n)=\xi^O_1.or.\xi^O_2   \label {B3} 
\end {equation} 
% and it can be shown to be the same for any $a_{1,2} \neq 0$. 
which means that in each event $\xi^O_{1}$ or $\xi^O_2$  appears at random.
%
%  It's important to stress that as the individual states
%  $xi^O_s \ne\xi^O_i$, despite that they can give the same $Q_O$ value
%  in the separate event.
%
% Now we consider the proof of the pure/mixture indistinguishability
% by $O$ for an arbitrary $a_i$. 
To reproduce the correct expectation values $\bar{Q}^n_O$,
the corresponding $\xi^O_i$ probabilities are $P'_i=|a_i|^2$.
Then for any $\xi^{MS}$ there is always the probabilistic ensemble 
 $W^O=W^O_{mix}$  of (\ref {AA55}) which describes
 consistently the properties of MS  restricted state. 
% Hence $MS\rightarrow O$
% map is  in general ambiguous and
%  breaks  $Q_O$ symmetry of $\xi^{MS}$.
%
One must also show that no 
 alternative nonprobabilistic solution $\xi^O_f$ 
for MS restriction exists.
% which in distinction from $W^O$
% doesn't change from event to event
% for the pure MS state. 
  $\xi^O_f$ should have   
 correct the expectation values $\bar{Q}^n_O$,
%       $\bar{Q}^'_O=0$,
the only nonstochastic
state $\xi^O_f \in \Omega_O$ which obeys this conditions is
$\xi^O_f=\sum|a_i|^2\xi^O_i$, the analog of Breuer state $R_O$.
%
% From the obvious reasons
% there no other $Q_O$ eigenstates $\xi^O_c$ with  $q^O \neq^O_{1,2}$,  
%  and there are also no state $\xi^O$ in $\Omega_O$
Such $\xi^O_f$ should be a nonlocalized state on $Q_O$ for which 
 $q^O_1 \le \tilde{q}^O \le q^O_2$, i.e. $\tilde{q}^O$ is uncertain.
 As was explained above, in that case
%  such $\xi^O$ should be an eigenstate of some IT 
some observable  $B'\in \cal U_O$ should exist
  which eigenvalue or an expectation value  $\bar{B}'$ for $\xi^O_f$
 would reveal its difference from $\xi^O_i$ states and
 demonstrate   $\tilde{q}^O$  genuine uncertainty,
i.e the simultaneous presence in several   $Q_O$ points.
Yet $\cal{U}_O$ doesn't include any   other observables beside $Q_O$,
thereon $\xi^O_f$ isn't feasible as $O$ individual state 
and  $\xi^O_{s}$ is the only suitable solution. 

The same arguments are applicable for an arbitrary $a_i$,
 the only difference  is that another IT $B^a \in \{B^{MS}\}$ is
 involved in the derivation; eventually the same unique solution $\xi^O_s$ of 
  (\ref {AA55}) exists, and $W^O$ with the 
given $|a_i|^2$  describes its ensemble.
 In general 
he nonexistence of any $O$ individual
states $\xi^O_f \ne \xi^O_i$ directly follows from
$\cal U_O$ elementarity, because as was shown, all individual 
states should be the eigenstates of some observables and in this
case it can be only $Q_O$.
% But the obtained solution 
%  $\xi^O_f$ is equivalent to $\varphi^O$, which means that $\xi^O_f$
% ensemble  is indistinguishable from $W^O_{mix}$ ensemble,  
%  as was already noted above, there is 
% no other observable $B' \in \cal U_O$
%  which value can indicate their difference
% due to the superposition in MS state.
% $\xi^O_i$ should appear at random  with some probabilities $P'_i$.
%  defined by $\bar{Q}^O$.
% It turns out that
% for such $\varphi^O$ content
%
% Hence  MS restricted AIS $\xi^O$ coincides with
% some $\xi^O_i$ in the individual event,
%  consequently as AIS $\xi^O_f=\xi^O_s$ which corresponds to 
% the randomness of outcomes observed by $O$. 
%  statistically described by $W^O_{mix}$.
Consequently in Algebraic QM  
$\xi^{MS}\rightarrow \xi^O$ inference map is stochastic 
and results in  the  subjective state collapse  observed by $O$.
 MS AIS can be expressed as the doublet
 $\xi^{MS}$, $\xi^O$, which corresponds to DSF dynamical and 
information components. Correspondingly MS evolution operator $\hat{Z}(t)$
is also the multiplet (doublet) which includes the unitary component
for $\xi^{MS}$ and nonunitary stochastic one for $\xi^O$. 
 In the regarded situation
 $O'$ doesn't interact with MS,  and so
analogously to DSF formalism  $O'$ information 
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
of chap.2.  Note that the same results can be obtained if in place 
of $\cal U_O$
we shall exploit MS restriction to $\cal U_R$ - the subalgebra 
of all $O$ observables.

For illustration 
let's regard the opposite hypothesis: namely that $O$ 
 percepts pure MS state $\xi^{MS}$ as $Q_o$ 'pointer superposition'
and it differs from $O$ perception of regarded MS mixture;
the analogous situations were widely discussed 
(Von Neumann,1934; Wigner,1961).
 If to assume that $O$ perception doesn't violate QM laws,
   in particular, that all $O$ observations are related to some 
  $O$ observables, then
 it should be some $O$ observable $G_O$ -
'the number of $Q_O$ peaks'  for which $\bar{G}_O=1,2$ for MS
mixture and pure state correspondingly. But 
such $O$ observable doesn't exist not only for $\cal U_O$, but
  even for $\cal U_R$ subalgebra. 

% 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.
%5$\xi^O_i \in \Omega _R$, 
%$\xi^O_s$ Since 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

% 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 symmetric individual states $\xi^{MS}_s$ possess the
reflection symmetry $Q_O \to  - Q_O$, but no
  $O$ states $\xi^O_i$  have that property.
%  In particular, the
% states $\xi_i^O$ doesn't possess the symmetry of the incoming measured state
% $\Psi^{in}_{MS}$.
In $C^*$algebras formalism of Quantum Field Theory
 such symmetry reduction results in
 the phenomena of Spontaneous Symmetry Breaking; in particular, it leads to 
the randomness of outcomes for some models of  measurements 
in the collective
systems (Guilini,1996; Mayburov,1998).
The  self-description approach  permits to extend
such randomness mechanism on the finite systems
measurements. Its most outstanding feature is the simultaneous
coexistence of two individual states $\xi^{MS}, \xi^O_i$ which
 seems contradictive.
From  the mathematical point of view the
 Algebraic QM  contains the generic structure -
the restricted AIS set $\Omega^I_R$ defined on  the
elementary subalgebra
 $\cal {U}^I_R \in \cal U$ 
(in our model $ \cal {U}^I_R=\cal{U}_O$).
  $\Omega^I_R$ extremal points are 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. Eventually MS, $O$ states
are defined on different algebras which corresponds
to observations in $O,O'$ RFs,
 and so they are unitarily nonequivalent. 
% it  results in the drastic difference of that states.


% Plainly, in this theory analogously to DSF
% 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.
% Note that the formal  nonseparability of pure/mixed states  
% for the restricted or incomplete set of observables
% was considered extensively earlier (Busch,1996),
% but the problem was to prove it for the feasible measuring devices.

% 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.


If to analyze obtained results  in the Information-Theoretical framework,
 remind that
 the difference between the pure and
mixed MS states reflected by the expectation values
$B$  of (\ref {AA5}) and other IT observables.
Therefore $O$ possible observation of S pure/mixed
 states difference would mean
that $O$ can acquire the information on   $B$  expectation value.
 But $B \notin \cal {U}_R$ 
% and isn't correlated with any $O$ observable
% alike  $Q_O$ and $Q$ of S are correlated, 
because of it S,$O$ IT correlations
 are unobservable from inside by $O$ (Mittelstaedt,1998).
 %The obtained $\xi^O_s$ describes in fact the upper limit 
% of S information available for $O$ in a single event.
Therefore the information  which describes the difference of the pure
and mixed S states principally can't be transferred in such MS scheme
  from $S$ to IGUS $O$ 
for the consequent processing.
It corresponds with 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).
 Our  calculations can be regarded as  the kind of 'no-go'
theorem which prohibit   such superpositions observations.
In this context the obtained $\xi^O_s$ state describes
      the upper limit of S information available to $O$. 
Note that the formal  impossibility  of pure/mixed states discrimination  
for the restricted  set of observables
was considered  earlier (Busch,1996),
but its applicability to
 the feasible measuring schemes wasn't  proved.


 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.
As was shown 
the formalism of Algebraic QM  extracts PB
 $\xi^O_i \sim |O_i\rangle $ in $\cal {H}_O$ 
even without  the account of E decoherent interaction,
but only from  the uniqueness $O$ subalgebra $\cal {U}_O$
for a given S, $O$ interaction. 
 The $O$-E decoherence, in fact, only duplicates this effect
        resulting  in the optimal case in the same PB solution.  
 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,
and the final $O$ states structure is analogous to the obtained above.
%
%  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.
%

%As was argued,  the formalism of standard QM doesn't permit 
%  to calculate   the states restriction for an arbitrary systems   
%   from its first principles. Yet if our MS its individual  
%   state $\Psi_{MS}$ restriction   should be some $O$ individual state  

As was shown in chap. 2 standard QM doesn't contain the unambiguous
      restriction  ansatz for individual states,
therefore some   additional assumptions are necessary
to introduce it, if one prefers to work inside its framework.
From our analysis it's sensible to assume that   
      for MS $\to O$ restriction   
   to any $\Psi_{MS}$ corresponds some $O$ individual state in any event.  
Undoubtly any pure state $\psi^O_j \in \cal H_O$ is $O$ individual
   state with some real property.
 Meanwhile  from the arguments given in the beginning of the
chapter, Breuer state $R_O$ of (\ref {AA4}) is the convex state and so
 can't be $O$ individual  state. But this is the only nonprobablistic  
solution which can dispatch the correct $Q_O, Q'_O$ expectation values.
Hence they can be restrored only by the random mixture of several
$\psi^O_j$ states, and the stochastic ensemble of
$|O_i\rangle$ states with the probability $P_i=|a_i|^2$ is the only solution.
Note that it
reproduces   the results of  our DSF ansatz which earlier was obtained 
phenomenlogically. Eventually under the simple minimal assumptions
the effect of state collapse can be obtained in the standard QM framework,
yet  it seems that Algebraic QM on whole proposes more consistent
 approach to the problem.

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). 
 Its possible application to the systems self-description 
and the measurement from inside  deserves detailed investigation,
here only few  notices are given.   
If one assumes that in this framework the
measurement restrictions  also are defined by the set of
$O$ local  observables -
i.e propositions available for $O$ observations, then 
 the results can be comprehended easily for the case when
only single observable $Q_O$ is available.
In this case the corresponding lattice $\cal L^O$ of propositions includes only 
$Q_O$ projectors   set - $\{P^O_i\}$,
 (together with $\ominus,I$) and hence is atomic
$\sigma$-continuous Boolean lattice. Then it follows that  
the set of restricted  $O$ individual states $\eta^O_i$ which
corresponds to this lattice
 is isomorphic
to classical $O$ states set $\{ \xi^O_i\}$ (Jauch, 1968). 
For any other state $\eta^O_R$ no proposition from $\cal L^O$
can be put in correspondence, and therefore such state  isn't feasible.
In general the search of alternative and 
   more simple formalism of self-description  
 deserves the thorough investigation. Really the only principal 
feature of Algebraic QM, which was used in our approach, 
is the strict correspondence of the restricted states and the set
of local observables, but it doesn't   demand that such set 
should constitute  Segal algebra.



Despite of the acknowledged achievements of Algebraic QM
   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, this problem
discussed thoroughly in (Mayburov, 2005). 
This question is  important by itself 
and  can be essential for our formalism feasibility (Primas, 1983).
% 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.



% 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 $O$.
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.
  At the phenomenological level the appearance of the
 state collapse described by  DSF which exploits 
 the doublet states $\Phi$ ansatz,
where one of its components $\phi^I$ corresponds to $O$
 subjective information.
%  for which 
%  the stochastic  outcomes in the quantum measurements  appear
%  in the consistent way.
  Algebraic QM  
presents the additional arguments in favor of this
 collapse mechanism; its formalism reflects the strict correspondence
between the states and observables -  the principal distinction of
QM from Classical Waves theory.
 In our approach
 $O$  subjective state after the measurements
 is  defined on the set of $O$ internal observables  $\cal {U}_O$.
 Yet the observable $B$, which value characterizes the 
pure/mixed states difference, or more precisely the class of IT observables
 doesn't belongs to $\cal {U}_O$, hence $O$ internal state
can't react on this difference.
%  which defines the multiplet states set analogous to $\Phi$. 
% 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 this paper Algebraic QM  was applied for 
the simple 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 $O$
 states $R_O(n)$ calculations.
Note also that no compelling mathematical arguments proves that  
 for the individual states $R_O$ 
 ansatz in standard QM framework
 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.
 Algebraic QM  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).



% Consequently, it's possible that Information Theory rules
% by itself can lead
%  to the subjective state collapse in the quantum measurements.


 
%  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.  

On the whole
the decoherence of macroscopic objects is a very important effect,
under the realistic conditions the rate of E atoms interactions with 
macroscopic detector D is very high, and because of  it
 in a very short time $t_d$
S,D partial state $\rho_{SD}=Tr_E \rho_{SDE}$ 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 further development of decoherence approach
which accounts the observer was  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 
  the 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 $O'$,
However the system S,$O$,E 
 is still in the pure state even at $t\gg t_d$, and there is  IT observable
$B$ analogous to (\ref {AA5})
  which proves it.  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,
hence 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 leads to the results which are  equivalent to
 Existential Interpretation of QM.


From the formal point of view
 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 quantum measurement
process was discussed first by London and Bauer (London,1939). They supposed
that  Observer Consciousness (OC) due to 'introspection
action' violates in fact Schr$\ddot{o}$dinger equation  
and results in the state reduction.
%         This idea was seriously  criticized  by Wigner \cite {Wig}.
In our  theory  the process  of $O$  perception      
 doesn't violate  Schrodinger evolution 
relative to external RF $O'$.
%   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.
Basing on  obtained results, our approach to it 
formulated here as the semiqualitative  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 Algebraic QM  approach, IM assumes that
 $O$ perception affected only by 
$O$ internal states defined on $O$ observables subalgebra $\cal {U}_O$. 
For  $O$ perception   the  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 $J^O$ 
corresponding  to $J^O_i=q^O_i$ 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 the  human perception. 
Impression $J^O$ 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, it's sensible to 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 -
the information pattern  $J^O=\{q_l^{sup}(n)\}$
which is expressed as some finite sequence of real numbers. 
 In Algebraic formalism the corresponding
  $O$ subjective information - impression in the individual
event  is equal to  $J^O(n)=q^O_i$ 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  $J^O(n)=q^O_i$.
In Algebraic QM  framework they corresponds 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
Eventually  in this approach IM doesn't need to exploit
  Von Neuman psychophysical parallelism hypothesis
(Von Neumann,1933).
%  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 general  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  the self-description
 restrictions  on the information acquisition
are shown to be important in Quantum 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 evidences
% 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
%\\
\\
(2003) Ph.Blanchard, P.Lugiewicz, R.Olkiewicz Phys. Lett A314, 29
\\
% \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 Universal 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}
 (2004) S.Mayburov Int. J.Theor. Phys,(to appear);
 Quant-ph 0205024; Quant-ph 0212099
%\\
\\
 (2005) S.Mayburov, Proc. of 'Foundations of Physics and Computations'
conference, Udine, 2004 (to appear )
\\
% \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-ph0506065.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
%\\
\\
(2003) Ph.Blanchard, P.Lugiewicz, R.Olkiewicz Phys. Lett A314, 29
\\
% \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 Universal 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}
 (2004) S.Mayburov Int. J.Theor. Phys,(to appear);
 Quant-ph 0205024; Quant-ph 0212099
%\\
\\
 (2005) S.Mayburov, Proc. of 'Foundations of Physics and Computations'
conference, Udine, 2004 (to appear )
\\
% \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}