Determination of quantum instrument parameters for a Stern-Gerlach non-ideal device
Ioan Sturzu
The paper identifies and determines some parameters with experimental relevance, which could describe the influence of the non-ideality for the measurement of the intrinsic spin of an atom, using a real Stern-Gerlach device.
quant-ph0205167.tex

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\title{Determination of quantum instrument parameters for a Stern-Gerlach
non-ideal device}
\author{Ioan Sturzu}
\affiliation{"Transilvania" University, Department of Physics\\
Eroilor 29, Brasov, Romania; email: sturzu@unitbv.ro}
\keywords{non-ideal measurements, quantum instrument}
\pacs{03.65.Ta, 02.60.Ed }

\begin{abstract}
The paper identifies and determines some parameters with experimental
relevance, which could describe the influence of the non-ideality for the
measurement of the intrinsic spin of an atom, using a real Stern-Gerlach
device.
\end{abstract}

\date[Date text]{date}
\maketitle

%\preprint{HEP/123-qed}

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%\received[Received text]{date}
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%\tableofcontents

\section{Introduction}

Stern-Gerlach experiment is usually considered as a prototype for a
filter-measurement, i.e. a measurement with two possible answers, which
practically consist in the disparition of the measured object or in its
preservation. For a filter-measurement it corresponds a projector $\hat{E}%
(B) $ from the spectral family of the selfadjoint operator associated to the
measured physical quantity:
\[
\hat{A}=\int \lambda \hat{E}(d\lambda )
\]%
Actually, these filter-measurement are only idealizations for those realized
using real devices. A major epistemological problem for the Quantum Theory
is the influence of the non-ideality of the device on the mathematical
formalism. This aim is mathematically-coherent followed in Operational
Quantum Physics (OQP) \cite{Davies,Schroeck}. Here one will use OQP
formalism in order to identify and determine some parameters with
experimental relevance, which would describe the influence of the
non-ideality for the measurement of the intrinsic spin of an atom, using a
real Stern-Gerlach device. A similar method was used for imperfect
polarizers in the paper \cite{Formanek}.

If $\hat{\rho}_{s}$ and $\hat{\rho}_{b}$ are state-operators for the atom
and, respectively for the device at the initial moment $t_{0}$ when the
measurement did not start yet, the state-operator for the compound system
can be written in a factorized form:%
\[
\hat{\rho}(t_{0})=\hat{\rho}_{s}(t_{0})\otimes \hat{\rho}_{b}(t_{0})
\]%
Consider that the measurement starts at the subsequent moment $t_{i}$ and
and ends at the moment $t_{f},$ so the compound system Hamiltonian is:
\[
\widehat{H}\left( t\right) =%
\begin{array}{l}
\widehat{H}_{0},\quad t<t_{i},\quad t>t_{f} \\
\widehat{H}_{0}+\widehat{H}_{int}\left( t\right) ,\quad t\in \left[
t_{i},t_{f}\right]%
\end{array}%
\]%
The compound system state for $t>$ $t_{f}$ is not anymore factorizable, so
it is considered that the state of the atom is given by the operator
obtained after the partial trace operation on the device degrees of freedom:%
\[
\hat{\rho}_{s}(t)=\mathrm{tr}_{b}(\hat{\rho}(t))
\]%
where:
\[
\hat{\rho}(t)=\hat{U}(t,t_{0})\hat{\rho}_{s}(t_{0})\otimes \hat{\rho}%
_{b}(t_{0})\hat{U}^{\dagger }(t,t_{0})
\]%
\[
\hat{U}(t,t_{0})=\mathbb{T}\exp \left( -\frac{i}{\hbar }\int_{t_{0}}^{t}dt^{%
\prime }\widehat{H}\left( t^{\prime }\right) \right)
\]%
The result of this operation is that the state $\hat{\rho}_{s}(t)$ is not
anymore pure.

Generally, any mixed state can be decompose in a convex combination of pure
statistical operators. An orthogonal decomposition is given by the spectral
theorem for the selfadjoit operator which is $\hat{\rho}_{s}(t)$. Apart from
some degeneracies of the spectral values, it is also unique \cite%
{Beltrametti}. The condition which an interaction Hamiltonian at the
measurement process for the a physical magnitude has to obey, is that $\hat{%
\rho}_{s}(t)$ has to be convexly orthogonally decomposable after a complete
set of spectral operators (projectors). In the Stern - Gerlach case, if one
works with the usually bidimensional 1/2 spin representation (with the base
given by vectors $|\uparrow \rangle $ and $|\downarrow \rangle $), the
alluded condition is:
\[
\hat{\rho}_{s}(t)=w_{\uparrow }(t)\hat{\rho}_{\uparrow }(t)+(1-w_{\uparrow
}(t))\hat{\rho}_{\downarrow }(t)
\]

\section{\protect\bigskip Description of the non-ideal measurement}

Transformation:
\[
\hat{\rho}_{s}(t_{0})\mapsto \hat{\rho}_{s}(t),\quad t>t_{f}
\]%
is a completely positive map, which in the bidimensional case we refer, can
be written in the form \cite{Kraus}:
\begin{equation}
\hat{\rho}_{s}(t)=\sum\limits_{m=\uparrow ,\downarrow }\widehat{\mathcal{A}}%
_{m}^{\dagger }\widehat{\rho }(t_{0})\widehat{\mathcal{A}}_{m}
\label{sgdinstr}
\end{equation}%
where the operators $\left\{ \widehat{\mathcal{A}}_{m}\right\} _{m=\uparrow
,\downarrow }$ fulfill the condition:
\begin{equation}
\sum\limits_{m=\uparrow ,\downarrow }\widehat{\mathcal{A}}_{m}^{\dagger }%
\widehat{\mathcal{A}}_{m}=\hat{1}  \label{normeff}
\end{equation}%
In OQP one postulates the existence of another type of transformation:%
\begin{equation}
\hat{\rho}_{s}(t)=\left( \widehat{\mathcal{A}}_{m}^{\dagger }\widehat{\rho }%
(t_{0})\widehat{\mathcal{A}}_{m}\right) _{\uparrow ,\downarrow }
\label{sgdinstrsel}
\end{equation}%
which correspond to the situation when the measurement was done, and the
result is already known as one of the values $\uparrow $ or $\downarrow $. (%
\ref{sgdinstr}) is named non-selective measurement, while (\ref{sgdinstrsel}%
) are selective measurements.

The two terms of (\ref{normeff}):%
\begin{equation}
\hat{F}_{m}=\widehat{\mathcal{A}}_{m}^{\dagger }\widehat{\mathcal{A}}_{m}
\label{sgeff}
\end{equation}%
are positive operators, which generalize the projectors on the pure states $%
|\uparrow \rangle $ and $|\downarrow \rangle .$ In OPQ they are named
\textit{effects}, while the applications (\ref{sgdinstr}) and (\ref%
{sgdinstrsel}) are the duals of some applications on the observables, which
are named \textit{quantum instruments.}

Of course, in the bidimensional case, the state and the instrument operators
can be decomposed after the Pauli matrices basis:
\[
\left\{ \hat{1},\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z}\right\}
\]%
where we shall use the usual vectorial notation $\overrightarrow{\hat{\sigma}%
}=\left( \hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z}\right) $:
\[
\hat{\rho}=\frac{1}{2}\left( \hat{1}+\vec{r}\cdot \overrightarrow{\hat{\sigma%
}}\right) ,\quad \vec{r}=\vec{r}^{\ast },\quad \left| \vec{r}\right| \leq 1
\]%
\begin{equation}
\widehat{\mathcal{A}}=\alpha \hat{1}+\vec{\beta}\cdot \overrightarrow{\hat{%
\sigma}},\quad \alpha =\alpha _{1}+i\alpha _{2}\in \mathbb{C},\mathbb{\quad
\vec{\beta}=\vec{\beta}}_{1}\mathbb{+}i\mathbb{\vec{\beta}}_{2}
\label{sginstr}
\end{equation}%
where the spin index was omitted.

The condition that a filter Stern-Gerlach experiment has to fulfill is to
yield a spatial separation for the two spinorial components, and to place in
the zone where one of the components is localized an absorbing screen. In
this case, the atoms which are passed after the device can be associated
with the other spinorial component.

A major problem is to determine the coefficients which define the
instrument. From (\ref{normeff}) one gets:
\[
\left( \left( \alpha \hat{1}+\vec{\beta}\cdot \overrightarrow{\hat{\sigma}}%
\right) \cdot \left( \alpha ^{\ast }\hat{1}+\vec{\beta}^{\ast }\cdot
\overrightarrow{\hat{\sigma}}\right) \right) _{\uparrow }+\left(
.....\right) _{\downarrow }=
\]%
\begin{equation}
=\left( \left( \left| \alpha \right| ^{2}+\left| \vec{\beta}\right|
^{2}\right) \left( \hat{1}+\vec{\xi}\cdot \overrightarrow{\hat{\sigma}}%
\right) \right) _{\uparrow }+\left( .....\right) _{\downarrow }=\hat{1}
\label{condflt}
\end{equation}%
where:
\[
\vec{\xi}_{\uparrow (\downarrow )}=\frac{\alpha \vec{\beta}^{\ast }+\alpha
^{\ast }\vec{\beta}+i\left( \vec{\beta}\times \vec{\beta}^{\ast }\right) }{%
\left| \alpha \right| ^{2}+\left| \vec{\beta}\right| ^{2}}|_{\uparrow
(\downarrow )}
\]%
is a real vector. From (\ref{condflt}) it comes:
\[
\left( \left| \alpha \right| ^{2}+\left| \vec{\beta}\right| ^{2}\right)
_{\uparrow }+\left( .....\right) _{\downarrow }=1
\]%
\begin{equation}
\left( \alpha \vec{\beta}^{\ast }+\alpha ^{\ast }\vec{\beta}+i\left( \vec{%
\beta}\times \vec{\beta}^{\ast }\right) \right) _{\uparrow }+\left(
.....\right) _{\downarrow }=0  \label{sgnorm}
\end{equation}

The state transformation (\ref{sgdinstr}) for the initial state $\frac{1}{2}%
\left( \hat{1}+\vec{k}\cdot \overrightarrow{\hat{\sigma}}\right) $ gives:
\[
\hat{\rho}_{s}(\vec{k})=\frac{1}{2}\{\hat{1}[\left| \alpha \right|
^{2}+\left| \vec{\beta}\right| ^{2}+\left( \alpha \vec{\beta}^{\ast }+\alpha
^{\ast }\vec{\beta}\right) \vec{k}+
\]%
\begin{equation}
+i\left( \vec{\beta}\times \vec{\beta}^{\ast }\right) \vec{k}]+%
\overrightarrow{\hat{\sigma}}[\left( \alpha \vec{\beta}^{\ast }+\alpha
^{\ast }\vec{\beta}\right) +\left( \left| \alpha \right| ^{2}-\left| \vec{%
\beta}\right| ^{2}\right) \vec{k}+
\end{equation}

\[
+i\left( \alpha \vec{\beta}^{\ast }-\alpha ^{\ast }\vec{\beta}\right) \times
\vec{k}+i\left( \vec{\beta}^{\ast }\times \vec{\beta}\right) ]\}_{\uparrow
}+\left( .....\right) _{\downarrow }
\]%
while the state transformations\ (\ref{sgdinstrsel}) gives only one of the
upwards terms. For the last measurements, the probability of obtaining the
answer ''yes'' is given by:%
\begin{equation}
f(\vec{k})_{\uparrow (\downarrow )}=\mathrm{tr}\left( \hat{\rho}_{s}(\vec{k}%
)_{\uparrow (\downarrow )}\right) =[\left| \alpha \right| ^{2}+\left| \vec{%
\beta}\right| ^{2}+  \label{sgrotinstr}
\end{equation}%
\[
+\left( \alpha \vec{\beta}^{\ast }+\alpha ^{\ast }\vec{\beta}\right) \vec{k}%
+i\left( \vec{\beta}\times \vec{\beta}^{\ast }\right) \vec{k}]_{\uparrow
(\downarrow )}
\]%
In (\ref{sgrotinstr}), using (\ref{sgnorm}) one gets:%
\[
\hat{\rho}_{s}(\vec{k})=\frac{1}{2}\{\hat{1}+\overrightarrow{\hat{\sigma}}%
[[\left( \left| \alpha \right| ^{2}-\left| \vec{\beta}\right| ^{2}\right)
\vec{k}+
\]%
\[
+i\left( \alpha \vec{\beta}^{\ast }-\alpha ^{\ast }\vec{\beta}\right) \times
\vec{k}+2i\left( \vec{\beta}^{\ast }\times \vec{\beta}\right) ]_{\uparrow
}+\left( ...\right) _{\downarrow }]
\]%
where one can introduce the notation:%
\begin{equation}
\vec{A}(\vec{k})=[\left( \left| \alpha \right| ^{2}-\left| \vec{\beta}%
\right| ^{2}\right) \vec{k}+i\left( \alpha \vec{\beta}^{\ast }-\alpha ^{\ast
}\vec{\beta}\right) \times \vec{k}+  \label{sgvectnsel}
\end{equation}%
\[
+2i\left( \vec{\beta}^{\ast }\times \vec{\beta}\right) ]_{\uparrow }+\left(
.....\right) _{\downarrow }
\]%
so the state is given by:%
\begin{equation}
\hat{\rho}_{s}(\vec{k})=\frac{1}{2}\left( \hat{1}+\overrightarrow{\hat{\sigma%
}}\cdot \vec{A}(\vec{k})\right)
\end{equation}

If the device is rotated with the angle $\phi $ around the $\vec{n}$ axis,
the quantum instrument has to fulfill a covariance condition\cite%
{Davies,Schroeck,Formanek}:
\begin{equation}
\widehat{\mathcal{A}}(\theta )_{m}=\hat{U}^{\dagger }(\theta )\widehat{%
\mathcal{A}}_{m}\hat{U}(\theta ),\quad \hat{U}(\theta )=\exp \left( -i\theta
\frac{\vec{n}\cdot \widehat{\vec{S}}}{\hbar }\right)   \label{sgcov}
\end{equation}%
where $\hat{U}(\theta )$ is the symmetry group of the rotations, whose
generator is the total kinetic moment operator, which can be identified here
with the intrinsic spin kinetic moment.
\[
\hat{U}(\phi )=\cos \left( \frac{\ \phi }{2}\right) \hat{1}+i\sin \left(
\frac{\phi }{2}\right) \vec{n}\cdot \overrightarrow{\hat{\sigma}}
\]%
The operators (\ref{sgdinstr}) are cf. (\ref{sgcov}):
\begin{equation}
\widehat{\mathcal{A}}_{\uparrow (\downarrow )}(\phi )=[\alpha \hat{1}+\cos
\phi \left( \vec{\beta}\cdot \overrightarrow{\hat{\sigma}}\right) +
\label{sgrotop}
\end{equation}%
\begin{equation}
+\sin \phi \left( \left( \vec{n}\times \vec{\beta}\right) \cdot
\overrightarrow{\hat{\sigma}}\right) +2\sin ^{2}\frac{\phi }{2}\left( \vec{n}%
\cdot \vec{\beta}\right) \left( \vec{n}\cdot \overrightarrow{\hat{\sigma}}%
\right) ]_{\uparrow (\downarrow )}
\end{equation}%
Introducing the notation:

\[
\vec{B}(\phi )=\cos \phi \vec{\beta}+\sin \phi \left( \vec{n}\times \vec{%
\beta}\right) +2\sin ^{2}\frac{\phi }{2}\left( \vec{n}\cdot \vec{\beta}%
\right) \vec{n}
\]%
one has:%
\begin{equation}
\widehat{\mathcal{A}}_{\uparrow (\downarrow )}(\phi )=\left( \alpha \hat{1}+%
\vec{B}(\phi )\cdot \overrightarrow{\hat{\sigma}}\right) _{\uparrow
(\downarrow )}
\end{equation}%
If one makes first the nonselective measurement (\ref{sgdinstr}) followed by
a selective one using a rotated instrument $\phi $\ around the axis $\vec{n}$%
, the final states are given by:%
\[
\hat{\rho}_{s}(\vec{k},\phi )=\frac{1}{2}\{\hat{1}[\left| \alpha \right|
^{2}+\left| \vec{B}(\phi )\right| ^{2}+\left( \alpha \vec{B}(\phi )^{\ast
}+\alpha ^{\ast }\vec{B}(\phi )\right) \vec{A}(\vec{k})+
\]%
\[
+i\left( \vec{B}(\phi )\times \vec{B}(\phi )^{\ast }\right) \vec{A}(\vec{k}%
)]+\overrightarrow{\hat{\sigma}}[\left( \alpha \vec{B}(\phi )^{\ast }+\alpha
^{\ast }\vec{B}(\phi )\right) +
\]

\[
+\left( \left| \alpha \right| ^{2}-\left| \vec{B}(\phi )\right| ^{2}\right)
\vec{A}(\vec{k})+i\left( \alpha \vec{B}(\phi )^{\ast }-\alpha ^{\ast }\vec{B}%
(\phi )\right) \times
\]%
\[
\times \vec{A}(\vec{k})+i\left( \vec{B}(\phi )^{\ast }\times \vec{B}(\phi
)\right) ]\}_{\uparrow (\downarrow )}
\]%
and the probabilities by:%
\begin{equation}
f(\vec{k},\phi )_{\uparrow (\downarrow )}=\mathrm{tr}\left( \hat{\rho}_{s}(%
\vec{k},\phi )_{\uparrow (\downarrow )}\right) =  \label{sgrotinstrrep}
\end{equation}%
\[
=\{\left| \alpha \right| ^{2}+\left| \vec{B}(\phi )\right| ^{2}+[\alpha \vec{%
B}(\phi )^{\ast }+\alpha ^{\ast }\vec{B}(\phi )\vec{A}(\vec{k})+
\]%
\[
+i\left( \vec{B}(\phi )\times \vec{B}(\phi )^{\ast }\right) \vec{A}(\vec{k}%
)]\}_{\uparrow (\downarrow )}
\]%
An interesting case is that of the angle $\phi _{m}=m\cdot \frac{2\pi }{3}$ (%
$m=0,1,2$) rotation around $\vec{n}=\frac{1}{\sqrt{3}}(1,1,1),$ which yield
even circular permutations for the 3 axis:
\[
\vec{B}_{0}=\vec{\beta}
\]%
\begin{equation}
\vec{B}_{1}=\beta _{z}\vec{e}_{x}+\beta _{x}\vec{e}_{y}+\beta _{y}\vec{e}_{z}
\label{sgrotvect}
\end{equation}%
\[
\vec{B}_{2}=\beta _{y}\vec{e}_{x}+\beta _{z}\vec{e}_{y}+\beta _{x}\vec{e}%
_{z}
\]%
Due to problems like determination or compatibility for the system of
equations (which, actually are quadratic ones) the general problem is very
difficult to solve. However, a straightforward simplification is given by
the small non-ideality case.

\section{Results for the small non-ideality case}

In the small non-ideality case, the system of equations becomes linear, and
the determination and compatibility conditions can be studied, either by
direct inspection of the equations, or using a specialized software for
algebraic computations. In the ideal case, the measurement of the $Oz$
component of the intrinsic spin is given by the projectors:
\[
\hat{E}_{\uparrow (\downarrow )}=\frac{1}{2}\left( \hat{1}\pm \hat{\sigma}%
_{z}\right)
\]%
The parameters:
\[
\alpha _{\uparrow (\downarrow )}=\frac{1}{2}
\]%
\begin{equation}
\vec{\beta}_{\uparrow (\downarrow )}=\pm \frac{1}{2}\vec{e}_{z}
\label{sgideal}
\end{equation}%
are the most simple solution for the equation:%
\begin{equation}
\hat{E}_{\uparrow (\downarrow )}=\left( \left| \alpha \right| ^{2}+\left|
\vec{\beta}\right| ^{2}\right) \hat{1}+\{[\alpha \vec{\beta}^{\ast }+\alpha
^{\ast }\vec{\beta}+
\end{equation}%
\begin{equation}
+i\left( \vec{\beta}\times \vec{\beta}^{\ast }\right) ]\overrightarrow{\hat{%
\sigma}}\}_{\uparrow (\downarrow )}
\end{equation}%
so, one will take as a small non-ideality measurement that given by the
parameters:
\begin{equation}
\alpha _{\uparrow (\downarrow )}=\frac{1}{2}+\eta a_{\uparrow (\downarrow )}
\label{sgsmid1}
\end{equation}%
\begin{equation}
\vec{\beta}_{\uparrow (\downarrow )}=\pm \frac{1}{2}\vec{e}_{z}+\eta \vec{b}%
_{\uparrow (\downarrow )}  \label{sgsmid2}
\end{equation}%
where $\eta $ is a small real parameter, while $a_{\uparrow (\downarrow )}$
and $\vec{b}_{\uparrow (\downarrow )}$ are complex magnitudes. Introducing (%
\ref{sgsmid1}) and (\ref{sgsmid2}) in (\ref{sgrotinstr}) and (\ref%
{sgrotinstrrep}), and developing in the first order of parameter $\eta $,
one can obtain the predictions $f$ for the probabilities of obtaining the
positive result at various experimental setups. Of course, one will consider
only the deviations from ideality, noted with $\delta $ $f$. Taking the
spherical angle representation for the unit vector $\vec{k}=\sin \theta \cos
\varphi \cdot \vec{e}_{x}+\sin \theta \sin \varphi \cdot \vec{e}_{y}+\cos
\theta \cdot \vec{e}_{z}$:
\[
\delta f(\vec{k})_{\uparrow (\downarrow )}=[a_{r}+b_{rz}+\left(
a_{r}+b_{rz}\right) \sin \theta \cos \varphi +
\]%
\begin{equation}
+\left( b_{rx}-b_{iy}\right) \sin \theta \sin \varphi +\left(
b_{ry}+b_{ix}\right) \cos \theta ]_{\uparrow (\downarrow )}  \label{sgpredic}
\end{equation}%
Fitting the experimental data for the deviations $\delta f(\vec{k}%
)_{\uparrow (\downarrow )}$ with the function:%
\begin{equation}
\delta \mathtt{f}(\vec{k})_{\uparrow (\downarrow )}=[c_{0}+c_{1}\sin \theta
\cos \varphi +  \label{sgexper}
\end{equation}%
\[
+c_{2}\sin \theta \sin \varphi +c_{3}\cos \theta ]_{\uparrow (\downarrow )}
\]%
from (\ref{sgnorm}) and identification between (\ref{sgpredic}) and (\ref%
{sgexper}) one gets the following equations for the parameters $a_{\uparrow
(\downarrow )}$ and $\vec{b}_{\uparrow (\downarrow )}$ (a total of 16 real
parameters):

\begin{center}
\[
\left( a_{r}+b_{rz}\right) _{\uparrow }+\left( a_{r}+b_{rz}\right)
_{\downarrow }=0
\]%
\[
\left( b_{rx}-b_{iy}\right) _{\uparrow }+\left( b_{rx}-b_{iy}\right)
_{\downarrow }=0
\]%
\[
\left( b_{ry}+b_{ix}\right) _{\uparrow }+\left( b_{ry}+b_{ix}\right)
_{\downarrow }=0
\]%
\[
a_{r\uparrow }+b_{rz\uparrow }=c_{0\uparrow }
\]%
\[
b_{rx\uparrow }-b_{iy\uparrow }=c_{2\uparrow }
\]%
\[
b_{ry\uparrow }+b_{ix\uparrow }=c_{3\uparrow }
\]
\end{center}

An important\ test for OQP is the compatibility between predictions and
experimental results, which can be expressed by the confidence parameters of
the fitting operation. In the case of an acceptable compatibility, the
upward system of equations is enough for the identification of the
parameters for the effects (\ref{sgeff}), which are given, in the small
non-ideality case, by:%
\[
\hat{F}_{\uparrow (\downarrow )}=\frac{1}{2}\left( \hat{1}\pm \hat{\sigma}%
_{z}\right) +\eta \left( \left( a_{r}+b_{rz}\right) \hat{1}+\left(
\begin{array}{c}
a_{r}+b_{rz} \\
b_{rx}-b_{iy} \\
b_{ry}+b_{ix}%
\end{array}%
\right) \overrightarrow{\hat{\sigma}}\right) _{\uparrow (\downarrow )}
\]%
\[
\hat{F}_{\uparrow (\downarrow )}=\frac{1}{2}\left( \hat{1}\pm \hat{\sigma}%
_{z}\right) \pm \eta \left( c_{0}\hat{1}+\left(
\begin{array}{c}
c_{0} \\
c_{2} \\
c_{3}%
\end{array}%
\right) \overrightarrow{\hat{\sigma}}\right)
\]

In the case of simple measurements , the effects are sufficient for the
determination of any probability. For successive measurements (non-selective
measurement, followed by a selective one), one needs supplementary
informations about the parameters. Equating the predictions given by (\ref%
{sgrotinstr}) with the corresponding experimental values, for the 3
rotations (\ref{sgrotvect}) one gets other 7 independent equations:%
\[
\left( a_{r}+b_{rz}+b_{iy}\right) _{\uparrow }+b_{iy\downarrow
}=c_{0z\uparrow }
\]%
\[
\left( a_{r}+b_{rz}-b_{ix}\right) _{\uparrow }-b_{ix\downarrow
}=c_{0x\uparrow }
\]%
\[
a_{r\uparrow }+b_{ry\uparrow }=c_{0y\uparrow }
\]%
\[
b_{iy\uparrow }+b_{iy\downarrow }=2c_{0z\uparrow }
\]%
\[
b_{ix\uparrow }+b_{ix\downarrow }=2c_{0x\uparrow }
\]%
\[
\left( a_{r}-b_{rz}\right) _{\uparrow }+\left( a_{r}-b_{rz}\right)
_{\downarrow }=2c_{1z\uparrow }
\]%
\[
\left( a_{i}-b_{iz}\right) _{\uparrow }+\left( a_{i}-b_{iz}\right)
_{\downarrow }=2c_{1x\uparrow }
\]%
where the second index from the experimental parameters stay for the main
axis of the device after the rotation (\ref{sgrotvect}), which, again, is
expected to be sufficient for the description of the two-step successive
measurements. For more-step successive measurements one can proceed in the
same manner.

\begin{thebibliography}{9}
\bibitem{Davies} E.B. Davies, \emph{Quantum Theory of Open Systems}
(Academic, London, 1976).

\bibitem{Schroeck} F. Schroek, \emph{Quantum Mechanics on Phase Space},
(Kluwer, Dordrecht, 1996).

\bibitem{Formanek} J. Formanek, Found. Phys. \textbf{25} (1995) 851.

\bibitem{Beltrametti} E. Beltrametti and G. Cassinelli, \emph{The Logic of
Quantum Mechanics} (Addison--Wesley, Reading, Massachusetts 1981).

\bibitem{Kraus} K. Kraus, Annals Phys. \textbf{64} (1977) 311.
\end{thebibliography}

\end{document}

quant-ph0205167.extracted_bib
\begin{thebibliography}{9}
\bibitem{Davies} E.B. Davies, \emph{Quantum Theory of Open Systems}
(Academic, London, 1976).

\bibitem{Schroeck} F. Schroek, \emph{Quantum Mechanics on Phase Space},
(Kluwer, Dordrecht, 1996).

\bibitem{Formanek} J. Formanek, Found. Phys. \textbf{25} (1995) 851.

\bibitem{Beltrametti} E. Beltrametti and G. Cassinelli, \emph{The Logic of
Quantum Mechanics} (Addison--Wesley, Reading, Massachusetts 1981).

\bibitem{Kraus} K. Kraus, Annals Phys. \textbf{64} (1977) 311.
\end{thebibliography}