A Geometrical Approach to Quantum Estimation Theory

Keiji Matsumoto

CHAPTER: INTRODUCTION

§ THE PURPOSES OF THE THESIS

The most important purpose of the thesis is
pursuit for  the geometrical theory of
statistical  estimation of the quantum mechanical state.

In the  statistical theory of the probability distribution,
S. Amari and his coworkers have formulated
a geometrical theory,  so called information geometry,
and successfully applied to various statistical problems [3].
H. Nagaoka, one of Amari's  coworkers, pointed out that
the duality between $e$- and $m$-connections sits at the heart of
the information geometry.
He also formulated quantum information geometry by use of
this idea of the mutually dual connections, and applied to
characterization of the model which has the efficient estimator

However, his information geometry does not give any insight
into the problem
of determination of the attainable CR (Cramer-Rao) type bound,
nor characterize the condition for the SLD CR bound,
which is of special interest for some reasons,
is attainable.
After all, Nagaoka's geometry deal with
the global properties of the model,
while the attainable CR type bound is related local properties
of the model.
Hence, it semms that another geometric structure
is needed for the thorough description
of the quantum estimation theory.

On the other hand, Berry's phase, discovered by M. V. Berry
as the  non-integrable phase factor in the adiabatic motion[4],
is naturally understood as a curvature of
the natural connection in the principle fiber bundle
over the space of pure states whose structure group is $U(1)$ [1].
In 1986, Uhlmann generalized the geometry to the space of mixed states.
Though the physical meaning of Uhlmann's geometry
is not known, Berry's phase is
applied to the explanation of various phenomina[28].

The author conjectures
that Berry-Uhlmann's curvature
reflects local properties of the model.
To prove the statement, it is needed to determine
the attainable CR type bound for arbitrary models,
which is far out of our reach.
However, for the 2-parameter pure state model,
the author  presents complete answer to the problem.
In addition, for  the faithful state model and pure state model,
it is shown that SLD CR bound is attainable if and only if
the model is free of Berry-Uhlmann curvature.

Furthermore, we try a kind of unification of the two geometries,
Nagaoka's information geometry and Uhlmann's parallelism.

Second most important purpose is the determination of
the attainable CR type bound, and the development  of new methodology
for that purpose. In the pure state models,
this purpose is achieved, though not completely,
to large extent.
A new methodology direct approach is formulated and
successfully applied to the 2-parameter pure state model,
the coherent model.
For the arbitrary pure state model,
calculated is the attainable CR type bound whose weight matrix is
the SLD Fisher information matrix.
Looking back, no one has ever determined
the attainable CR type bound for this wide range of models.

Third, some considerations about such physical problems
as the uncertainty principle are done.
The time-energy uncertainty is nicely formulated
as a hypothesis test, and the position-momentum uncertainty
as a estimation of the mean values
of the position and the momentum operators.
In this formulation of the position-momentum uncertainty ,
it is shown that the mean values
of the position and the momentum operators
are simultaneously estimated up to arbitrarily high efficiency,
if the particle is prepared carefully.

§ ORGANIZATION OF THE THESIS

The thesis is divided into three parts:
the faithful model theory, the pure state theory, the general model theory;
The reason for this organization is that
the extent of the achievement of the purposes is
different in these three cases.

Before these three parts,
chapter 2 gives brief review of the estimation theory of
probability distributions, the quantum mechanical theory of the measurement,
and the quantum estimation theory,
and chapter 3 gives
the geometrical and the estimation-theoretical
framework, commonly used in any of the following three parts.

CHAPTER: PRELIMINARIES

In this chapter, statistical estimation theory and
quantum mechanics are reviewed briefly.
For the thorough description of estimation theory,
see, for example,
Ref. [17].
As for quantum mechanics, see Ref. [26], or other text books.
Some basic concepts in quantum estimation theory are  introduced also.

§ CLASSICAL ESTIMATION THEORY

Throughout the thesis, the usual estimation theory,
or the estimation theory of the probability distribution
is called classical estimation theory,
in the sense that the theory is not quantum mechanical.

The theme of the classical estimation theory is
identification of the probability distribution
from which the $N$ data $x_1, x_2, ..., x_N $ is produced.
Usually, the probability distribution is assumed to be a member of
a model, or a family

\begin{eqnarray}
{\cal M}=\{p(x|\theta)|\theta\in\Theta\subset\R^m\}
\nonumber
\end{eqnarray}

of probability distributions
and that the finite dimensional parameter $\theta\in\Theta\subset\R^m$
is to be estimated statistically.

Unbiased estimator $\hat\theta=\hat\theta(x_1,x_2,...,x_N)$
of parameter $\theta$ the estimate which satisfies

\begin{eqnarray}
&\equiv &
\int dx_1dx_2,...,dx_N
\hat\theta(x_1,x_2,...,x_N)\prod_{i=1}^{N} p(x_i|\theta)\nonumber\\
\label{eqn:clunbiased}
\end{eqnarray}

that is, the estimate which gives the true value of parameter
in average.
For the technical reason, we also define
locally unbiased estimator $\hat\theta$ at $\theta_0$

\begin{eqnarray}
&&\left. \partial_j E_{\theta}[\hat\theta^i]\right|_{\theta=\theta_0}
\nonumber
\end{eqnarray}

The estimator is unbiased iff it is locally unbiased
at every $\theta\in\Theta$.

For the variance of locally unbiased estimator at $\theta$, the following
theorem gives bound of efficiency of the estimation.

(Cramer-Rao inequality)
For any locally unbiased estimate $\hat\theta$ at $\theta$,

\begin{eqnarray}
V_{\theta}[\hat\theta]\ge \frac{1}{N} J^{-1}(\theta).
\label{eqn:clcrN}
\end{eqnarray}

Here, $N$ is the number of the data and $J(\theta)$
is $m\times m$ real symmetric matrix defined by

\begin{eqnarray}
\left[\int dx p(x|\theta)
\partial_i \ln p(x|\theta)\partial_j \ln p(x|\theta)
\right],
\label{eqn:clbest}
\end{eqnarray}

where $\partial_i$ stands for $\partial/\partial\theta^i$.

The best estimator, or the estimator $\hat\theta$ satisfying
$(\ref{eqn:clcr})$, is given by

\begin{eqnarray}
\hat\theta^i(x_1,...,x_N)&=&\hat\theta_{(\theta)}^i(x_1,...,x_N)\nonumber\\
&\equiv& \theta^i+
\sum_{j=1}^m [J^{-1}(\theta)]^{ij}\partial_j \ln\,\prod_{k=1}^N p(x_k|\theta).
\nonumber
\end{eqnarray}

$J(\theta)$ is called Fisher information matrix,
because the larger the $J(\theta)$ is, the more precise estimate
can be done with the same number of data. Metaphorically speaking,
we obtain as much information as $J(\theta)$ per data.
Actually, as easily seen by putting $N=1$ in Cramer-Rao (CR) inequality,
we can obtain $J(\theta)$ as the minimum variance of
locally unbiased estimate when only one data is given.

\begin{eqnarray}
V_{\theta}[\hat\theta]\ge  J^{-1}(\theta).
\label{eqn:clcr}
\end{eqnarray}

The trouble with  the $(\ref{eqn:clcr})$ is
that the best estimator $\hat\theta_{(\theta)}$ is
dependent on the true value of the parameter $\theta$,
which is unknown to us.
When the true value of parameter is not $\theta_0$,
the estimate $\hat\theta_{(\theta_0)}$ is not even locally unbiased
at the true value of the parameter.
To avoid this dilemma,
we give up with the unbiased estimator,
and focus on the consistent estimator defined by

\begin{eqnarray}
\lim_{N\rightarrow\infty} E_{\theta}[\hat\theta(x_1,x_2,...,x_N)]=\theta.
\nonumber
\end{eqnarray}

For the consistent estimator, we also have the following theorem.

If the estimator is consistent,

\begin{eqnarray}
\frac{1}{N} J^{-1}(\theta)+o\left(\frac{1}{N}\right)
\label{eqn:clcscr}
\end{eqnarray}

holds true.

The maximum likelihood estimator $\hat\theta_{MLE}$,
which is defined by,

\begin{eqnarray}
\hat\theta_{MLE}\equiv
\argmax \left\{
\left.\sum_{j=1}^N \ln p(x_i|\theta)\:
\right|\:\theta\in\Theta\subset\R^m
\right\}.
\nonumber
\end{eqnarray}

is consistent and achieves the equality in $(\ref{eqn:clcscr})$.

Notice that to obtain $\hat\theta_{MLE}$,
we need no information about the true value of the parameter beforehand.
Hence, the Fisher information matrix is
a good measure of the efficiency of the optimal consistent estimator.

§  QUANTUM MECHANICS AND MEASUREMENT THEORY

In the quantum mechanics,
the state of physical system is described
by the density operator $\rho$, which is
a non-negative Hermitian operator whose trace is equal to 1,
in a separable Hilbert space ${\cal H}$, whose dimension is
denoted by $d\leq\infty$ hereafter.
We denote by ${\cal P}({\cal H})$
the space of  density operators in ${\cal H}$,
by  ${\cal P}_r ({\cal H})$
the the space of density operators whose rank is $r$,
and by ${\cal P}_+({\cal H})$ the space of strictly positive definite
density operators.
${\cal P}({\cal H})$, ${\cal P}_r ({\cal H})$, and  ${\cal P}_+({\cal H})$
are often simply denoted by  ${\cal P}$,
${\cal P}_r $, and  ${\cal P}_+$, respectively .

Let $\Omega$ be a space of all possible outcomes of
an experiment,
and $\sigma(\Omega)$ be a $\sigma$- field
in $\Omega$.
When the density operator of the system is $\rho$,
the probability  that the data $\omega \in \Omega$
lies in $B\in\sigma(\Omega)$ writes

\begin{eqnarray}
{\rm Pr}\{ \omega \in B|\rho \} =\tr \rho M(B),
\label{eqn:pdm}
\end{eqnarray}

by use of the map  $M$ from $\sigma(\Omega)$
to  nonnegative Hermitian operator which satisfies

\begin{eqnarray}
&&M(\phi)=O, M(\Omega)=I,\nonumber\\
&&M(\bigcup_{i=1}^{\infty} B_i),
\;\;(B_i\cap B_j=\phi,i\neq j),
\label{eqn:maxiom}
\end{eqnarray}

so that $(\ref{eqn:pdm})$ define a probability measure
(see Ref.[11], p.53
and Ref.[12], p.50).
We call the map $M$ the measurement, because
there always exist an physical experiment corresponds to the map $M$
which satisfies $(\ref{eqn:maxiom})$

§ UNBIASED ESTIMATOR IN QUANTUM ESTIMATION THEORY

The purpose of the quantum estimation is to
identify the density operator of the given physical system
from the data obtained by the appropriately designed experiment.
For simplicity, we usually assume that
the density operator is a member of a model, or a manifold of
${\cal M}
=\{\rho(\theta)|\theta\in \Theta \subset {\bf R}^m\}\subset{\cal P}$,
and that
the parameter $\theta$ is to be estimated statistically.
For example, ${\cal M}$ is
the set of spin states with given wave function part and unknown spin part.

To estimate the parameter, we performs an experiment
to obtain the data $\omega$ by which we calculate
an estimator $\hat\theta$ by estimator  $\hat\theta(\omega)$.
A pair $(\hat\theta,M,\Omega)$ of a space $\Omega$ of data, a measurement $M$,
and an estimator $\hat\theta(*)$ is also called an estimator.
The expectation of $f(\omega)$ with respect to the probability measure
$(\ref{eqn:pdm})$ is denoted by $E_\theta[f(\omega)|M]$.

We have seen that the locally unbiased estimator played
a key role in classical estimation theory.
Hence we try to keep the same track also
in the quantum estimation theory.

The estimator $(\hat\theta,M,\Omega)$ is said to be  unbiased

\begin{eqnarray}
\label{eqn:unbiased}
\end{eqnarray}

holds for all $\theta\in\Theta$.
If $(\ref{eqn:unbiased})$ and

\begin{eqnarray}
\partial_i E_\theta[\theta^j(\omega)|M]=\delta^j_i\:(i,j=1,...,m)
\nonumber
\end{eqnarray}

hold at a
particular  $\theta$, $(\hat\theta,M,\Omega)$ is called locally unbiased
at $\theta$.

It is also reasonable to include calculation of the estimate from data
into the process of measurement.
In this point of view,
the estimate $\hat\theta$ itself is produced
by the measurement process, and the data space $\Omega$
is $\R^m$.
Therefore, by the term `estimator'
we also  mean  the measurement which takes value on  $\R^m$.
In this case,
the unbiased estimator is a measurement
which takes value on  $\R^m$ which satisfies

\begin{eqnarray}
\label{eqn:unbiased2}
\end{eqnarray}

holds for all $\theta\in\Theta$,

\begin{eqnarray}
E_\theta[M]\equiv\int \hat\theta \tr\rho(\theta)M(d\hat\theta ).
\nonumber
\end{eqnarray}

If $(\ref{eqn:unbiased2})$ and

\begin{eqnarray}
[\partial_i E_\theta[M]]^j=\delta^j_i\:(i,j=1,...,m)
\nonumber
\end{eqnarray}

hold at a
particular  $\theta$, $M$ is called locally unbiased
at $\theta$.
We denote by $V_{\theta}[M]$ the
covariance matrix of the estimator $M$
when the true value of the parameter is $\theta$.

Obviously, these two definition of the estimator
in some situations, we prefer the former to the latter,
while in other situations the latter is
preferred for the sake of simplicity.

CHAPTER: CONCEPTUAL FRAMEWORK

§ HORIZONTAL LIFT AND SLD

In this thesis, except for the pure state model theory,
$d\equiv\dim{\cal H}$ is
assumed to be finite  for the sake of clarity.
The author believe the essence of the discussion will not be damaged
by this restriction.

Let ${\cal W}_r$ be
the space of  $d\times r$ complex and full-rank matrix $W$ such that

\begin{eqnarray}
\tr WW^*=1,
\nonumber
\end{eqnarray}

${\cal P}_r$ the space of density operators whose rank is $r$,
$\pi$ the map from ${\cal W}_r$
to ${\cal P}_r$ such that

\begin{eqnarray}
\rho=\pi(W)\equiv WW^*.
\nonumber
\end{eqnarray}

$\pi(WU)$ is identical to $\pi(W)$ iff $U$ is a $r \times r$
unitary matrix,
it is natural to see the  space
${\cal W}_r$ as the total space of
the principal fiber bundle
with the base space ${\cal P}_r$
and the ${\it structure group}$ $U(d)$ [15].
One possible physical interpretation of
$W$ is a representation of a state vector
$|W\rgl$ in a bigger
Hilbert space ${\cal H}\otimes{\cal H}'$.
Here, the dimension of ${\cal H}'$ is $r$ and
the operation $\pi(*)$ corresponds to  the partial trace of
$|W\rgl\lgl W|$ over ${\cal H}'$.

In this section, basic concepts about
the tangent bundle ${\cal T}({\cal W}_r)$ over
${\cal W}_r$, which is a real manifold
with the real parameter
are introduced.

The matrix representation
${\bf M}(\partial/\partial \zeta^i)$ of
the tangent vector $\partial/\partial \zeta^i$
(throughout the thesis, the tangent vector is understood as
the differential operator) is
a $d\times r$ complex matrix  such that

\begin{eqnarray}
{\bf M}\left(\frac{\partial}{\partial\zeta^i}\right)\equiv
2\frac{\partial}{\partial\zeta^i} W(\zeta).
\nonumber
\end{eqnarray}

The real span of the  matrix representations

\begin{eqnarray}
\{X\: |\: {\rm Re}\,\tr XW^*(\zeta)=0,X\in M(d,r,\C)\}.
\nonumber
\end{eqnarray}

We introduce the inner product $\lgll *,*\rgll_W$ to
${\cal T}({\cal W}_r)$ such that,

\begin{eqnarray}
&\equiv&\sum_{i,j}({\rm Re}({\bf M}\hat{X})_{ij}{\rm Re}({\bf M}\hat{Y})_{ij}
+{\rm Im}({\bf M}\hat{X})_{ij}{\rm Im}({\bf M}\hat{X})_{ij})
\nonumber\\
&=&{\rm Re}\,\tr (\,({\bf M}\hat{X})({\bf M}\hat{Y})^*\,),
\nonumber
\end{eqnarray}

which is invariant under the action of
$U\in U(r)$ to the matrix representation  of
the tangent vector from right side,

\begin{eqnarray}
&&U\in U(r),\nonumber\\
&&\lgll\, ({\bf M}\hat{X}) U, ({\bf M}\hat{Y})U\rgll_{WU}
=\lgll {\bf M}\hat{X}, {\bf M}\hat{Y}\rgll_W
\nonumber
\end{eqnarray}

Let us decompose ${\cal T}_W({\cal W}_r)$ into
the direct sum of the horizontal subspace ${\cal LS}_W$ and
the vertical subspace ${\cal K}_W$
where ${\cal LS}_W$ is defined by

\begin{eqnarray}
{\cal LS}_W\equiv
\{\hat{X}\: |\: W^*({\bf M}\hat{X})=({\bf M}\hat{X})^*W\},
\label{eqn:defLS2}
\end{eqnarray}

${\cal K}_W$ is the orthogonal complement space
${\cal T}_W({\cal W}_r)\ominus{\cal LS}_W$
with respect to the inner product $\lgll *,* \rgll_W$.
$\hat{X}\in{\cal K}_W$ satisfies

\begin{eqnarray}
({\bf M}\hat{X})W^*+ W({\bf M}\hat{X})^*
\label{eqn:defk2}
\end{eqnarray}

or its equivalence,

\begin{eqnarray}
\pi_*(\hat{X})=0,
\label{eqn:pi*=0}
\end{eqnarray}

where $\pi_*$ is the differential map of $\pi$.
A member of the horizontal subspace and the vertical subspace are
called a horizontal vector and vertical vector,
The image of $\hat{X}\in{\cal T}_W({\cal W}_r)$ by the projection
onto the horizontal subspace ${\cal LS}_W$
is called the horizontal component,
while the image by the projection onto
the vertical  subspace  ${\cal K}_W$ is called
vertical component.

The horizontal lift $\hlift_W$
is a mapping from  ${\cal T}_{\pi(W)}({\cal P}_r) $
to ${\cal T}_W({\cal W}_r) $ such that

\begin{eqnarray}
\pi_*\left(\,\hlift_W(X)\,\right)=X\nonumber,\\
\hlift_W(X)\in {\cal LS}_W.
\nonumber
\end{eqnarray}

Because of the following theorem,
the matrix representation of the horizontal lift $\pi_*(\hlift_W(X))$
is a representation of
the tangent vector $X\in{\cal T}_{\pi(W)}({\cal P}_r) $.

$\hlift_W$ is a isomorphism from
${\cal T}_{\pi(W)}({\cal P}_r)$ to ${\cal LS}_W$.

First, notice that
for any $\hat{Y}\in{\cal T}_{W}({\cal W}_r)$,
$W+ \varepsilon {\bf M}(\hat{Y})$ also is a member
of ${\cal W}_r$ if $\varepsilon$ is small enough.
Therefore, we have

\begin{eqnarray}
\pi_*(\, {\cal T}_{W}({\cal W}_r)\,)\subset{\cal T}_{\pi(W)}({\cal P}_r).
\nonumber
\end{eqnarray}

Second, we prove that
the map $\pi_*|_{{\cal LS}_W}$ is a one to one map
from ${\cal LS}_W$ to ${\cal T}_{\pi(W)}({\cal P}_r)$.
For that, it is sufficient to prove that
$\hat{X}=0$ when  $\hat{X}\in {\cal K}_W$.
This statement is proved to be true because
${\cal K}_W$ is orthogonal to ${\cal LS}_W$.

Finally, checking the dimension of ${\cal T}_{\pi(W)}({\cal P}_r)$
is equal to ${\cal LS}_W$, we have the theorem.

Using the horizontal lift, the inner product $\lgl *,*\rgl$
in ${\cal T}({\cal P}_r)$
is deduced from
$\lgll *,*\rgll$:

\begin{eqnarray}
\lgl X,Y \rgl_{\pi(W)}=\llgll  \hlift_W(X),\hlift_W(Y) \rrgll_W.
\nonumber
\end{eqnarray}

The horizontal lift $\hlift$ satisfies
the following equality so that the above definition of
the inner product $\lgl *,*\rgl$ is self-consistent:

\begin{eqnarray}
\llgll \hlift_W X,\hlift_{W} Y \rrgll
=\llgll \hlift_{WU} X ,\hlift_{WU} Y \rrgll,\:
(U,U'\in U(n)\;).
\nonumber
\end{eqnarray}

The symmetrized logarithmic derivative (SLD, in short)
of $X\in {\cal T}_{\pi(W)}({\cal P}_r)$ is
the Hermitian operator $L^S_X$ in ${\cal H}$ defined by the equation

\begin{eqnarray}
\label{eqn:defsldx}
\end{eqnarray}

where $\theta$ is a real parameter which is assigned to
a member of  ${\cal P}_r $.
Iff the density operator is strictly positive,
SLD is uniquely defined  by $(\ref{eqn:defsldx})$.
$L^S_{\partial/\partial \theta^i}$ is often denoted simply by

SLD is closely related to the horizontal lift by the following equation:

\begin{eqnarray}
{\bf M}\left(\hlift_W{X}\right)=L^S_X W.
\label{eqn:hsld}
\end{eqnarray}

§ DEFINITION OF UHLMANN'S PARALLELISM

Berry's phase, by far confirmed by several experiments,
is a holonomy of a natural connection in the line bundle
over the space of pure states [1][4].
In 1986, Uhlmann generalized the theory to include
mixed states in the Hilbert space ${\cal H}$
Throughout this chapter, for the sake of clarity,
$d\equiv\dim{\cal H}$ is
assumed to be finite.
For notational simplicity, the argument $\theta$ is omitted,
as long as the omission is not misleading.

Define a horizontal lift of
a curve $C=\{\rho(t)|t\in {\bf R}\}$ in ${\cal P}_r$
as a curve $C_h=\{W(t)|t\in {\bf R}\}$ in ${\cal W}_r$
which satisfies
$C=\pi(C_h)$ and

\begin{eqnarray}
\frac{dW(t)}{dt}
={\bf M}\left(\hlift_{W(t)}\left(\frac{d}{dt}\right)\, \right).
\label{horizontal}
\end{eqnarray}

Then, the relative phase factor (RPF)
between  $\rho(t_0)$ and $\rho(t_1)$
along the curve $C$
the unitary matrix $U$ which satisfies
the equation

\begin{eqnarray}
W(t_1)=\hat{W}_1 U,
\nonumber
\end{eqnarray}

satisfies $\rho(t_1)=\pi(\hat{W}_1)$ and

\begin{eqnarray}
\hat{W}_1^*W(t_0)=W^*(t_0)\hat{W}_1.
\nonumber
\end{eqnarray}

RPF is said to vanish when it is equal to the identity.

§ RPF FOR INFINITESIMAL LOOP

The RPF for the infinitesimal loop

\begin{eqnarray}
\begin{array}{ccc}
(\theta^1,..., \theta^i,..., \theta^j+d\theta^j,..., \theta^m)
(\theta^1,..., \theta^i+d\theta^i,..., \theta^j+d\theta^j,...., \theta^m)
\\
\downarrow& &\uparrow
\\
\theta=(\theta^1,..., \theta^i,..., \theta^j,...., \theta^m)
(\theta^1,..., \theta^i+d\theta^i,..., \theta^j,...., \theta^m)
\end{array}
\label{loop}
\end{eqnarray}

is calculated up to the second order of $d\theta$
by expanding the solution of
the equation $(\ref{horizontal})$
to that order:

\begin{eqnarray}
I+\frac{1}{2}W^{-1}F_{ij}W\;d\theta^i d\theta^j
=(\partial_i L^S_j-\partial_j L^S_i)
\label{eqn:Fij}
\end{eqnarray}

Note that $F_{ij}$ is a representation of  the curvature form,
and that RPF  for any closed loop vanishes  iff  $F_{ij}$ is zero
at any point in ${\cal M}$.

§ THE SLD CRAMER-RAO INEQUALITY

In parallel with the classical estimation theory,
in the quantum estimation theory,
we have the following SLD CR inequality,
which is proved for the faithful state model
by Helstrom [10][11],
for the pure state model by Fujiwara and Nagaoka
and for the general case by Fujiwara and Matsumoto [6]:

\begin{eqnarray}
V_{\theta}[\hat\theta(\omega)\, |\, M]\geq(J^S(\theta))^{-1},
\label{eqn:genCR}
\end{eqnarray}

i.e., $V_{\theta}[\hat\theta(\omega)\, |\, M]-(J^S(\theta))^{-1}$
is non-negative definite.
Here $ V_{\theta}[\hat\theta(\omega)\, |\, M]$ is a covariance matrix of
an unbiased estimator  $(\hat\theta, M, \Omega)$,
and $J^S(\theta)$ is called SLD Fisher information matrix,
and is defined by

\begin{eqnarray}
J^S(\theta)&\equiv &
\left[\left\lgl \frac{\partial}{\partial \theta^i},
\frac{\partial}{\partial \theta^j}\right\rgl_{\rho(\theta)} \right],
\nonumber\\
&=& [{\rm Re}\, \tr \rho(\theta)L^S_i(\theta))L^S_j(\theta)],
\label{eqn:defgenfshr}
\end{eqnarray}

which is nothing but
the metric tensor of the inner product $\lgl *, *\rgl$.

The inequality $(\ref{eqn:genCR})$ is of special interest,
$J^{S-1}$, called SLD CR bound,
is the one of the best bounds in the sense
explained later.

To prove the inequality $(\ref{eqn:genCR})$,
we set some notations, and present some lemmas
For unbiased estimator $(\hat\theta, M, \Omega)$,
we define  the notation ${\bf M}$ as follows,

\begin{eqnarray}
{\bf M}^i(\hat\theta,\, M,\,W)
\equiv\int (\hat\theta^i(\omega)-\theta^i) M(d\omega)W.
\nonumber
\end{eqnarray}

$Z[\hat\theta, M]$ is $m\times m$ matrix defined by

\begin{eqnarray}
Z[\hat\theta, M]=
\left[\; \tr\:{\bf M}^i(\hat\theta,\, M,\,W)\,
({\bf M}^j(\hat\theta,\, M,\,W)\,)^*\; \right]
\label{eqn:defz}
\end{eqnarray}

Following two inequalities are valid:

\begin{equation}
V[\hat\theta\, |\, M]\ge {\rm Re}\, Z[\hat\theta,M].
\label{eqn:genVZ}
\end{equation}

\begin{equation}
V[\hat\theta\, |\, M] \ge Z[\hat\theta,M].
\label{eqn:genVZ2}
\end{equation}

\begin{eqnarray}
{\rm Re}\, Z[\hat\theta,M]\ge J^{S-1}
\label{eqn:genZJ}
\end{eqnarray}

holds. The equality is valid iff

\begin{eqnarray}
{\bf M}^j(\hat\theta,\, M,\,W)
&=&\sum_k [J^{S-1}]^{j,k} L^S_k W,
\nonumber\\
&=&\sum_k [J^{S-1}]^{j,k}
{\bf M}\left(\hlift_W \frac{\partial}{\partial \theta^k} \right),
\label{eqn:genxx=J}
\end{eqnarray}

They are proved in almost the same manner as the strictly positive case
(see Ref.[12] p.88 and p.274 respectively).
Lemma $\ref{lemma:genVZ}$ and $\ref{lemma:genZJ}$ lead to
the SLD CR inequality $(\ref{eqn:genCR})$.

SLD Fisher information gives a lower bound of covariance matrix
of an unbiased measurement, i.e.,
$(\ref{eqn:genCR})$ holds true.

The SLD CR bound $(\ref{eqn:genCR})$ is the best bound
in the following sense.

Letting $A$ be a real hermitian matrix which is larger than $J^{S-1}$,
that is, $A>J^{S-1}$, there exists
such an unbiased estimator $(\hat\theta, M, \Omega)$
that $V[\hat\theta\, |\, M]$ is not smaller than $A$.

Let ${\bf v}\in{\R}^m$ be the real vector
such that

\begin{eqnarray}
\exists C\subset \R\;\;\;
\frac{\partial}{\partial \theta^i}
\int_C \tr \rho(\theta) E_{\bf v}(dt)\neq 0,
\: (i=1,...,m),
\label{eqn:dpneq0}
\end{eqnarray}

where $E_{\bf v}$ is a projection valued measure obtained
by the spectral decomposition
of $\sum_{i,j} v_j(\theta^j I + [J^{S-1}]^{i,j} L_j^S)$,
where $I$ and $v_i$ denotes identity and $i$th component of ${\bf v}$.

The condition $(\ref{eqn:dpneq0})$ is
implies the existence of  an estimator  $\hat\theta_{\bf v}(\omega)$
which makes
the triplet $(\hat\theta_{\bf v}, E_{\bf v}, \Omega)$
locally unbiased at $\theta$.
For that triplet $(\hat\theta, E_{\bf v}, \Omega)$,
we have

\begin{eqnarray}
{\bf v}^T \left(V[\hat\theta\, |\, E_{\bf v}]-J^{S-1}\right){\bf v}=0.
\nonumber
\end{eqnarray}

If $\varepsilon$ is small enough,
for any real vector ${\bf v}\in{\R}^m$,
${\bf v}+\varepsilon{\bf v}_0 $ satisfies
the condition $(\ref{eqn:dpneq0})$,
or its equivalence,

\begin{eqnarray}
({\bf v}+\varepsilon{\bf v}_0 )^T
\left(V[\hat\theta_{{\bf v}+\varepsilon{\bf v}_0 }\, |
\, E_{{\bf v}+\varepsilon{\bf v}_0 }]-J^{S-1}\right)
({\bf v}+\varepsilon{\bf v}_0) =0.
\label{eqn:vevjve}
\end{eqnarray}

Let us assume that there exists a real matrix $A$ which satisfies

\begin{eqnarray}
V[\hat\theta|M]\ge A >J^{S-1}
\nonumber
\end{eqnarray}

for any unbiased estimator.
Then, by virtue of $(\ref{eqn:vevjve})$, we have
for any real vector ${\bf v}_0\in{\R}^m$
and enough small $\varepsilon$,

\begin{eqnarray}
({\bf v}+\varepsilon{\bf v}_0 )^T
\left(A-J^{S-1}\right)
({\bf v}+\varepsilon{\bf v}_0) =0,
\nonumber
\end{eqnarray}

whose second derivative with respect to $\varepsilon$

\begin{eqnarray}
{\bf v}_0 ^T\left(A-J^{S-1}\right){\bf v}_0 =0.
\nonumber
\end{eqnarray}

Because  ${\bf v}_0\in{\R}^m$ is arbitrary, we have

\begin{eqnarray}
\nonumber
\end{eqnarray}

which contradicts with the assumption $A>J^{S-1}$.

If the model ${\cal M}$ has only one parameter,
the equality in $(\ref{eqn:genCR})$ is achievable.

Let $E$ and $\hat\theta(*)$ be a projection valued measurement
and an estimator which satisfies,

\begin{eqnarray}
\left[\int(\hat\theta(\omega)-\theta)M(d\omega)\right]^i
=\sum_j \left[J^{S-1}\right]^{ji}L^S_j.
\nonumber
\end{eqnarray}

Then, the triplet $(\hat\theta, E, \Omega)$ is locally unbiased at $\theta$
and attains SLD CR bound.

implies the statistical significance of the natural metric
$\rgl *,*\lgl$.
A possible geometrical interpretation of $(\ref{th:gen1pCR})$:
the closer
two states
$\rho(t)$ and $\rho(t+dt)$
the harder
it is
to distinguish $\rho(t)$
from $\rho(t+dt)$.

The SLD CR inequality $(\ref{eqn:genCR})$ looks quite analogical to
CR inequality in classical estimation theory.
However, as will be  found out later,
the equality is not generally attainable.

§ THE ATTAINABLE CRAMER-RAO TYPE BOUND

In the previous section,
SLD CR bound  is proved to be the best bound.
However, as will be turned out, this best bound
is attainable only in the special cases, that is,
the case when the model is locally quasi-classical.
In general case, therefore, we must give  up
to find a tight lower bound of covariance matrix
in the form of the matrix inequality.
Instead, we determine the region ${\cal V}_{\theta}({\cal M})$
of the map $V[*]$
from unbiased estimators
to $m \times m$ real positive symmetric matrices
(so far as no confusion is expected, we write ${\cal V}$
for ${\cal V}_{\theta}({\cal M})$).
Especially, the boundary $bd {\cal V}$ is of  interest,
because  ${\cal V}$ is convex as in the following lemma.

${\cal V}$ is convex.

In this proof, we define the estimator to be
the  measurement which takes value in $\R^m$.
Let $M$ and $M'$ be an unbiased estimator.

\begin{eqnarray}
\lambda V[M] + (1-\lambda) V[M']=V[\lambda M+(1-\lambda)M']
\nonumber
\end{eqnarray}

holds true and $\lambda M+(1-\lambda)M'$ is an unbiased estimator,
we have the lemma.

If a matrix $V$ is a member of ${\cal V}$,
$V+V_0$ is also a member of  ${\cal V}$
for any arbitrary nonnegative real symmetric matrix $V_0$.

Let $(\hat\theta, M, \Omega)$ be a locally unbiased estimator
whose covariance matrix is $V$,
and define $\varepsilon(\omega)$ by

\begin{eqnarray}
\varepsilon(\omega)\equiv
\nonumber
\end{eqnarray}

$\hat\theta^*(\omega)=\hat\theta(\omega)+\varepsilon(\omega)$ is
also a locally unbiased estimator, and its covariance matrix
is  equal to $V+V_0$.

To obtain $bd {\cal V}$,
the following procedure is used in this thesis.
Define an inner product of two real symmetric matrices
$A$ and $B$ by $\Tr AB$.
Then, the set $\{V| \Tr VG=const.\}$ is a hyperplain
perpendicular to the vector $G$.
Because of lemmas $\ref{lemma:vconvex}$-$\ref{lemma:vunbound}$,
$bd{\cal V}$ is the collection of all the matrices
$V\in{\cal V}$
which  achieve the minimum of $\Tr GV$
for a certain symmetric real nonnegative definite matrix $G$.

However, when the model has too many parameters,
the dimension of the space $Sym(m)$ of real symmetric matrices
is so  large that  ${\cal V}_{\theta}({\cal M})$
is extremely hard to  determine.
In such cases, we calculate

\begin{eqnarray}
\CR (G,\theta,{\cal M})\equiv
\min\{\Tr GV\: |\: V\in{\cal V}_{\theta}({\cal M})\}
\label{eqn:crtype}
\end{eqnarray}

for an arbitrary nonnegative symmetric real matrix $G$,
and call it the attainable CR (Cramer-Rao) type bound.
The matrix $G$ is called weight matrix.
Often, we drop $G$,  $\theta$ and/or ${\cal M}$
when no confusion is expected.
If $\CR (G,\theta,{\cal M})$ is smaller than
$\CR (G,\theta',{\cal M}')$ for any weight matrix $G$,
the ${\cal V}_{\theta}({\cal M})$ of
is located in the `lower part' of $Sym(m)$
compared with that of ${\cal V}_{\theta'}({\cal M}')$.

To make the estimational meaning of $(\ref{eqn:crtype})$ clear, let us
consider a  diagonal weight matrix $G=diag(g_1,g_2...,g_m)$.
Letting $v_{ii}$ be the $(i,i)$-th component of $V[M]$,

\begin{eqnarray}
\Tr GV[M]=\sum_i g_i v_{ii},
\nonumber
\end{eqnarray}

is the weighed sum of the variances of the estimations of
the parameters $\theta^i\,(i=1,...m)$.
If the accuracy of estimation of, for example, the parameter $\theta^1$
is required
more than other parameters,
then $g_1$ is set larger than any other $g_i$,
and the estimator which minimize $\sum_i g_i v_{ii}$ is to be used.

§ A CONJECTURE ABOUT THE QUANTUM MLE

In the classical estimation theory,
the inverse of the Fisher information matrix is attained
globally up to the order of $1/N$, where $N$ is the number of data.
An example of estimators which attain the bound is
the maximum likelihood estimator.
On the other hand, in the quantum estimation theory,
Nagaoka $[21]$ conjectured that
the $G$-quantum maximum likelihood estimate,
defined below, has similar property.

Suppose that $N$ copies of the state of the system are given.
Then, we  define $\hat\theta_{(k)}$ recursively by the equation

\begin{eqnarray}
\hat\theta_{(k)}
= \argmax\left\{
\left.
\sum_{i=1}^k \ln p_i(x_i\: | \: \theta)\: \right|
\: \theta\in\R^m
\right\},
\nonumber
\end{eqnarray}

\begin{eqnarray}
&&p_i(x\: | \: \theta)dx = \tr \rho(\theta)M_{(i)}(dx),\\
\begin{array}{cc}
\argmin\{\tr GV_{\theta_{(i-1)}}[M]\: |\: \mbox{ $M$ is locally unbiased}\},
& (i\geq 2)\\
\mbox{an arbitrary locally unbiased measurement}& (i=1)
\end{array}
\right.
\nonumber
\end{eqnarray}

and $x_i$ is the data produced by  $M_{(i)}$ from $i$th copy.
When the number of sample is equal to $N$ and the weight matrix is $G$,
the $G$-quantum maximum likelihood estimate $\hat\theta_{GqMLE}(x_1,...,x_N)$
is defined to be $\hat\theta_{GqMLE}(x_1,...,x_N)=\hat\theta_{(N)}$.

$\hat\theta_{GqMLE}$ is consistent, or

\begin{eqnarray}
\lim_{N\rightarrow\infty} E_{\theta}[\hat\theta_{GqMLE}(x_1,...,x_N)]=\theta.
\nonumber
\end{eqnarray}

\begin{eqnarray}
\Tr G V_{\theta}[\hat\theta_{GqMLE}(x_1,...,x_N)]=
\frac{1}{N}\CR (G, \theta)+o\left(\frac{1}{N} \right).
\nonumber
\end{eqnarray}

*Part I

The faithful model theory
Part I :
The faithful model theory

CHAPTER: THE ESTIMATION THEORY OF THE FAITHFUL MODEL

§ THE LOCALLY QUASI-CLASSICAL MODEL AND THE QUASI-CLASSICAL MODEL

In this section,
the condition for SLD CR bound to be attainable is reviewed briefly,
in the case of
the faithful model,
any member of which is
faithful, i.e., a reversible operator.
We denote the space of the faithful states by
${\cal P}_+({\cal H})$, or simply by ${\cal P}_+$.
For mathematical simplicity, the dimension $d$ of the Hilbert space
${\cal H}$ is assumed to be finite.
The author believes the essence of the discussion will not be damaged
by this restriction.

As for the equality in  $(\ref{eqn:genCR})$
in the faithful model,
we have the following theorem, which is proved
by Nagaoka [23].

The equality in $(\ref{eqn:genCR})$ is attainable at $\theta$
iff $[L^S_i(\theta),L^S_j(\theta)]=0$ for any $i,j$.
$|\omega\rgl$ be a simultaneous eigenvector of
the matrices $\{L^S_j (\theta)|j=1,...,m\}$
$\lambda_i(\omega)$ be the eigenvalue of $L^S_i (\theta)$
corresponding to $|\omega\rgl$,
the equality is attained by the estimator
such that

\begin{eqnarray}
&&\Omega=\{\omega| \omega=1,...,n\},\nonumber\\
&&M_{(\theta)}(\omega)=|\omega\rgl\lgl \omega|,\nonumber\\
= \theta^j + \sum_{k=1}^d [(J^S)^{-1}]^{jk} \lambda_k(\omega).
\label{eqn:bests}
\end{eqnarray}

The model ${\cal M}$ is said to be
locally quasi-classical at $\theta$ iff
$L^S_i(\theta)$ and $L^S_j(\theta)$ commute for any $i,j$,
because in this case $(\ref{eqn:genCR})$
gives the attainable lower bound of the covariance matrix
of the unbiased estimator,
as its classical counterpart does.
However, it should be noted that
even if the model is locally quasi-classical at any $\theta\in\Theta$,
theorem $\ref{theorem:mpCR}$ do not tell us
the optimal experiment scheme,
the measurement $M_{(\theta)}$ in $(\ref{eqn:bests})$
generally depends on the unknown parameter $\theta$
(so does the best experiment scheme).

Therefore, let us move to easier case.
Suppose  that
$L^S_i(\theta)$ and $L^S_j(\theta')$
commute even when $\theta\neq\theta'$,
in addition to being locally quasi-classical at any $\theta\in\Theta$.
the measurement $M_{(\theta)}$ in $(\ref{eqn:bests})$,
denoted by $M_{best}$ in the remainder of the section,
is uniformly optimal for all $\theta$
(so is the corresponding scheme).
We say such a model is quasi-classical [35].

After the best experiment is done,
the rest of our task is to estimate the value of the parameter $\theta$
in the probability distribution $(\ref{eqn:pdm})$,
where $M_{best}$ is substituted into $M$.
Hence, in this case, the quantum estimation  reduces to
the classical estimation.

§ ESTIMATION OF THE TEMPERATURE

Because quantum mechanics is applicable to
the measuring process  of the temperature,
the measurement of the temperature can be described
with the generalized measurement.
Therefore, we formulate the measurement of the temperature
as estimation of the parameter $T$
in the canonical  model
${\cal M}$  which is defined by

\begin{eqnarray}
{\cal M}=\left\{\rho(T)\:
\left|\:
\rho (T)= \exp\left(-\frac{1}{k_B T}(H-F(T)\,)\right),\,
T\in \R \right.\right\},
\nonumber
\end{eqnarray}

where $k_B$ is the Boltzmann constant, $H$ the Hamiltonian of the system,
$F(T)$ the free energy.

Simple calculations yield he SLD Fisher information $J^S(T)$,

\begin{eqnarray}
J^S(T)=\frac{1}{k_B T^2}C(T),
\nonumber
\end{eqnarray}

and the best estimator,

\begin{eqnarray}
&&\Omega =\{\omega \: |\: \omega=1,...,d\},\\
&&M_{T} (\omega) = |\omega \rgl\lgl \omega |, \\
&&\hat{T}_{T}(\omega) = T+ \frac{E_{\omega}-\lgl H \rgl_T}{C(T)},
\nonumber
\end{eqnarray}

where  $C(T)$ is the specific heat of the system,
$|\omega\rgl$ the $\omega$th eigenvector of the Hamiltonian,
$E_{\omega}$ the $\omega$th eigenvalue of the Hamiltonian,
and $\lgl H \rgl_T$ denotes $\tr \rho(T) H$.

Because the model is quasi-classical,
by use of the maximum likelyhood estimator of the parameter $T$
in  the induced family of probability distributions,

\begin{eqnarray}
\{p(\omega|T)\: |\: p(\omega|T)=\lgl \omega| \rho(T)|\omega \rgl \},
\nonumber
\end{eqnarray}

we can attain asymptotically the SLD CR bound.

When the temperature is high,
the specific heat behaves like constant independent of $T$,
which implies that SLD Fisher information tends to zero
as $T$ tends to infinity.
In other words, in the high temperature, the parameter
$T$ is hard to estimate.

In the low temperature, the dependency of the specific heat on $T$
differs from system to system,
and so does  the dependency of $J^S(T)$.

For example, if the system is the mass of the phonon,

\begin{eqnarray}
C(T)\propto T^3,
\nonumber
\end{eqnarray}

which implies that SLD Fisher information tends to zero
as $T$ tends to zero.
On the other hand, the electronic specific heat is

\begin{eqnarray}
C(T)\propto T ,
\nonumber
\end{eqnarray}

which implies that SLD Fisher information tends to infinity
as $T$ tends to zero.

CHAPTER: UHLMANN CONNECTION AND THE ESTIMATION THEORY

§ SOME NEW FACTS ABOUT RPF

In this section,
we derive conditions for RPF to vanish, which is used to
the classes of manifold defined in the previous section.
For notational simplicity, the argument $\theta$ is omitted,
as long as the omission is not misleading.

RPF  for any closed loop vanishes
iff $[\, L^S_i(\theta),\, L^S_j(\theta)\,]=0$ for any $\theta \in \Theta$.
In other words,

\begin{eqnarray}
\left[\, L^S_i(\theta),\, L^S_j(\theta)\,\right]=0.
\label{eqn:f0lsls0}
\end{eqnarray}

If $F_{ij}$ equals zero,
then both of the two terms in the left-hand side of $(\ref{eqn:Fij})$
must vanish,
because the first term is Hermitian and the second term is skew Hermitian.
Hence, if $F_{ij}=0$, $[\, L^S_i,\, L^S_j\, ]$ vanishes.

On the other hand, the identity
or its equivalence

\begin{eqnarray}
\left(\partial_iL^S_j-\partial_jL^S_i
-\frac{1}{2}[ L^S_i,\, L^S_j ]\right)\rho+
\rho\left(\partial_iL^S_j-\partial_jL^S_i +\frac{1}{2}[ L^S_i,\, L^S_j ]\right)
\label{eqn:didj}
\end{eqnarray}

$\partial_i L^S_j-\partial_j L^S_i$ vanishes if $[\, L^S_i,\, L^S_j\, ]=0$,
$\partial_i L^S_j-\partial_j L^S_i$ is Hermitian
and $\rho$ is positive definite.
Thus we see $F_{ij}=0$  if $L^S_i$ and $L^S_j$ commute.

A manifold ${\cal M}$ is said to be parallel
when the RPF between any two points along any curve vanishes.
From the definition, if ${\cal M}$ is parallel, RPF along
any closed loop vanishes, but the reverse is not necessarily true.
The following theorem is a generalization of Uhlmann's
theory of $\Omega$-horizontal real plane [33].

The following three conditions are equivalent.

(1) ${\cal M}$ is parallel.

(2) Any element $\rho(\theta)$ of ${\cal M}$ writes
$M(\theta)\,\rho_0\, M(\theta)$,
where $M(\theta)$ is Hermitian  and
for any $\theta_0, \theta_1 \in\Theta$.

\forall i,j,\:\forall \theta_0,\,\theta_1\in\Theta,\:\;
\left[\,L^S_i(\theta_0),\, L^S_j(\theta_1)\,\right]=0.

Let $W(\theta_t)=M(\theta_t)W_0$ be a horizontal lift of
$\{\rho(\theta_t)\, |\, t\in{\bf R}\}\subset{\cal M}$.
Then, $W^*_0\, W(\theta_t)=W^*(\theta_t)\, W_0$ implies
$M(\theta_t)=M^*(\theta_t)$, and
$W^*(\theta_{t_0})\, W(\theta_{t_1})
=W^*(\theta_{t_1})\, W(\theta_{t_0})$ implies
$M(\theta_{t_0})\, M(\theta_{t_1})=M(\theta_{t_1})\, M(\theta_{t_0})$.
Thus we get $(1)\Rightarrow(2)$. Obviously, the reverse also holds true.
For the proof of $(2)\Leftrightarrow(3)$,
see Ref. [35], pp.31-33.

§ UHLMANN'S PARALLELISM IN THE QUANTUM ESTIMATION THEORY

To conclude the chapter, we present the theorems which
geometrically characterize
the locally quasi-classical model and
quasi-classical model,
by the  vanishing conditions of RPF,
implying the close tie between
Uhlmann parallel transport and the quantum estimation theory.
They are straightforward consequences of
the definitions of the terminologies  and
theorems <ref> -<ref>.

${\cal M}$ is locally quasi-classical at $\theta$
iff $F_{ij}(\theta)=0$ for any $i,j$.
${\cal M}$ is locally quasi-classical at any $\theta\in\Theta$
iff the RPF for any loop vanishes.

${\cal M}$ is quasi-classical
iff ${\cal M}$ is parallel.

CHAPTER: NAGAOKA'S QUANTUM INFORMATION GEOMETRY
AND UHLMANN'S PARALLELISM

§ NAGAOKA'S QUANTUM INFORMATION GEOMETRY

In this section, we give brief review of
Nagaoka's quantum information geometry,
which is another geometrical theory of the quantum
statistical model than Uhlmann's parallelism.

In the Nagaoka's geometry, metric tensor is
chosen to be SLD Fisher information matrix.
Letting ${\bf L}^S$ denote the linear mapping  from
the tangent vector to its SLD,
$e$-parallel transport $\pte$ is
defined as follows:

\begin{eqnarray}
{\bf L}^S\left(\pte_{\rho\rightarrow\sigma} X\right)
&\equiv&{\bf L}^SX
-\tr (\sigma {\bf L}^S X).
\label{eqn:ptte}
\end{eqnarray}

Note that in the faithful model, ${\bf L}^S$ is one to one mapping,
and the equation $(\ref{eqn:ptte})$ defines the connection
The dual $\ptm$ of $e$-parallel transport
with respect to SLD inner product
is called $m$-parallel transport,

\begin{eqnarray}
\forall X,Y\in{\cal T}_{\rho}({\cal P}_+)\:\:
\left\lgl \ptm_{\rho\rightarrow\sigma}X,
\pte_{\rho\rightarrow\sigma}Y \right\rgl_{\sigma}
=\lgl X,Y \rgl_{\sigma}.
\nonumber
\end{eqnarray}

For the  autoparallel manifold in $e$-connection,
the $e$- covariant derivative is calculated as

\begin{eqnarray}
{\bf L}^S\left(\,\nablae_X Y\,\right)=XL^S_Y-\tr \rho XL^S_Y,
\nonumber
\end{eqnarray}

the tortion of $e$-connection $\Te$ as,

\begin{eqnarray}
{\bf L}^S
\left(\,\Te(X,Y)\,\right)
&=&{\bf L}^S\left(\,\nablae_X Y-\nabla_Y X -[X,Y]\,\right)\nonumber\\
&=& X{\bf L}^S(Y)-Y{\bf L}^S(X),
\label{eqn:etortion}
\end{eqnarray}

or, equivalently,

\begin{eqnarray}
\Te(X,Y)\rho(\theta)
&=&\frac{1}{4}[[L^S_X,L^S_Y], \rho],
\label{eqn:etortion2}
\end{eqnarray}

where $X$ and $Y$ are understood as differential operators.

Nagaoka showed that
these $e$- and $m$-connections
nicely characterize the estimation theoretical properties of models,
in a different manner than Uhlmann's parallelism.
Is there any relation between the two geometrical structures?

§ $W$-CONNECTION IN ${\CAL W}$

To elucidate the relations between Uhlmann's parallelism and
Nagaoka's information geometry, we consider
the geometry of the tangent bundle over the total space ${\cal W}_d$.

the logarithmic derivative ${\bf L}(\hat{X}) $ of
$X\in{\cal T}_W({\cal W}_d)$ is a $d\times d$ complex matrix
which satisfies the equation,

\begin{eqnarray}
{\bf L}(\hat{X}) W
={\bf M}(\hat{X}).
\label{eqn:deflogd}
\end{eqnarray}

${\bf L}(\partial/\partial\zeta^i)$ is often denoted by
$L_{\partial/\partial \zeta^i}$ for simplicity.
Notice the logarithmic derivative is uniquely defined iff
the rank $r$ of $W$ is equal to the dimension $d$ of
the Hilbert space ${\cal H}$.

The real span of the logarithmic derivatives of at $\zeta$ is

\begin{eqnarray}
\{L\: |\: {\rm Re}\,\tr(L \pi(W(\zeta)))=0,
L\in M_d(\C)\}.
\nonumber
\end{eqnarray}

We introduce the $w$-connection in ${\cal T}({\cal W})$
by the $w$-parallel transport defined by

\begin{eqnarray}
{\bf L}\left(\pttw_{W_0\rightarrow W_1}\hat{X}\right)
\equiv {\bf L}\hat{X}  -{\rm Re}\,\tr(\pi(W_1){\bf L}\hat{X}).
\label{eqn:defpttw}
\end{eqnarray}

This $w$-parallel transport is left invariant
under the action of $U\in U(d)$ in the following sense:

\begin{eqnarray}
{\bf L}\left(\pttw_{W_0\rightarrow W_1}\hat{X}\right)
={\bf L}\left(\pttw_{W_0U\rightarrow W_1}\hat{X}\right)
={\bf L}\left(\pttw_{W_0\rightarrow W_1U}\hat{X}\right).
\nonumber
\end{eqnarray}

The covariant derivative $\nablaw$
and the tortion $w$-tortion $\Ttw(\hat{X},\hat{Y})$
are  easily calculated for the $w$-autoparallel
submanifold of ${\cal W}$ as

\begin{eqnarray}
{\bf L}\left(\nablaw_{\hat{X}}\hat{Y} \right)
&=&\hat{X}{\bf L}(\hat{Y})
-{\rm Re}\,\tr \left(\pi(W)\hat{X}{\bf L}(\hat{Y})\right)I
\nonumber\\
{\bf L}\Ttw(\hat{X},\hat{Y})
{\bf L}\left(\,\nablaw_{\hat{X}} \hat{Y}-\nablaw_{\hat{Y}}\hat{X}
-{\rm Re}\,\tr \left\{\pi(W)(\hat{X}L_Y-\hat{Y}L_X)\right\}I,
\nonumber\\
\label{eqn:defTtwmtrx}
\end{eqnarray}

the tangent vector $\hat{X}$ is understood as a differential operator,
and $I$ is the identity in the Hilbert space ${\cal H}$.
The latter equation is equivalent to

\begin{eqnarray}
{\bf M}\Ttw(\hat{X},\hat{Y})
\frac{1}{2}[L_X,L_Y]W
-{\rm Re}\,\tr \left\{ \pi(W)(\hat{X}L_Y-\hat{Y}L_X)\right\}W,
\label{eqn:mttw}
\end{eqnarray}

which is of use when the theory is generalized to non-faithful models.

§ PROJECTION OF GEOMETRIC STRUCTURES

In the beginning, we show that the Nagaoka's information geometry
is naturally induced from the geometry of the ${\cal T}({\cal W})$.

As in the definition,
the metric $\lgl *,* \rgl$ in Nagaoka's quantum information
geometry is  induced from
the natural metric $\lgll *,* \rgll$ in ${\cal T}({\cal W})$.

Not only the metric $\lgl *,* \rgl$, but also
the transport $\pte$ is  induced from
$\pttw$ :

\begin{eqnarray}
\begin{array}{ccc}
\rho=\pi(W)&         &\sigma=\pi(V)\\
&\pttw&               \\
\ds{\hat{X}=\hlift_{W}(X)\in{\cal T}_W({\cal W})}&--\longrightarrow
&\ds{\pttw_{W\rightarrow V}}\hat{X}\in{\cal T}_V({\cal W})
\\
\hlift\uparrow& &\downarrow\pi_*
\\
X\in{\cal T}_{\rho}({\cal P}_+)&--\longrightarrow
X\in{\cal T}_{\sigma}({\cal P}_+)
\\
\end{array}
\label{eqn:ptwte}
\end{eqnarray}

The horizontal lift $\hlift$ satisfies
the following requirements so that the definition of
the transport $\pte$ by the diagrams above are

\begin{eqnarray}
\pi_*\left(\pttw_{W\rightarrow V}\hlift_{W}(X)\:\right)
=\pi_*\left(\pttw_{WU\rightarrow VU'}\hlift_{WU}(X)\;\right),
\nonumber
\end{eqnarray}

Because of the diagram,
it is quite easy to see that
if the submanifold ${\cal N}$ of ${\cal W}$
is $w$-autoparallel,
the model ${\cal M}=\pi({\cal N})$ is $e$-autoparallel.

Some elementary calculations leads to  the following theorem,
which illustrates the relation between the two geometries.

Let ${\cal M}$ be a submanifold of ${\cal P}_+$
which is induced from $w$-autoparallel submanifold
${\cal N}$ of ${\cal W}$ by
${\cal M}=\pi({\cal N})$
$X$ and $Y$  tangent vectors to ${\cal P}_+$ such that,

\begin{eqnarray}
X&=&\sum_i x^i\frac{\partial}{\partial\theta^i},\nonumber\\
Y&=&\sum_i y^i\frac{\partial}{\partial\theta^i}.
\nonumber
\end{eqnarray}

$\Ttw (\hlift(X),\hlift(Y))$
is decomposed into the sum such that

\begin{eqnarray}
{\bf L}\Ttw (\hlift(X),\hlift(Y))
&=&{\bf L}^S\Te(X,Y) - F_{XY},\nonumber\\
F_{XY}&=& \sum_{i,j}F_{i j}x^i y^j,
\nonumber
\end{eqnarray}

where ${\bf L}^{-1}({\bf L}^S\Te(X,Y))$ is a horizontal vector
and ${\bf L}^{-1} F_{XY}$ is a vertical subspace.

In other words, the horizontal component of the $w$-tortion  is
the $e$-tortion and the vertical component is
the curvature form of the Uhlmann parallelism.

CHAPTER: THE DUALITY BETWEEN THE OBSERVED SYSTEM AND THE HIDDEN SYSTEM

§ THE DUALITY OF SLD AND RLD

First, we define the right logarithmic derivative (RLD, in short),
which played quite important role
in the estimation theory of the Gaussian model,
which is a superposition of the coherent states by the Gaussian kernel
( see Ref. [34]).

RLD $L^R_i(\theta)$ of the parameter $\theta^i$ is defined by
the equation

\begin{eqnarray}
\frac{\partial \rho(\theta)}{\partial \theta^i}
\nonumber
\end{eqnarray}

and RLD ${\bf L}^R X$ of the tangent vector
by the equation

\begin{eqnarray}
\hat{X}\rho(\theta)
=({\bf L}^R X)\rho(\theta).
\nonumber
\end{eqnarray}

Our question is why we need two types of logarithmic derivatives,
SLD and RLD namely, and what the relations between them.
To answer the question, we interpret
the total space ${\cal W}_d$,
like in the section $\ref{sec:defuhlmann}$,
as the space of the state vector $|\Phi\rgl$
in the bigger Hilbert space ${\cal H}\otimes{\cal H}'$,
where we took the dimension of ${\cal H}'$ to be $d$,
the dimension of ${\cal H}$.
Let us call ${\cal H}$ the observed system, and ${\cal H}'$ the hidden system,
and the partial trace over ${\cal H}$ and ${\cal H}'$ are
denoted by $\pi$ and $\pi'$, respectively.

In terms of $W$, $\pi$ and $\pi'$ write

\begin{eqnarray}
\pi(W)&=&WW^*=\rho,\nonumber\\
\pi'(W)&=&W^*W=\sigma,
\nonumber
\end{eqnarray}

and $\pi_*$ and $\pi'_*$ write

\begin{eqnarray}
\pi_*(\hat{X})\rho&=&
\frac{1}{2}(\:
({\bf M}\hat{X})W^*+W({\bf M}\hat{X})^*\: ),\nonumber\\
\pi'_*(\hat{X})\sigma&=&
\frac{1}{2}(\:
({\bf M}\hat{X})^*W+W^*({\bf M}\hat{X})\: ),
\label{eqn:defpi'*}
\end{eqnarray}

The RLD subspace ${\cal LR}_W$ of ${\cal T}_W({\cal W}_d)$ is
the space of all vectors which satisfies,

\begin{eqnarray}
{\bf L}(\hat{X})={\bf L}^R(\pi_*(\hat{X})),
\label{eqn:defLR1}
\end{eqnarray}

or, its equivalence,

\begin{eqnarray}
({\bf M}\hat{X})W^*=W({\bf M}\hat{X})^*.
\label{eqn:defLR2}
\end{eqnarray}

Then, from $(\ref{eqn:defpi'*})$,
$(\ref{eqn:defLR2})$ and $(\ref{eqn:defLS2})$,
we have

\begin{eqnarray}
{\cal LR}_W&=&
\{\hat{X}\; |\;{\bf L}(\hat{X})={\bf L}^S(\pi'_*(\hat{X}))\},\\
{\cal LS}_W&=&
\{\hat{X}\; |\;{\bf L}(\hat{X})={\bf L}^R(\pi'_*(\hat{X}))\}.
\label{eqn:dualsr}
\end{eqnarray}

In other words, looking from the hidden system,
the RLD subspace looks like the horizontal subspace,
and the horizontal subspace looks like the RLD subspace.
We call the fact $(\ref{eqn:dualsr})$
the duality between SLD and RLD.

The orthogonal complement subspace ${\cal SK}_W$ of the RLD subspace
is also
dual of   ${\cal K}_W$ in the following sense.
${\cal SK}_W$ is the space of all the tangent vectors which satisfies

\begin{eqnarray}
{\bf L}\hat{X}= -({\bf L}\hat{X})^*
\label{eqn:defsk1}
\end{eqnarray}

or, equivalence,

\begin{eqnarray}
({\bf M}\hat{X})W^*+W({\bf M}\hat{X})^*=0,
\label{eqn:defsk2}
\end{eqnarray}

which yields

\begin{eqnarray}
\pi'_*(\hat{X})=0.
\label{eqn:pi'*=0}
\end{eqnarray}

Metaphorically speaking,
$(\ref{eqn:pi*=0})$ and $(\ref{eqn:pi'*=0})$
implies that ${\cal SK}_W$ looks like ${\cal K}_W$
seen from the hidden system.

$(\ref{eqn:defsk1})$ means that
for any member $\hat{X}$ of ${\cal SK}_W$,
$\pi_* (\hat{X})$ corresponds to  a unitary motion of the observed system.
The dual of this statement is also valid:
for any member $\hat{X}$ of ${\cal K}_W$,
$\pi'_*(\hat{X})$ corresponds to a unitary motion of the hidden system.
This statement reflects the physical fact that the unitary motion of
the hidden system do not affect the observed system.

The intersection of the ${\cal LS}_W$ and ${\cal LR}_W$ is
given by

\begin{eqnarray}
&&{\cal LS}_W\cap{\cal LR}_W\nonumber\\
&=&\{\hat{X}\: |\:  [{\bf L}^S (\pi_*(\hat{X})),\pi(W)]=0,\,
({\bf L}\hat{X})^*={\bf L}\hat{X},\,\}\nonumber\\
&=&\{\hat{X}\: |\:  [{\bf L}^S (\pi'_*(\hat{X})),\pi'(W)]=0,\,
({\bf L}\hat{X})^*={\bf L}\hat{X},\,\}.
\label{eqn:lscaplr}
\end{eqnarray}

The intersection of the ${\cal K}_W$ and ${\cal SK}_W$ is
given by

\begin{eqnarray}
&&{\cal K}_W\cap{\cal SK}_W\nonumber\\
&=&\{\hat{X}\: |\:  [{\bf L}(\hat{X}),\pi(W)]=0,\,
({\bf L}\hat{X})^*=-{\bf L}\hat{X},\,\}\nonumber\\
&=&\{\hat{X}\: |\:  [{\bf L}'(\hat{X}),\pi'(W)]=0,\,
({\bf L}\hat{X})^*=-{\bf L}\hat{X},\,\},
\label{eqn:kcapsk}
\end{eqnarray}

where ${\bf L}'(\hat{X})$ is defined by

\begin{eqnarray}
{\bf L}'(\hat{X})\equiv W^{-1}{\bf M}(\hat{X})
\nonumber
\end{eqnarray}

The former statement of the theorem
means that for any vector $\hat{X}\in {\cal LS}_W \cap {\cal LR}_W$,
$\pi_*(\hat{X})$ and $\pi'_*(\hat{X})$ correspond
to the change of eigenvalues of $\pi(W)$ and $\pi'(W)$, respectively.
The latter statement implies that
for any vector $\hat{X}\in {\cal K}_W\cap {\cal SK}_W$,
$\pi_*(\hat{X})$ and $\pi'_*(\hat{X})$ correspond
to the change of the phase of the eigenvectors of of
$\pi(W)$ and $\pi'(W)$, respectively.

\begin{eqnarray}
\pi_*(\hat{X})=({\bf L}\hat{X})\pi(W)=\pi(W)({\bf L}\hat{X})^*,
\nonumber
\end{eqnarray}

which, combined with $(\ref{eqn:hsld})$, yields

\begin{eqnarray}
[({\bf L}\hat{X}),\pi(W)]=0
\nonumber
\end{eqnarray}

Because  ${\bf L}(\hat{X})={\bf L}^S (\pi_*(\hat{X})$ holds true
for any $\hat{X}\in{\cal LS}_W$, we have the first equality
in $(\ref{eqn:lscaplr})$. The second equality in $(\ref{eqn:lscaplr})$
and the equalities in
$(\ref{eqn:kcapsk})$ are obtained in the same manner.

§ CANONICAL DISTRIBUTION

In this section, as an application of the duality of SLD and RLD,
we try an estimation theoretical characterization
of the canonical  model.

One conspicuous feature of the canonical distribution model
is that only the eigenvalue of the density matrix
is dependent on the parameter, and that the eigenvector
is left unchanged even if the parameter changed.
Other thermodynamical models,
for example, the $T-p$ model

\begin{eqnarray}
\left\{\rho(T,p)\: \left|\:
\rho(T,p)=
\sum_{\omega} |\omega\rgl\lgl \omega|
\exp\left[-\frac{1}{k_B T}
\left(E_{\omega}-pV_{\omega} -G(T,p)\,\right)\right]
\right.\right\}
\nonumber
\end{eqnarray}

the grand canonical model

\begin{eqnarray}
\left\{\rho(T,\mu)\: \left|\:
\rho(T,\mu)=
\sum_{\omega} |\omega\rgl\lgl \omega|
\exp\left[ -\frac{1}{k_B T}
\left(E_{\omega}-\mu N_{\omega}+Y(T, \mu)\,\right)\right]
\right.\right\}
\nonumber
\end{eqnarray}

also share this feature.
Here,  $H$ is the Hamiltonian and
$E_{\omega}$  the   $\omega$th eigenvalue of $H$,
and $|\omega\rgl$ the  $\omega$th  eigenvector of $H$.
We call the model which has this feature the classical model.

Let us require  first that the canonical distribution is
a pure state in the composite Hilbert space
${\cal H}\otimes {\cal H}'$,
where ${\cal H}$ is for the system and ${\cal H}'$  for the heat bath
( taking trace over the heat bath, we have the canonical distribution).
In other words, we assume that  the pure state model

\begin{eqnarray}
{\cal N}\equiv
\left\{|W(T)\rgl\lgl W(T)|\:\: |\: |W(T)\rgl\in {\cal H}\otimes {\cal H}'
\right\},
\label{eqn:totmdl}
\end{eqnarray}

the partial trace $\pi$ over the hidden system ${\cal H}'$
reduces to the canonical model ${\cal M}$.
We denote by ${\cal M}'$ the model induced from
${\cal N}$ by the partial trace $\pi'$
over the observed system ${\cal H}$.

Second, we assume the following situation:
the optimization of the measurement in the estimator of
the temperature over any of the following three range

* all the measurements in ${\cal H}$

* all the measurements in ${\cal H}'$

* all the measurements in ${\cal H}\otimes{\cal H}'$

achieves  exactly the same extent of the efficiency.
In usual situation, we can achieve more efficiency in the case of $3$ than
the other cases, for the range of the optimization is larger.
However, as for the macroscopic parameter like
the temperature, it is natural to assume that
the measurement of the total system do not bring about more information
than the measurement of the system.
In addition, the measurement of the temperature in the system and
the heat bath must yield same amount of information,
because they are in the thermal equilibrium.
In this situation, we  say that
the model ${\cal M}$ and ${\cal N}$ are maximally entangled.

The  model ${\cal M}$ and ${\cal M}'$
induced by the projection $\pi$ and $\pi'$ from
the pure state model ${\cal N}$ are classical
iff they are maximally entangled.

To prove the theorem, we need the following fact
in the pure state estimation theory,
which is explained in the later chapters in detail.

The attainable lower bound of the variance of
the unbiased estimator of the pure state model $(\ref{eqn:totmdl})$
is given by $1/\tr A(T)A^*(T)$, where $A(T)$ is
the matrix representation of
the tangent vector $d/dT|_{W}\in {\cal T}_W({\cal W})$
to the ${\cal W}$.

Here, we assume $\dim {\cal H}'$ is equal to $d=\dim {\cal H}$.
For the efficiency of the estimation in the case of $1$
is equal to that in the case of $3$,
$d/dT|_{W}\in {\cal T}_W({\cal W})$
need to be a member of the horizontal subspace ${\cal LS}_W$,
the length of the horizontal component
of $d/dT|_{W}\in {\cal T}_W({\cal W})$ gives
the SLD Fisher information of the model ${\cal M}$.

Mostly in the same manner, it can be proved that
$d/dT|_{W}\in {\cal T}_W({\cal W})$ is the member of ${\cal LR}_W$
for the efficiency of the estimation in the case of $2$
to be equal to that in the case of $3$.

\begin{eqnarray}
\left. \frac{d}{dT}\right|_{W} \in{\cal LS}_W\cap {\cal LR}_W,
\nonumber
\end{eqnarray}

which, mixed with lemma $\ref{lemma:lscaplr}$
leads to the statement of the theorem.

This theorem,
which also applies to the grand canonical model
and the $T-p$ model,
characterize the entanglement between
the heat bath and the system.

*Part II

The pure state  model theory
Part II :
The pure state  model theory

CHAPTER: THE PURE STATE ESTIMATION THEORY

§ HISTRICAL REVIEW OF THE THEORY AND THE PURPOSE OF THE CHAPTER

First, we review the history of quantum estimation theory
to clarify the purpose of the chapter.

In parallel with the classical estimation theory,
in 1967, Helstrom show that in the faithful state model,
the covariance matrix is larger than or equal to the inverse of
SLD Fisher information matrix, and that in the one parameter faithful model,
the bound is attainable [10][11].

On the other hand, in the multi-parameter model,
it is proved
that there is no matrix
which makes attainable lower bound of covariance matrix,
because of non-commutative nature of quantum theory.
Therefore, we deal with the attainable CR type bound defined
in the section $\ref{sec:nonclassical}$.
Lower bounds of $\Tr GV_{\theta}[M]$
is, attainable or not,  called Cramer-Rao (CR) type bound.

Five years after Helstrom's work,
Yuen and Lax found out the exact form of the attainable CR type bound
of the Gaussian state model,
which is a faithful 2-parameter model obtained by superposition of
coherent states by Gaussian kernel[34].
Their work is remarkable not only
because it was first calculation of the attainable CR type bound of
a multi-parameter model,
but also because
they established a kind of methodology,
which we call indirect approach hereafter,
to calculate the attainable CR type bound.
First an auxiliary bound which is not generally attainable
is found out
and then it is proved to be attained in the specific cases.
In their work, they used so-called RLD bound,
which was used and generalized by several authors.

Holevo completed their work by solving analytically subtle problems
and generalizing RLD bound
In 1998, Nagaoka calculated the attainable CR type bound of
the faithful 2-parameter spin-$1/2$ model
using Nagaoka bound, which is also another auxiliary bound

The models had been assumed to be faithful
Fujiwara and Nagaoka
formulated the problem in the pure state model,
and calculated the CR type bound for the 1-parameter model
by use of generalized SLD bound
and that of the 2-parameter coherent model
by use of generalized RLD bound in 1995

The approach in this chapter,  called direct approach
in contrast with indirect approach,
is essentially different from the approaches of other authors.
We reduce the given minimization problem to the problem
which is easy enough to be solved directly
by elementary calculus.
The methodology is successfully applied to
the general 2-parameter pure state model,
coherent model with arbitrary number of parameters,
and the minimization of $\Tr J^S(\theta)V_{\theta}[M]$
for  arbitrary sure state model.
are relatively general category in comparison with the cases
treated by other authors.
In the 2-parameter pure state model,
the existence of the order parameter $\beta$ which
is a good index of noncommutative nature between the parameters.

As a by-product, we have remarkable corollary,
which asserts that even for non-commutative cases,
a simple measurement attains the lower bound.

§ NOTATIONS

In this chapter, we consider data space $\Omega$ to be $\R^m$,
and each $\omega\in\Omega$ to be the estimate $\hat\theta$.

The pure state  model ${\cal M}$ is assumed to
be induced by ${\cal M}=\pi({\cal N})$ from the manifold

\begin{eqnarray}
{\cal N}=\{ |\phi(\theta)\rgl\, |\, |\phi(\theta)\rgl\in \tilde{\cal H}\},
\nonumber
\end{eqnarray}

where $\tilde{\cal H}$ is the set of the members of ${\cal H}$
with unit length,

\begin{eqnarray}
\tilde{\cal H}=\{|\phi\rgl\:|\:|\phi\rgl\in{\cal H},\lgl\phi|\phi\rgl=1\}.
\nonumber
\end{eqnarray}

If this assumption is made, the horizontal lift of the tangent vector
is taken for granted.
Uniqueness is proved mostly in the same way as
the proof of theorem $\ref{theorem:uqhl}$.

We denote by $|l_X\rgl$
the matrix representation of the horizontal lift
of $X\in {\cal T}_{\rho(\theta)}({\cal M})$,
and $|l_i\rgl$ is short for  $|l_{\partial/\partial \theta^i}\rgl$.
$|l_X\rgl$ satisfies

\begin{eqnarray}
\frac{1}{2}(|l_X\rgl\lgl\phi(\theta)|+|\phi(\theta)\rgl\lgl l_X|),
\label{eqn:sld:lift}
\end{eqnarray}

\begin{eqnarray}
\lgl l_X|\phi(\theta)\rgl=0.
\label{eqn:sld:horizontal}
\end{eqnarray}

Notice that
$span_{\R}\{|l_i\rgl\;|\: i=1,...,m\}$ is a representation of
${\cal T}_{\rho(\theta)}({\cal M})$
because of unique existence of $|l_X\rgl$.
Therefore, we often also call the matrix representation $|l_X\rgl$
horizontal lift.

We call ${\bf M}^i(M,\, |\phi(\theta)\,)$
estimation vector of the parameter $\theta^i$
by a measurement $M$ at $|\phi(\theta)\rgl$,
An estimation vector ${\bf M}^i(M,\, |\phi(\theta)\,)$
is said to be locally unbiased
iff $M$ is locally unbiased.
The local unbiasedness conditions for estimating vectors writes

\begin{eqnarray}
\lgl \phi(\theta)|{\bf M}^i(M,\, |\phi(\theta)\rgl \,)&=&0,\\
{\rm Re}\lgl l_j(\theta)|{\bf M}^i(M,\,|\phi(\theta)\rgl\,)
\:(i,j=1,...,m).
\label{eqn:sld:hunbiased}
\end{eqnarray}

Often, we omit the argument  $\theta$ is
$| l_j(\theta)\rgl,|\phi(\theta)\rgl, \rho(\theta)$, and $J^S(\theta)$
and denote them simply by $| l_j\rgl,|\phi\rgl, \rho, J^S$.

We denote the ordered pair of vectors

\begin{eqnarray}
\nonumber
\end{eqnarray}

by ${\sf L}$> In this notation,
the SLD Fisher information matrix $J^S$ writes

\begin{eqnarray}
J^S={\rm Re}{\sf L}^*{\sf L}\equiv {\rm Re}\,\left[\,\lgl l_i|l_j\rgl\,\right],
\nonumber
\end{eqnarray}

and the imaginary part of ${\sf L}^*{\sf L}$
is denoted by $\tilde{J}$.
for the ordered pairs

\begin{eqnarray}
{\sf X}=[\,|x^1\rgl, \,|x^2\rgl,...,\,|x^m\rgl\,],\;\;
{\sf Y}=[\,|y^1\rgl, \,|y^2\rgl,...,\,|y^m\rgl\,],\nonumber
\nonumber
\end{eqnarray}

of vectors,
we define

\begin{eqnarray}
{\sf X}^*{\sf Y}=\left[\,\lgl x^i|y^j\rgl\,\right]\nonumber
\end{eqnarray}

for notational simplicity.

Then, letting ${\sf X}$ be

\begin{eqnarray}
\left[\,{\bf M}^1(M,\,|\phi\rgl\,), {\bf M}^2(M,\,|\phi\rgl\,),
...,{\bf M}^m(M,\,|\phi\rgl\,)\,\right],
\nonumber
\end{eqnarray}

the unbiasedness conditions  $(\ref{eqn:sld:hunbiased})$

\begin{eqnarray}
{\rm Re}\,{\sf X}^*{\sf L}=I_m,\:
{\rm Re}\,\left\{\,({\bf M}^i(M,\, |\phi\rgl\,)\,)^*|l^j \rgl\,\right\}=I_m,
\label{eqn:lagrange:restriction1}
\end{eqnarray}

where $I_m$ is the $m\times m$ unit matrix,
the matrix $Z_{\theta}[M]$ defined
in the equation $(\ref{eqn:defz})$ writes

\begin{eqnarray}
Z_{\theta}[M]={\sf X}^*{\sf X}
\nonumber
\end{eqnarray}

§ THE COMMUTING THEOREM AND THE LOCALLY QUASI-CLASSICAL MODEL

In this section,
the necessary and sufficient condition
for SLD CR bound to be attainable is studied.
Fujiwara proved the following theorem [7].

(Fujiwara [7])
SLD CR bound is attainable iff
SLD's $\{L^S_i|i=1,...,m\}$ can be chosen so that

\begin{eqnarray}
[L^S_i, L^S_j]=0, (i,j=1...,m).
\nonumber
\end{eqnarray}

We prove another necessary and sufficient condition which is much easier
to check for the concrete examples, by use of
the following commuting theorem, which
plays key role in our direct approach to pure state estimation.

If there exists a unbiased measurement $M$ such that

\begin{eqnarray}
V[M]={\rm Re}{\sf X}^*{\sf X},
\label{eqn:V=Z}
\end{eqnarray}

the ordered pair ${\sf X}$ is

\begin{eqnarray}
{\sf X}=\left[{\bf M}^1(M,\,|\phi\rgl\,), {\bf M}^2(M,\,|\phi\rgl\,),
...,{\bf M}^m(M,\,|\phi\rgl\,) \right],
\label{eqn:xm}
\end{eqnarray}

\begin{eqnarray}
{\rm Im}{\sf X}^*{\sf X}=0
\label{eqn:imxx=0}
\end{eqnarray}

holds true.
conversely, if $(\ref{eqn:imxx=0})$ holds true for some ordered pair ${\bf X}$
of vectors, then
there exists  a simple, or projection valued,
unbiased estimator $E$ which satisfies
$(\ref{eqn:V=Z})$, $(\ref{eqn:xm})$, and

\begin{eqnarray}
&&E(\{\hat\theta_{\kappa}\})^2 = E(\{\hat\theta_{\kappa}\}),\nonumber\\
&&E(\{\hat\theta_{0}\}) = E_0,\nonumber\\
&&E\left( {\R}^m/\bigcup_{\kappa=0}^m \{\hat\theta_{\kappa}\}
\right)=0,
\label{eqn:commes}
\end{eqnarray}

for some
$\{\hat\theta_{\kappa}|\hat\theta_{\kappa}\in {\R}^m,\kappa = 0,...,m+1\}$,
where $E_0$ is a projection onto orthogonal complement subspace
of $span_{\C}\{{\sf X}\}$.

If $(\ref{eqn:V=Z})$ holds,
inequality $(\ref{eqn:genVZ2})$ in lemma $\ref{lemma:genVZ}$
leads to

\begin{eqnarray}
{\rm Re}{\sf X}^* {\sf X}\ge {\sf X}^*{\sf X},
\nonumber
\end{eqnarray}

\begin{eqnarray}
0 \ge  i{\rm Im}{\sf X}^*{\sf X},
\nonumber
\end{eqnarray}

which implies  ${\rm Im}{\sf X}^*{\sf X}=0$.

Conversely, Let us assume that  $(\ref{eqn:imxx=0})$ holds true.
Applying Schmidt's orthogonalization to the system
$\{|\phi\rgl, {\sf X}\}$ of vectors,
we obtain the orthonormal system $\{|b^i\rgl\;|\; i=1,...,m+1\}$
by which the system ${\sf X}$ of vectors write
of vectors such that,

\begin{eqnarray}
{\sf X}=\left[\sum_{j=1}^{m+1} \lambda^1_j|b^j\rgl,
\sum_{j=1}^{m+1} \lambda^2_j|b^j\rgl,...,
\sum_{j=1}^{m+1} \lambda^m_j|b^j\rgl\right],
\nonumber
\end{eqnarray}

where $\lambda^i_j\, (i=1,...,m,\,j=1,...,m+1)$ are
real numbers.
Letting  $O=[o^i_j]$ be a $(m+1)\times (m+1)$ real orthogonal matrix
such that

\begin{eqnarray}
\lgl\phi|\sum_{j=1}^{m+1} o^i_j |b^j\rgl\neq 0,
\nonumber
\end{eqnarray}

and denoting $\sum_{j=1}^{m+1} o^i_j |b^j\rgl$ by $|b'^{i}\rgl$,
$i$th member of the ordered pair ${\sf X}$ writes

\begin{eqnarray}
&&\sum_{j=1}^{m+1} \lambda^i_j\sum_{k=1}^{m+1} o^k_j |b'^k\rgl\nonumber\\
\left(\sum_{j=1}^{m+1} \lambda^i_j o^k_j \right)|b'^k\rgl\nonumber\\
\frac{\sum_{j=1}^{m+1} \lambda^i_j o^k_j }{\lgl b'^k|\phi\rgl}
|b'^k\rgl\lgl b'^k|\phi\rgl.
\nonumber
\end{eqnarray}

Therefore, since the system
$\{|b'^i\rgl\:|\: i=1,...,m+1\}$ of vectors
is orthonormal,
we obtain an unbiased measurement which satisfies $(\ref{eqn:commes})$
as follows:

\begin{eqnarray}
\frac{\sum_{j=1}^{m+1} \lambda^i_j o^{\kappa}_j }{\lgl b'^{\kappa}|\phi\rgl},
\:\:\kappa =1,...,m+1,\nonumber\\
&&E(\{\hat\theta_{\kappa}\})=|b'^{\kappa}\rgl\lgl b'^{\kappa}|,\:\:
\kappa=1,...,m+1\nonumber,\\
I-\sum_{\kappa=1}^{m+1}|b'^{\kappa}\rgl\lgl b'^{\kappa}|.
\nonumber
\end{eqnarray}

Here, $I$ is the identity in ${\cal H}$.

SLD CR bound is attainable iff

\begin{eqnarray}
\tilde{J}={\rm Im}{\sf L}^*{\sf L}=0
\label{eqn:lsls0}
\end{eqnarray}

$\lgl l_j|l_i\rgl$ is real for any $i,j$.
Conversely, if $(\ref{eqn:lsls0})$ holds true, SLD CR bound is achieved
by a simple measurement, i.e., a projection valued measurement.

If SLD CR bound is attainable,
by virtue of lemma $\ref{lemma:genVZ}$-$\ref{lemma:genZJ}$,
we have $(\ref{eqn:imxx=0})$ and
$(\ref{eqn:genxx=J})$, which lead directly to $(\ref{eqn:lsls0})$.

Conversely, if ${\rm Im}\lgl l_j|l_k\rgl =0$ for any $j,k$,
by virtue of commuting theorem,
there exists such a simple measurement $E$  that

\begin{eqnarray}
\sum_k [J^{S-1}]^{j,k}|l_k\rgl={\bf M}(E,\,|\phi\rgl).
\nonumber
\end{eqnarray}

Elementary calculations show that the covariance matrix of this measurement
equals  $J^{S-1}$.

Our theorem is equivalent to Fujiwara's one,
by virtue of commuting theorem,
$\lgl l_j|l_i\rgl$ is real iff there exist such SLD's that
$L_i^S$ and $L_j^S$ commute for any $i,j$.
However, our condition is much easier to be checked,
SLD's are not unique in the pure state model.

Often, a model is defined by an initial state and  generators,

\begin{eqnarray}
\end{eqnarray}

Then, $\lgl l_j|l_i\rgl$ is real iff
$\lgl\phi(\theta)|[\, H_i(\theta),\, H_j(\theta)\, ]|\phi(\theta)\rgl$ is $0$.
Because of theorem $\ref{theorem:commute}$,

$\lgl\phi(\theta)|[\, H_i(\theta),\, H_j(\theta)\, ]|\phi(\theta)\rgl=0$
is equivalent to the existence of
generators which commute with each other, $[H_i(\theta),H_j(\theta)]=0$.

Because of this example  and the theorem by Fujiwara,
we may metaphorically say
that SLD CR bound is attainable iff any two parameter has
`classical  nature' at $\theta$,
because often classical limit of a quantum system is obtained
by taking such a limit that commutation relations of observables tend to 0.
Throughout the paper, we say that a manifold ${\cal M}$ is
locally quasi-classical at $\theta$
iff $\lgl l_j|l_i\rgl$ is real at $\theta$.
The following remark describes another `classical' aspect of
the condition ${\rm Im}\lgl l_j|l_i\rgl=0$.

The model ${\cal M}=\pi({\cal N})$, where
${\cal N}$ is a real span of some orthonormal basis of ${\cal H}$,
is locally quasi-classical at any point.

As is illustrated in this example,
when the model ${\cal M}=\pi({\cal N})$
is locally quasi-classical at $\theta_0$,
${\cal N}$
behaves like an element of real Hilbert space around  $\theta_0$.
Metaphorically speaking,
$|\phi(\theta)\rgl$'s phase parts  don't change around $\theta_0$
at all, and ${\cal N}$ looks like the family of
the square root of the probability distributions.

§ THE REDUCTION THEOREM AND THE DIRECT APPROACH

(Naimark's theorem, see Ref. $[12]$, pp. 64-68.)

Any generalized measurement $M$ in ${\cal H}$ can be dilated
to a simple measurement $E$
in a larger Hilbert space ${\cal K}\subset{\cal H}$,
so that

\begin{eqnarray}
\label{eqn:naimark}
\end{eqnarray}

will hold,
where $P$ is the projection from ${\cal K}$ onto ${\cal H}$.

Naimark's theorem, mixed with commuting theorem,
leads to the following reduction theorem,
which sits at the heart of our direct approach.

Let  ${\cal M}$ be a $m$-dimensional manifold in ${\cal P}_1$,
and ${\sf B}_{\theta}$ be a system
$\{|\phi'\rgl,\,|l'_i\rgl\: |\; i=1,...,m\}$
of vectors in $2m+1$-dimensional Hilbert space ${\cal K}_{\theta}$
such that

\begin{eqnarray}
\lgl \phi'| l'_j\rgl&=&\lgl \phi| l_j\rgl=0,\nonumber\\
\lgl l'_i| l'_j\rgl&=&\lgl l_i| l_j\rgl,\nonumber\\
\lgl \phi'| \phi'\rgl&=&\lgl \phi | \phi\rgl=1
\nonumber
\end{eqnarray}

for any $i,j$.
for any locally unbiased estimator $M$ at $\theta$ in ${\cal H}$,
there is a simple `locally unbiased' measurement $E$ in ${\cal K}_{\theta}$,

\begin{eqnarray}
|x^i\rgl={\bf M}^i(E, |\phi'\rgl)\in{\cal K}_{\theta}
\label{eqn:xe}
\end{eqnarray}

\begin{eqnarray}
\lgl x^i|\phi'\rgl&=&0,\\
{\rm Re}\lgl x^i| l'_j\rgl&=&\delta^i_j
\:(i,j=1,...,m),
\label{eqn:naimark:unbiased}
\end{eqnarray}

whose  `covariance matrix' $V[E]$ equals $V[M]$,

\begin{eqnarray}
\left[\int (\hat\theta^i-\theta^i)(\hat\theta^j-\theta^j)
\lgl \phi'| E(d\hat\theta )| \phi'\rgl\right],
\label{eqn:ve}
\end{eqnarray}

For any locally unbiased measurement $M$,
there exists a Hilbert space ${\cal H}_{M}$ and a simple measurement
$E_{M}$ in ${\cal H}_{M}$ which satisfies $(\ref{eqn:naimark})$
by virtue of Naimark's theorem. Note that $E_{M}$ is also locally unbiased.
$span_{\bf C}\{|\phi\rgl, |l_i\rgl,{\bf M}^i(M, |\phi\rgl)\: |\:i=1,...,m\}$
isometrically onto ${\cal K}_{\theta}$
so that $\{|\phi\rgl,|l_i\rgl\: |\:i=1...m\}$
are mapped to $\{|\phi'\rgl,|l'_i\rgl\: |\: i=1,...,m\}\:$,
we denote the images of $\{{\bf M}^i(M, |\phi\rgl)\: |\:i=1,...,m\}$ by
$\{|x^i\rgl\: |\: i=1,...,m\}$.

Then, by virtue of the commuting theorem,
we can construct a simple measurement $E$
in ${\cal K}_{\theta}$ satisfying the equations
$(\ref{eqn:xe})$ - $(\ref{eqn:ve})$.

In our direct approach,
the reduction theorem reduces
the determination of ${\cal V}$ to
the determination of the set of matrices

\begin{eqnarray}
V={\rm Re}{\sf X}^*{\sf X}
\nonumber
\end{eqnarray}

where a system
${\bf X}=[|x^1\rgl, |x^2\rgl, ..., |x^m\rgl]$ of
elements of ${\cal K}_{\theta}\ominus\{|\phi'\rgl\}$
which satisfies
$(\ref{eqn:imxx=0})$ and $(\ref{eqn:naimark:unbiased})$.
In the same way,
we minimize

\begin{eqnarray}
\Tr G{\rm Re}{\sf X}^*{\sf X},
\nonumber
\end{eqnarray}

$\{|x^i\rgl\: |\: i=1,...,m\}\in {\cal K}_{\theta}\ominus\{|\phi'\rgl\}$,
under the restriction such that
the equations $(\ref{eqn:imxx=0})$ and  $(\ref{eqn:naimark:unbiased})$
are satisfied,
instead of minimization of $\tr GV$
where $V$ runs through ${\cal V}$.

Now, the problem is simplified to the large extent,
we only need to treat
with vectors $\{|x^i\rgl\: |\: i=1,...,m\}$
in finite dimensional Hikbert space  ${\cal K}_{\theta}$
instead of measurements, or operator valued measures.

We  conclude this  section with
a corollary of reduction theorem,
which is seemingly paradoxical, since
historically, non-projection-valued measurement
is introduced to describe  measurements of
non-commuting observables.

When the dimension of ${\cal H}$ is larger than or equal to $2m+1$,
for any unbiased measurement $M$ in ${\cal H}$,
there is a simple measurement $E$ in ${\cal H}$
which has the same covariance matrix as that of $M$.

$\{ |l'_i\rgl\: |\:i=1,...,m\}$
to be $\{ |l_i\rgl\: |\:i=1,...,m\}$.

Especially, if  ${\cal H}$ is infinite dimensional,
as is the space of wave functions,
the assumption of the corollary is always satisfied.

§ LAGRANGE'S METHOD OF INDETERMINATE
COEFFICIENTS IN THE PURE STATE ESTIMATION THEORY

Now, we apply our direct approach to the problems presented in the end of
section $\ref{sec:nonclassical}$.
To minimize the functional
$\Tr G{\rm Re}{\sf X}^*{\sf X}$ of vectors in ${\cal K}_{\theta}$,
Langrange's indeterminate coefficients method is employed.
denoting an ordered pair $\{|l_i'\rgl\: |\: i=1,...,\}$
of vectors in ${\cal K}_{\theta}$ also by ${\sf L}$,
the symbol which is used also for
an ordered pair $\{|l_i\: |\: i=1,...,\}$
of vectors in ${\cal H}$,
we define a function $Lag({\sf X})$ by

\begin{eqnarray}
Lag({\sf X})\equiv
{\rm Re}{\Tr}{\sf X}^*{\sf X}G
-2{\Tr}(({\rm Re}{\sf X}^*{\sf L}-I_m)\Xi)
-{\Tr}{\rm Im}{\sf X}^*{\sf X}\Lambda,
\label{eqn:lagrangean}
\end{eqnarray}

where $\Xi, \Lambda$ are matrices whose components are
Langrange's indeterminate coefficients.
Here, $\Lambda$ can be chosen to be antisymmetric, for

\begin{eqnarray}
{\Tr}{\rm Im}{\sf X}^*{\sf X}\Lambda
={\Tr}{\rm Im}{\sf X}^*{\sf X}(\Lambda-\Lambda^T)/2
\nonumber
\end{eqnarray}

holds true and only skew symmetric part of $\Lambda$ appears
in $(\ref{eqn:lagrangean})$.

From here, we follow the routine of
Langrange's method of indeterminate coefficients.
Differentiating $L({\sf X}+\varepsilon\delta{\sf X})$
with respect to $\varepsilon$
and substituting  $0$ into $\varepsilon$ in the derivative,
we get

\begin{eqnarray}
{\rm Re}{\Tr}(\delta{\sf X}^*
(2{\sf X}G-2{\sf L}\Xi-2i{\sf X}\Lambda))=0.
\nonumber
\end{eqnarray}

Because $\delta{\sf X}$ is arbitrary,

\begin{eqnarray}
{\sf X}(G-i\Lambda)={\sf L}\Xi
\label{eqn:xg-il}
\end{eqnarray}

is induced.

Multipling ${\sf X}^*$ to both sides of
the real part of the outcomming equation,
together with $(\ref{eqn:lagrange:restriction1})$, yields

\begin{eqnarray}
\Xi={\rm Re}{\sf X}^*{\sf X}G=VG.
\label{eqn:lagrange:xi}
\end{eqnarray}

Substituting $(\ref{eqn:lagrange:xi})$ into $(\ref{eqn:xg-il})$ ,
we obtain

\begin{eqnarray}
{\sf X}(G-i\Lambda)={\sf L}VG.
\label{eqn:basic0.1}
\end{eqnarray}

If $(\ref{eqn:lagrange:restriction1})$,
$(\ref{eqn:basic0.1})$ and $V={\rm Re}{\sf X}^*{\sf X}$
are solved for $V$, ${\sf X}$ and
real skew symmetric matrix $\Lambda$,
our problems will be perfectly solved.
so far, solutions only for special cases are known.

The rest of this section is devoted to the proof of
the theorem which claim a little stronger assertion than
the corollary $\ref{corollary:ve=vm1}$.

Though the  real linear space $span_{\bf R}\{ {\sf L}\}$ is
always $m$-dimensional for the parameters not to be redundant,
the dimension of the complex linear space
$span_{\bf C}\{ {\sf L}\}$, or  the rank of ${\sf L}$,
is not necessarily equal to $m$.
If ${\rm rank}_{\C}{\sf L}={\rm rank}G=m$ is assumed,
since the rank of the matrix $V$ is $m$ as is proved soon,
the rank of the left hand side of $(\ref{eqn:basic0.1})$
is $m$, so is the rank of the right hand side,
that $G-i\Lambda$ is invertible.
The rank of the matrix $V$ is $m$
because $(\ref{eqn:lagrange:restriction1})$ implies that
the dimension of
$span_{\bf R}\{ {\sf X}\}$ is $m$.

${\sf X}$ is given by ${\sf L}VG(G-i\Lambda)^{-1}$,
we can conclude that $span_{\bf C}\{ {\sf X}\}$
should be a subspace of
$span_{\bf C}\{ {\sf L}\}$.
Therefore, by the same argument as in the proof of
the corollary $\ref{corollary:ve=vm1}$,
we obtain the following theorem.

Suppose that the dimension ${\cal H}$ is larger than or equal to $m+1$,
and that the dimension of the complex linear space
$span_{\bf C}\{ {\sf L}\}$ is $m$.
Then, for any strictly positive weight matrix $G$,
the attainable CR type bound $\CR(G)$
is attained by a simple measurement.

§ THE MODEL WITH TWO PARAMETERS

In this section, we determine ${\cal V}$ for the arbitrary 2-parameter
pure state model.

The equation$(\ref{eqn:basic0.1})$,
mixed  with $(\ref{eqn:imxx=0})$,  leads to

\begin{eqnarray}
=GV{\sf L}^*{\sf L}VG.
\label{eqn:basic1.0}
\end{eqnarray}

real part and imaginary part  are

\begin{eqnarray}
GVG-\Lambda V \Lambda =GV J^S VG,
\label{eqn:basic1.1}
\end{eqnarray}

\begin{eqnarray}
GV\Lambda +\Lambda VG=-GV\tilde{J}VG,
\label{eqn:basic1.2}
\end{eqnarray}

where $\tilde J$ denotes ${\rm Im}{\sf L}^*{\sf L}$,

We assert that
when the matrix $G$ is strictly positive,
$(\ref{eqn:basic1.0})$ is equivalent to the existence of ${\sf X}$
which satisfies
and $V={\rm Re}{\sf X}^*{\sf X}$.
If real positive symmetric matrix $V$ and
real antisymmetric matrix $\Lambda$ satisfying $(\ref{eqn:basic1.0})$ exist,
${\sf X}$ which satisfies  $(\ref{eqn:basic0.1})$ and
$(\ref{eqn:imxx=0})$ is given by
${\sf X}=UV^{1/2}$, where $U$ is a $2m+1$ by $m$  complex matrix such that
If $G$ is strictly positive, ${\sf X}=UV^{1/2}$ also satisfies

\begin{eqnarray}
VG={\rm Re}{\sf X}^*{\sf L}VG\nonumber
\nonumber
\end{eqnarray}

is obtained by
multiplication of ${\sf X}^*$ to
and taking real part of the both sides of $(\ref{eqn:basic0.1})$,
and our assertion is proved.

Hence, our task is to solve $(\ref{eqn:basic1.1})$ and  $(\ref{eqn:basic1.2})$
for real positive symmetric matrix $V$ and real antisymmetric matrix $\Lambda$,
if $G$ is strictly positive.
When $G$ is not strictly positive,
after solving $(\ref{eqn:basic1.1})$ and  $(\ref{eqn:basic1.2})$,
we must check
whether there exists an ordered pair ${\sf X}$ of vectors
which satisfies
$(\ref{eqn:lagrange:restriction1})$, $(\ref{eqn:imxx=0})$
$V={\rm Re}{\sf X}^*{\sf X}$.

In the remainder of  this section, we use the coordinate system
where  $J^S$ is equal to the identity $I_m$.
Given an arbitrary coordinate system $\{\theta^i|\: i=1,...,m\}$,
such a coordinate system $\{\theta'^i|\: i=1,...,m\}$
is obtained by the following coordinate transform:

\begin{eqnarray}
\theta'^i=\sum_{j=1}^m [(J^{S})^{1/2}]_{ij}\:\theta^j \:(i=1,...,m)
\label{eqn:2para:trans1}
\end{eqnarray}

By this coordinate transform,

$V$ is transformed as:

\begin{eqnarray}
%{\sf L}'&=&{\sf L}(J^{S})^{-1/2},\nonumber\\
%{\sf X}'&=&{\sf X}(J^{S})^{1/2},\nonumber\\
%G'&=&(J^{S})^{-1/2} G (J^{S})^{-1/2},\nonumber\\
%\Lambda'&=&(J^{S})^{-1/2} \Lambda (J^{S})^{-1/2},\nonumber\\
V'&=&(J^{S})^{1/2} V (J^{S})^{1/2}.
\label{eqn:2para:trans2}
\end{eqnarray}

Therefore, the result in the originally given coordinate is
obtained as a transformation of the result
in the coordinate system $\{\theta'^i|\: i=1,...,m\}$,
by using
$(\ref{eqn:2para:trans2})$ in the converse way.

So far, we have not assumed $\dim{\cal M}=2$.
When $\dim{\cal M}=2$, covariance matrices are members of
the space $Sym(2)$ of  $2\times 2$ symmetric matrices
which is parameterized by the real variables $x,y,$ and $z$, where

\begin{eqnarray}
\left\{V\left|V=
\left[
\begin{array}{cc}
z+x & y\\
y   & z-x
\end{array}
\right]
\right.
\right\}.
\nonumber
\end{eqnarray}

Before tackling the equations $(\ref{eqn:basic1.1})$
and  $(\ref{eqn:basic1.2})$, three useful facts about this parameterization
are noted.
First, letting $A$ is a symmetric real matrix
which is represented by $(A_x, A_y, A_z)$ in the $(x,y,z)$-space,
the set ${\cal PM}(A)$ of all matrices larger than $A$

\begin{eqnarray}
{\cal PM}(A)
=\{(x,y,z)\:|\: (z-A_z)^2-(x-A_x)^2-(y-A_y)^2\ge 0,\:(z-A_z)\ge 0 \},
\nonumber
\end{eqnarray}

that is, interior
of a upside-down corn with its vertex at $A=(A_x, A_y, A_z)$.
$\cal V$ is a subset of ${\cal PM}(I_m)$, or inside of
an upside-down corn with its vertex
at $(0,0,1)$ because of SLD CR bound.
When the model $\cal M$ is locally quasi-classical at $\theta$,
$\cal V$ coincides with  ${\cal PM}(I_m)$.

Second, an  action of
rotation matrix $R_{\theta}$ to $V$ such that
$R_{\theta}V R_{\theta}^T$,

\begin{eqnarray}
\begin{array}{cc}
\cos\theta & -\sin\theta\\
\sin\theta & \cos\theta
\end{array}
\right],
\nonumber
\end{eqnarray}

corresponds to the rotation in the $(x,y,z)$-space around
$z$-axis by the angle $2\theta$.

Third, we have the following lemma.

${\cal V}$ is rotationally symmetric around $z$-axis,
if the coordinate in $\cal M$ is chosen so that
$J^S$ writes the unit matrix $I_m$.

$\cal V$ is the set of every matrix  which writes
${\rm Re}{\sf X}^*{\sf X}$
using a $2m+1$ by $m$ complex matrix ${\sf X}$ satisfying
$(\ref{eqn:lagrange:restriction1})$ and $(\ref{eqn:imxx=0})$.
Therefore,  the  rotational symmetry of $\cal V$
around $z$-axis is equivalent to
the existence of  a $2m+1\times m$ complex matrix ${\sf Y}$ satisfying
$(\ref{eqn:lagrange:restriction1})$, $(\ref{eqn:imxx=0})$

\begin{eqnarray}
\forall\theta\;\; R_{\theta}{\sf X}^*{\sf X}R_{\theta}^T={\sf Y}^*{\sf Y},
\label{eqn:rxxr}
\end{eqnarray}

for any $2m+1\times m$ complex matrix ${\sf X}$
which satisfies
$(\ref{eqn:lagrange:restriction1})$ and $(\ref{eqn:imxx=0})$.
Because of ${\sf L}^*{\sf L}=I_m+i\tilde{J}$,
elementary calculation shows that

\begin{eqnarray}
{\sf L}^*{\sf L}={\rm Re}R_{\theta}{\sf L}^*{\sf L}R_{\theta}^T,
\nonumber
\end{eqnarray}

or, that for some unitary matrix $U$ in ${\cal K}_{\theta}$,

\[
{\sf L}^*U=R_{\theta}{\sf L}^*,
\]

which leads, together with $(\ref{eqn:lagrange:restriction1})$, to

\begin{eqnarray}
{\rm Re}{\sf L}^*U{\sf X}=R_{\theta}.
\end{eqnarray}

\[
{\sf Y}=U{\sf X}R_{\theta}^T
\]

satisfies $(\ref{eqn:rxxr})$, and we have the lemma.

Because of lemma $\ref{lemma:Vrot}$,
${\cal V}$ is determined
if the boundary of the intersection $\tilde{\cal V}$ of $\cal V$
and the $zx$-plane
is calculated.
Note that
the `inner product' $\Tr GV$
of $G$ and $V$
does not take its minimum at $V\in \tilde{\cal V}$
unless $G$ is in the $zx$-plane.
Therefore, to obtain $bd\tilde{\cal V}$,
only diagonal weight matrix is needed to be considered.

Let us begin with the case of a positive definite weight matrix.
In this case, we only need to deal with
$(\ref{eqn:basic1.1})$ and $(\ref{eqn:basic1.2})$.

\begin{eqnarray}
\tilde{J}=\left[
\begin{array}{cc}
0 & -\beta\\
\beta & 0
\end{array}
\right],
\nonumber
\end{eqnarray}

\begin{eqnarray}
\Lambda=\left[
\begin{array}{cc}
0 & -\lambda\\
\lambda & 0
\end{array}
\right],\:
\begin{array}{cc}
1 & 0 \\
0 & g
\end{array}
\right],\:
\begin{array}{cc}
u & 0 \\
0 & v
\end{array}
\right],
\nonumber
\end{eqnarray}

where $g,v,$ and $u$ are positive real real numbers.
Note that

\begin{eqnarray}
|\beta|\leq 1
\nonumber
\end{eqnarray}

holds, because ${\sf L}^*{\sf L}=I_m+\tilde{J}$
is nonnegative definite.
Then,  $(\ref{eqn:basic1.1})$ and $(\ref{eqn:basic1.2})$ writes

\begin{eqnarray}
vg\lambda+u\lambda+uv\beta g=0.
\label{eqn:2para:uvglmd}
\end{eqnarray}

The necessary and sufficient condition for
$\lambda$ and positive $g$ to exist is,
after some calculations,

\begin{eqnarray}
\label{eqn:2para:uv}
\end{eqnarray}

Note that  $u$ and $v$ are larger than  or equal to $1$,
because $V\ge J^{S-1}=I_m$.
Substitution of  $u=z+x$ and $v=z-x$ into $(\ref{eqn:2para:uv})$,
after some calculations, leads to

\begin{eqnarray}
\pm(1-\beta^2)^{1/2}((z+x-1)^{1/2}+(z-x-1)^{1/2})=|\beta|\nonumber\\
\label{eqn:zx+-1}
\end{eqnarray}

Fig.1 shows that
the lower sign in the equation $(\ref{eqn:zx+-1})$,

\begin{eqnarray}
\label{eqn:2para:zx1}
\end{eqnarray}

gives a part of $bd\tilde{\cal V}$.
In $(\ref{eqn:2para:zx1})$, $x$ takes value ranging
from $-\beta^2/(1-\beta^2)$ to  $\beta^2/(1-\beta^2)$ if $|\beta|$ is smaller
than $1$.
When $|\beta|=1$, $x$ varies from $-\infty$ to $\infty$.
This restriction on the range of $x$ comes from
the positivity of $z-x-1$ and $z+x-1$.

[Two stationary lines]
Two stationary lines;

$(i)\: |\beta|(z+x-1)^{1/2}(z-x-1)^{1/2}

$(ii)\: |\beta|(z+x-1)^{1/2}(z-x-1)^{1/2}

When the weight matrix $G$ is

\begin{eqnarray}
\begin{array}{cc}
1 & 0\\
0 & 0
\end{array}
\right],\:\:\mbox{or}\:\:
\begin{array}{cc}
0 & 0\\
0 & 1
\end{array}
\right],
\label{eqn:2para:G}
\end{eqnarray}

we must treat the case of  $|\beta|=1$ and
the case of $|\beta|<1$ separately.
If $|\beta|=1$,
there exists no  $(2m+1)\times m $ complex matrix ${\sf X}$
which satisfies  $V={\rm Re}{\sf X}^*{\sf X}$,
$(\ref{eqn:lagrange:restriction1})$, and $(\ref{eqn:imxx=0})$.
On the other hand, if $|\beta|<1$,
such complex matrix ${\sf X}$ always exists and
$V={\rm Re}{\sf X}^*{\sf X}$ is given by, in terms of $(x,y,z)$,

\begin{eqnarray}
z&=&-x+1,\: x\leq -\frac{\beta^2 }{1-\beta^2}\nonumber\\
z&=&x+1,\: x\ge \frac{\beta^2 }{1-\beta^2}
\label{eqn:2para:zx2}
\end{eqnarray}

With the help of  $(\ref{eqn:2para:zx1})$ and $(\ref{eqn:2para:zx2})$,
${\cal V}$ is depicted as Fig.2.
The intersection of $z$-axis and $bd{\cal V}$ gives

\begin{eqnarray}
\CR(J^S)=\min\{J^S V|V\in {\cal V}\}=\frac{4}{1+(1-|\beta|^2)^{1/2}},
\label{eqn:minvv}
\end{eqnarray}

where the equality holds in any  coordinate of the model ${\cal M}$.
Simple calculation leads to following theorem.

If a  model ${\cal M}$ has lager value of $|\beta|$ at $\theta$ than
another model ${\cal N}$ has at $\theta'$,
the ${\cal V}_{\theta}({\cal M})$
is a subset of ${\cal V}_{\theta'}({\cal N})$.

By virtue of this theorem,
$|\beta|$ can be seen as a measure of `uncertainty'
between the two parameters.
Two extreme cases are worthy of special attention;
When $|\beta|=0$, the model $\cal M$ is locally quasi-classical at $\theta$
and ${\cal V}$ is largest.
On the other hand,
if $|\beta|=1$ , ${\cal V}$ is smallest  and
uncertainty between  $\theta^1$ and $\theta^2$ is maximum.
In the latter case, we say that the model is coherent at $\theta$.

We define generalized spin coherent model [2] by

\begin{eqnarray}
&{\cal M}_{s,m}&=\pi({\cal N})\nonumber\\
&{\cal N}_{s,m}&\nonumber\\
&=&\{ |\phi(\theta)\rgl\, |\,
|\phi(\theta)\rgl=\exp i\theta^1(\sin\theta^2 S_x-\cos\theta^2 S_y)
|s, m\rgl,\,
\label{eqn:gscoherent}
\end{eqnarray}

where $S_x$, $S_y$, $S_z$ are spin operators,
and $|s,m\rgl$ is defined by,

\begin{eqnarray}
&&S_z|j,m\rgl=\hbar m |s, m\rgl\nonumber\\
&&(S^2_x+S^2_y+S^2_z)|s,m\rgl=\hbar^2 s(s+1)|s, m\rgl.
\nonumber
\end{eqnarray}

$s$ takes value of half integers,
and m is a half integer such that $-j\leq m\leq j$.
Then after tedious calculations, we obtain

\begin{eqnarray}
{\bf M}\left(\hlift_{|s, m\rgl}\,\partial_1\right)&=&
2i(\sin\theta^2 S_x-\cos\theta^2 S_y)|s, m\rgl\nonumber\\
{\bf M}\left(\hlift_{|s, m\rgl}\, \partial_2\right)&=&
2i\{-\sin\theta^1(\cos\theta^2 S_x + \sin\theta^2 S_y )
+(\cos\theta-1)(S_z-m\hbar)\}|s, m\rgl,\nonumber\\
\nonumber
\end{eqnarray}

\begin{eqnarray}
\begin{array}{cc}
1 & 0\\
0 & \sin^2\theta^1
\end{array}
\right],\nonumber\\
\tilde{J}&=&\left[
\begin{array}{cc}
0 & 2m\hbar^2\sin\theta^1 \\
-2m\hbar^2\sin\theta^1 & 0
\end{array}
\right],\nonumber\\
\beta_{s,m} &=& \frac{m}{s^2+ s -m^2}.
\nonumber
\end{eqnarray}

If $m=\alpha s$, where $\alpha<1$ is a constant,
$\beta_{s,m}$ tends to zero as $s\rightarrow\infty$,
and the model ${\cal M}_{s,m}$ becomes locally quasi-classical.
However, if  $m=s$, the model ${\cal M}_{s,m}$ is coherent for any $s$.

§ MULTIPLICATION OF THE IMAGINARY UNIT

As is shown in the previous section,
in the 2-parameter model,
a good index of `uncertainty' between two parameters,
or a measure of how distinct the model is from the classical model.
It can be easily shown that, whatever coordinate
of the model ${\cal M}$ is chosen,
$\pm i\beta$ are the eigenvalues of
the matrix $J^{S-1}\tilde{J}$,
is deeply related to the complex structure of the model.
that  matrix
stands for the linear map ${\bf D}$ from ${\cal T}_{\rho}({\cal M})$
onto ${\cal T}_{\rho}({\cal M})$ defined as in the followings;
First, we  multiply the imaginary unit $i$ to
$|l_X\rgl={\bf M}(\hlift(|l_X\rgl))$ and
and ${\bf M}^{-1}$ and $\pi_*$ are applied successively to
$\pi_*({\bf M}^{-1}(i|l_X\rgl))$
is not a member of ${\cal T}_{\rho}({\cal M})$ generally,
we project $\pi_*({\bf M}^{-1}(i|l_X\rgl))\in {\cal T}_{\rho}({\cal P}_1)$
to  ${\cal T}_{\rho}({\cal M})$
with respect to the metric $\lgl *|* \rgl_{\rho}$,
and we obtain  ${\bf D}X\in {\cal T}_{\rho}({\cal M})$.

\begin{eqnarray}
\begin{array}{ccc}
&\mbox{multiplication of}\:\: i&
\\
|l_X\rgl  \in span_{\bf R}{\sf L}&--\longrightarrow
&i|l_X\rgl  \in span_{\bf R}\{{\sf L}, i{\sf L}\}
\\
\uparrow& &\:\downarrow {\bf M}^{-1},\, \pi_*
\\
\hlift_{|\phi\rgl},\, {\bf M}& &\pi_*({\bf M}^{-1}(i|l_X\rgl))\in {\cal T}_{\rho}({\cal P}_1)
\\
\uparrow& &\mbox{project}\,\downarrow \mbox{w.r.t.\,  $\lgl*,*\rgl_{\rho}$}
\\
X\in{\cal T}_{\rho}({\cal M})&--\longrightarrow
& {\bf D}X\in{\cal T}_{\rho}({\cal M})
\\
&{\bf D}&
\end{array}
\nonumber
\end{eqnarray}

The following theorems are  straightforward
consequences of the above discussion.

The  absolute value
of an  eigenvalues of ${\bf D}$, or equivalently, of $J^{S-1}\tilde{J}$,
is smaller than or equal to $1$.

§ THE COHERENT MODEL

As for the model with arbitrary number of parameters,
the model is said to be coherent at $\theta$
iff all of the eigenvalues of $(J^S)^{-1}\tilde{J}$ are $\pm i$.
When the number of parameters is $2$,
this definition of coherency reduces to $|\beta|=1$.
It should be noted that the eigenvalues of $J^{S -1}\tilde{J}$ are ,
whether the model is coherent or not, of the form $\pm i\beta_j$ or 0.
Therefore, the number of parameters of the coherent model is even.

In this section, we determine the attainable CR type bound of
the coherent model with arbitrary numbers of parameters.
The coherent model is worthy of attention firstly
because the coherent model is `the maximal uncertainty' model,
secondly because there are many physically important coherent models.

Because $J^{S-1}\tilde{J}$ is
a representation of ${\bf D}$,
or of multiplication of the imaginary unit $i$,
the following theorem.

The model $\cal M$ is coherent at $\theta$ iff

\begin{eqnarray}
\label{eqn:dd=-1}
\end{eqnarray}

holds true.

The model $\cal M$ is coherent at $\theta$ iff
$span_{\R}\{i{\sf L}\}$ is identical to $span_{\R}{\sf L}$,
or equivalently, iff
$span_{\R}\{ {\sf L}, i{\sf L}\}$ is identical to
$span_{\R}{\sf L}$.

This theorem  leads to the following theorem.

The model $\cal M$ is coherent at $\theta$ iff
the dimension of $span_{\C}{\sf L}$ is $m/2$.

First, we  assume that

\begin{eqnarray}
\dim_{\C}span_{\C}{\sf L}=m/2.
\label{eqn:dimL=m/2}
\end{eqnarray}

Because $span_{\R}{\sf L}$ is a $m$-dimensional subspace of
$span_{\R}\{{\sf L},i{\sf L}\}$ whose dimension is
smaller than or equal to $m$ because of $(\ref{eqn:dimL=m/2})$,
we have
$span_{\R}\{{\sf L},i{\sf L}\}=span_{\R}{\sf L}$,
or coherency of the model at $\theta$.

Conversely, let us assume that the model is coherent at $\theta$.

If we take an orthonormal basis $\{e_j|\: j=1,...m\}$ of
${\cal T}_{\rho}({\cal M})$ such that
$e_{j+m/2}={\bf D}e_i\:(i=1,2,...,m/2)$,
${\bf M}(\hlift(e_{j+m/2}))=i{\bf M}(\hlift(e_{j}))\:(i=1,2,...,m/2)$
holds true,
and any element $|u\rgl$ of
$span_{\R}{\sf L}= span_{\R}\{{\sf L}, i{\sf L}\}$

\begin{eqnarray}
|u\rgl &=&\sum_{j=1}^m a_j {\bf M}(\hlift(e_{j})) \nonumber\\
&=&\sum_{j=1}^{m/2} (a_j+ i a_{j+(m/2)}){\bf M}(\hlift(e_{j})),
\nonumber
\end{eqnarray}

implying that the dimension of $span_{\C}{\sf L}$ is $m/2$.

In 1996, Fujiwara and Nagaoka [9]
determined the attainable CR type bound of
the two parameter coherent model.
In the following, more generally,
we calculate the bound of the coherent model with arbitrary
number of parameters.

In the case of the coherent model, ${\rm Re}{\sf L}^*{\sf X}=I_m$
or its equivalence ${\rm Re}{\sf L}^*({\sf X}-{\sf L}J^{S-1})=0$,

\begin{eqnarray}
{\sf L}^*{\sf X}=I_m+i{\tilde J}J^{S-1}.
\nonumber
\end{eqnarray}

\begin{eqnarray}
{\rm Im}{\sf L}^*({\sf X}-{\sf L}J^{S-1})&=&
-{\rm Re}i{\sf L}^*({\sf X}-{\sf L}J^{S-1})\nonumber\\
\label{eqn:coherent:imlx0=0}
\end{eqnarray}

Here, $span_{\R}{\sf L}=span_{\R}i{\sf L}$ is
used to deduce the last equality.
$(\ref{eqn:coherent:imlx0=0})$ and
${\rm Re}{\sf L}^*({\sf X}-{\sf L}J^{S-1})=0$

\begin{eqnarray}
{\sf L}^*{\sf X}={\sf L}^*{\sf L}J^{S-1}=I_m+i{\tilde J}J^{S-1}.
\nonumber
\end{eqnarray}

Multiplication of  ${\sf L}^*$ to the both sides of $(\ref{eqn:basic0.1})$,
together with the lemma  presented above, yields

\begin{eqnarray}
(I_m+i{\tilde J}J^{S-1})(G-i\Lambda)=(J^S+i\tilde{J})VG.
\label{eqn:coherent:g-il}
\end{eqnarray}

By virtue of $(\ref{eqn:dd=-1})$,
both of the real part and the imaginary part of
$(\ref{eqn:coherent:g-il})$ give the same equation,

\begin{eqnarray}
G+{\tilde J}J^{S-1}\Lambda=J^SVG,
\nonumber
\end{eqnarray}

\begin{eqnarray}
\left(\, G^{1/2}J^{S-1}{\tilde J}J^{S-1}G^{1/2}\,\right)
\left(\, G^{-1/2}\Lambda G^{-1/2}\,\right).
\nonumber\\
\label{eqn:vj=jl}
\end{eqnarray}

Therefore, letting $a_i$ and $b_i$ denote  the eigenvalues of
$G^{1/2}J^{S-1}{\tilde J}J^{S-1}G^{1/2}$ and
$G^{-1/2}\Lambda G^{-1/2}$ respectively, we have

\begin{eqnarray}
&&\left[\, (G^{1/2}J^{S-1}{\tilde J}J^{S-1}G^{1/2}),\,
(G^{-1/2}\Lambda G^{-1/2})\,\right]=0\nonumber\\
&&\Tr\,\left\{\,(G^{1/2}J^{S-1}{\tilde J}J^{S-1}G^{1/2})
(G^{-1/2}\Lambda G^{-1/2})\,\right\}
= \sum_i |a_i||b_i|,
\nonumber
\end{eqnarray}

because $(\ref{eqn:vj=jl})$ and  SLD CR inequality implies
that $(G^{1/2}J^{S-1}{\tilde J}J^{S-1}G^{1/2})(G^{-1/2}\Lambda G^{-1/2})$
is positive Hermitian.

On the other hand, $(\ref{eqn:basic0.1})$ or its equivalence,

\begin{eqnarray}
{\sf X}G^{1/2}(I_m-iG^{-1/2}\Lambda G^{-1/2})
={\sf L}VG^{1/2},
\label{eqn:coherent:i-iglg}
\end{eqnarray}

$|b_i|=1\,(i=1,...,m)$, because
the rank of $I_m-iG^{-1/2}\Lambda G^{-1/2}$ is
shown to be $m/2$
from $(\ref{eqn:coherent:i-iglg})$, and
the rank of matrices ${\sf L}$, ${\sf X}$, and $V$.

\begin{eqnarray}
\rank_{\C}{\sf X}=\rank_{\R}{\sf X}=m,
\nonumber
\end{eqnarray}

hold true,
where the last equation is valid by virtue of ${\rm Im}{\sf X}^*{\sf X}=0$.

After all, letting $\Tr\abs A$ denote
the sum of the absolute values of the eigenvalues of $A$,
we have the following theorem.

\begin{eqnarray}
\CR (G)
=\Tr\, GJ^{S-1}+\Tr\abs\, GJ^{S-1}{\tilde J}J^{S-1},
\nonumber
\end{eqnarray}

where letting $|A|=(AA^*)^{1/2}$,
the covariance matrix $V$ such that

\begin{eqnarray}
V=J^{S-1}+G^{-1/2}\left|G^{1/2}J^{S-1}{\tilde J}J^{S-1}G^{1/2}\right|G^{-1/2}.
\nonumber
\end{eqnarray}

attain the minimum.

To check the coherency of the model, the following theorem,
which is induced from
theorem $\ref{theorem:coherent:beta_i}$, is

the model is coherent at $\theta$

\begin{eqnarray}
|{\rm det} J^S|=|{\rm det} \tilde{J}|.
\nonumber
\end{eqnarray}

(squeezed state model)
Squeezed state model, which has four parameters, is defined by

\begin{eqnarray}
{\cal M}=\{\rho(z, \xi)\:|\; \rho(z,\xi)=\pi(|z,\xi\rgl),\; z,\xi\in\C\},
\nonumber
\end{eqnarray}

\begin{eqnarray}
D(z)&=&\exp(za^{\dagger}-\ol{z} a),\nonumber\\
\exp\left\{\frac{1}{2}(\, \xi a^{\dagger2}-\ol{\xi} a^2\,)\right\}.
\label{eqn:defsqz}
\end{eqnarray}

Here, the operator $a$ is defined as $a=(Q+iP)/\sqrt{2\hbar}$
where $P$ and $Q$ satisfy the canonical commutation relation
$[P,\, Q]=-i\hbar$.
Letting $z=\sqrt{\frac{2}{\hbar}}(\theta^1+i\theta^2)$,
and $\xi=\theta^3 e^{-2i\theta^4}\;(0\leq \theta^3, 0\leq\theta^4 \leq 2\pi)$,
we have

\begin{eqnarray}
{\bf M}\left(\hlift_{|z,\xi\rgl}\partial_{\theta^1}\right)
\nonumber\\
{\bf M}\left(\hlift_{|z,\xi\rgl}\partial_{\theta^2}\right)
\nonumber\\
{\bf M}\left(\hlift_{|z,\xi\rgl}\partial_{\theta^3}\right)
&=&-i(e^{2i\theta^4}a^2- e^{-2i\theta^4}a^{\dagger 2})|z,\xi\rgl,
\nonumber\\
{\bf M}\left(\hlift_{|z,\xi\rgl}\partial_{\theta^4}\right)&=&
i4(\sinh^2 \theta^3)(a^{\dagger}a+\frac{1}{2})|z,\xi\rgl,\nonumber\\
&&-i2(\sinh\theta^3 )(\cosh\theta^3)(e^{2i\theta^4}a^2-
+e^{-2i\theta^4}a^{\dagger 2}))
\end{eqnarray}

\begin{eqnarray}
\begin{array}{cccc}
\cosh 2\theta^3-\sinh 2\theta^3 \cos 2\theta^4 &
\sinh 2\theta^3\sin 2\theta^4 &0&0\\
\sinh 2\theta^3\sin 2\theta^4 &
\cosh 2\theta^3+\sinh 2\theta^3 \cos 2\theta^4 &0&0\\
0&0&0&\hbar\sinh^2 2\theta^3
\end{array}
\right],\nonumber\\
\tilde{J}&=&\frac{2}{\hbar}\left[
\begin{array}{cccc}
0&0&0&-\frac{\hbar}{2}\sinh 2\theta^3\\
0&0&\frac{\hbar}{2}\sinh 2\theta^3&0
\end{array}
\right]
\nonumber
\end{eqnarray}

Coherency of this model is easily checked
by theorem <ref>,

\begin{eqnarray}
|{\rm det} J^S|=|{\rm det} \tilde{J}|=\frac{4}{\hbar^2}\sinh^2 2\theta^3.
\nonumber
\end{eqnarray}

(spin coherent model)
As is pointed out by Fujiwara $[9]$,
spin coherent model ${\cal M}_{s,s}$,
where ${\cal M}_{s,m}$ is defined by $(\ref{eqn:gscoherent})$,
is coherent.

(total space model)
The total space model is the space of all the pure state ${\cal P}_1$
in finite dimensional Hilbert space ${\cal H}$.
By virtue of theorem <ref>,
the coherency of the model is proved by
checking that $span_{\C}{\sf L}$ is invariant by the multiplication of
the imaginary unit $i$.
Let $|l\rgl$ be a horizontal lift of a tangent vector at $|\phi\rgl$.
Then, $i|l\rgl$ is
also a horizontal lift of another tangent vector at $|\phi\rgl$,
because $|\phi\rgl+i|l\rgl dt$ is a member of $\tilde{\cal H}$.

§ INFORMATIONALLY EXCLUSIVE, AND INDEPENDENT PARAMETERS

In a $m$-parameter model ${\cal M}$,
we say parameter $\theta^i$ and $\theta^j$ are
informationally independent at $\theta_0$, iff

\begin{eqnarray}
{\rm Re}\lgl l_i|l_j\rgl |_{\theta=\theta_0}
={\rm Im}\lgl l_i|l_j\rgl |_{\theta=\theta_0}
\nonumber
\end{eqnarray}

because if the equation holds true,
in the estimation of  the parameters $\theta^1,\theta^2$ of
the $2$-parameter submodel ${\cal M}(1,2|\theta_0)$ of ${\cal M}$,

\begin{eqnarray}
{\cal M}(1,2|\theta_0)\equiv
\left\{\rho(\theta)\:\left|\:
\theta=(\,\theta^1,\,\theta^2,\,\theta^3_0 ,...,\,\theta^m_0\,),\,
(\theta^1,\,\theta^2\,)\in \R^2\right.\right\},
\label{eqn:calm12}
\end{eqnarray}

both of the parameters can be estimated up to the accuracy
which is achieved
in the estimation of the parameter of the $1$-parameter submodels
${\cal M}(1|\theta_0)$ and ${\cal M}(2|\theta_0)$,

\begin{eqnarray}
{\cal M}(1|\theta_0)&\equiv&
\{\rho(\theta)\:|\: \theta=(\theta^1,\theta^2_0,...,\theta^m_0),\,
\theta^1\in\R\},
\nonumber\\
{\cal M}(2|\theta_0)&\equiv&
\{\rho(\theta)\:|\: \theta=(\theta^1_0,\theta^2,...,\theta^m_0),\,
\theta^2\in\R\}.
\nonumber
\end{eqnarray}

On the other hand, iff

\begin{eqnarray}
{\rm Re}\lgl l_1|l_2\rgl |_{\theta=\theta_0}=0,
\label{eqn:def:excv}
\end{eqnarray}

and ${\cal M}(1,2|\theta_0)$ is coherent, or equivalently,

\begin{eqnarray}
{\rm Im}\lgl l_1|l_2\rgl |_{\theta=\theta_0}
=\left.(\lgl l_1|l_1\rgl\lgl l_2|l_2\rgl)^{1/2}\right|_{\theta=\theta_0}
\nonumber
\end{eqnarray}

hold true,
we say the parameters are informationally exclusive at $\theta_0$.

Fujiwara and Nagaoka $[9]$
showed that in the coherent model with
two orthogonal parameters, the attainable CR type bound is
achieved by applying the best measurement for each parameter
alternatively to the system.
This fact implies that if two parameters are informationally exclusive,
the one of them do not contain any information about the other.
In fact, we have the following theorem.

If two parameters $\theta^1$ and  $\theta^2$ are informationally exclusive,
any unbiased measurement $M$ in ${\cal M}(1,2|\theta^0)$
which estimates $\theta^1$ as accurately as possible,

\begin{eqnarray}
\int(\hat\theta^1-\theta^1)^2\Tr\rho(\theta_0)M(d\hat\theta)
\label{eqn:excv:1}
\end{eqnarray}

can extract no information about
$\theta^2$ from the system, i.e.,

\begin{eqnarray}
\forall B\subset {\R}^2 \:\:\:
\Tr \left(M(B)
\left. \frac{\partial \rho}{\partial \theta^2}
\right|_{\theta=\theta_0}
\right)={\rm Re}\lgl\phi|M(B)|l_2\rgl=0,
\label{eqn:excv:2}
\end{eqnarray}

and vice versa.

We prove the theorem only for the measurements which writes

\begin{eqnarray}
M(B)=\int_B |\hat\theta\rgl\lgl\hat\theta|\mu(d\hat\theta).
\nonumber
\end{eqnarray}

The proof for general case will be discussed elsewhere.
If $(\ref{eqn:excv:1})$ holds true,
as in the proof of lemma $\ref{lemma:genVZ}$
(see Ref.[12], p.88),

\begin{eqnarray}
\lgl\hat\theta|\left\{(\hat\theta^1-\theta^1)|\phi\rgl
-\frac{1}{\lgl l_1|l_1 \rgl}|l_1\rgl\right\}=0
\label{eqn:excv:3}
\end{eqnarray}

must hold.
On the other hand,
because of coherency of ${\cal M}(1,2|\theta^0)$
and $(\ref{eqn:def:excv})$,

for some real number $a$, we have

\begin{eqnarray}
\nonumber
\end{eqnarray}

by use of which it is shown that
$(\ref{eqn:excv:2})$ is equivalent to

\begin{eqnarray}
{\rm Im}(\,\lgl\phi|\hat\theta\rgl\lgl\hat\theta|l_1\rgl\,)=0.
\nonumber
\end{eqnarray}

This equation is obviously true if
$(\ref{eqn:excv:3})$ is true, and we have the theorem.

§ DIRECT SUM OF MODELS

For the submodels

\begin{eqnarray}
{\cal M}_1\equiv{\cal M}(1,2,...,m_1\,|\,\theta_0),\:
{\cal M}_2\equiv{\cal M}(m_1,m_1+1,...,m\,|\,\theta_0)
\nonumber
\end{eqnarray}

of ${\cal M}$,
where ${\cal M}(1,2,...,m_1\,|\,\theta_0)$ and
${\cal M}(m_1,m_1+1,...,m\,|\,\theta_0)$ are defined almost
in the same way as the definition $(\ref{eqn:calm12})$
${\cal M}(1,2\,|\,\theta_0)$,
we write

\begin{eqnarray}
{\cal M}|_{\theta_0}={\cal M}_1\oplus {\cal M}_2|_{\theta_0},
\nonumber
\end{eqnarray}

say that
${\cal M}$ is sum of ${\cal M}$ and ${\cal M}$ at $\theta_0$.
$m-m_1$ is denoted by $m_2$.

If any parameter of ${\cal M}_1$ is informationally independent
of any parameter of ${\cal M}_2$ at $\theta_0$, and
the weight matrix $G$ writes

\begin{eqnarray}
\begin{array}{cc}
G_1 & 0\\
0 & G_2
\end{array}
\right],
\nonumber
\end{eqnarray}

\begin{eqnarray}
\CR(G, \theta_0, {\cal M})=
\CR(G_1, \theta_0, {\cal M}_1)+\CR(G_2, \theta_0, {\cal M}_2)
\nonumber
\end{eqnarray}

When the assumption of the lemma is satisfied,
${\cal M}_1$ and ${\cal M}_2$ are said to be
informationally independent at $\theta_0$.

Let  $\hat\theta_1(\omega)$ and  $\hat\theta_2(\omega)$
be the vector whose components are
the estimates of  $\theta^1, \theta^2,...,\theta^{m_1}$,
and $\theta^{m_1+1}, \theta^{m_1+2},...,\theta^{m}$,

\begin{eqnarray}
\hat\theta_1(\omega)
\,\hat\theta^{m_1}(\omega)\,\right),
\nonumber\\
\hat\theta_2(\omega)
\left(\,\hat\theta^{m_1+1}(\omega),\,\hat\theta^2(\omega),...,
\,\hat\theta^{m}(\omega)\,\right).
\nonumber
\end{eqnarray}

Then, if $(M,\hat\theta, \Omega)$ is locally unbiased,
$(M,\hat\theta_i, \Omega)$ is locally unbiased.
Therefore, we have,

\begin{eqnarray}
\{ M\, |\, \mbox{$(M,\hat\theta_i, \Omega)$ is locally unbiased}\}
\subset\{ M\, |\, \mbox{$(M,\hat\theta, \Omega)$ is locally unbiased}\}
\nonumber
\end{eqnarray}

which yields,

\begin{eqnarray}
& &\min\left\{\left.\Tr GV[\hat{\theta}|M]\, \right|\,
\mbox{$(M,\hat\theta, \Omega)$ is locally unbiased}\right\}\nonumber\\
&=&\min\left\{\left.\Tr G_1 V[\hat{\theta}_1 |M]\,\right|\,
\mbox{$(M,\hat\theta, \Omega)$ is locally unbiased}\right\}\nonumber\\
& &+\min\left\{\left.\Tr G_2 V[\hat{\theta}_2 |M]\,\right|\,
\mbox{$(M,\hat\theta, \Omega)$ is locally unbiased}\right\}\nonumber\\
&\ge& \min\left\{\left.\Tr G_1 V[\hat{\theta}_1 |M]\,\right|\,
\mbox{$(M,\hat\theta_1, \Omega)$ is locally unbiased}\right\}\nonumber\\
& &+\min\left\{\left.\Tr G_2 V[\hat{\theta}_2 |M]\, \right|\,
\mbox{$(M,\hat\theta_2, \Omega)$ is locally unbiased}\right\},\nonumber
\end{eqnarray}

or its equivalence,

\begin{eqnarray}
\CR(G, {\cal M})\ge
\CR(G_1, {\cal M}_1)+\CR(G_2, {\cal M}_2).
\label{eqn:cr>cr+cr}
\end{eqnarray}

Because ${\cal M}_1$ and ${\cal M}_2$ are informationally independent,
${\sf L}$ for ${\cal M}$ writes

\begin{eqnarray}
{\sf L}=\left[
\begin{array}{cc}
{\sf L}_1 & 0 \\
0 & {\sf L}_2
\end{array}
\right],
\nonumber
\end{eqnarray}

in the appropriate coordinate,
where ${\sf L}_1=[|l_1\rgl,|l_2\rgl,...,|l_{m_1}\rgl]$,
and ${\sf L}_2$ is in the same manner.
In that coordinate,  ${\sf X}$ writes

\begin{eqnarray}
{\sf X}=\left[
\begin{array}{cc}
{\sf X}_1 & {\sf X}_{12}\\
{\sf X}_{21} & {\sf X}_2
\end{array}
\right].
\nonumber
\end{eqnarray}

Therefore, if

\begin{eqnarray}
{\rm Re} {\sf X}_i^*{\sf L}_i=I_{m_i},\,
{\rm Im}{\sf X}_i^*{\sf X}_i=0\,(i=1,2)
\nonumber
\end{eqnarray}

holds true,
the measurements corresponding to ${\sf X}$ is locally unbiased,

\begin{eqnarray}
& &\CR(G, {\cal M})\nonumber\\
&=&\min\left\{\, \Tr G {\sf X}^*{\sf X}\,\left|\,
{\rm Re} {\sf X}^*{\sf L}=I_m,\,{\rm Im}{\sf X}^*{\sf X}=0\right.
\,\right\}
\nonumber\\
&\leq&\min\left\{\left. \sum_{i=1}^2\Tr G_i {\sf X}_i^*{\sf X_i}\, \right| \,
{\rm Re} {\sf X}_i^*{\sf L_i}=I_{i_m},\,{\rm Im}{\sf X}_i^*{\sf X}_i =0,
\,(i=1,2)\,\right\}
\nonumber\\
&=&\CR(G_1, {\cal M}_1)+\CR(G_2, {\cal M}_2),
\nonumber
\end{eqnarray}

which, mixed with $(\ref{eqn:cr>cr+cr})$ leads to the lemma.

CHAPTER: BERRY'S PHASE IN QUANTUM ESTIMATION THEORY

§ BERRY'S PHASE

In this section, we review the geometrical theory of Berry's phase.

Berry's phase was discovered by M.V. Berry in 1984[4],
and confirmed by
many experimental facts[28].
In 1987,  Aharonov and Anandan [1] pointed out that
Berry's phase is naturally interpreted as a curvature
in the fiber bundle over ${\cal P}_1$.
Actually, Berry's phase is nothing but the Uhlmann's curvature
restricted to ${\cal P}_1$ [31][33].

Uhlmann's RPF in the space ${\cal P}_1$ takes value
in the set of unimodular complex number.
On the other hand,
Berry's phase takes value in real numbers.
They are related as

\begin{eqnarray}
\mbox{Berry's phase}=-i\ln (\mbox{Uhlmann's RPF}).
\nonumber
\end{eqnarray}

The Berry's phase for the infinitesimal loop $(\ref{loop})$
is calculated up to the second order of $d\theta$

\begin{eqnarray}
\frac{1}{2i}\lgl\phi(0)|F_{ij}|\phi(0)\rgl d\theta^i d\theta^j + o(d\theta)^2
\nonumber
\end{eqnarray}

Because Berry's phase is independent of the choice of SLD,
we can take $L^S_i$ to be $2\partial_i\rho$.
Then, the phase is equal to

\begin{eqnarray}
\frac{1}{2}\tilde{J}_{ij}  d\theta^i d\theta^j + o(d\theta)^2,
\nonumber
\end{eqnarray}

where $\tilde{J}_{ij}$ is equal to
$\frac{1}{2}{\rm Im}\lgl l_i|l_j \rgl$ .

\begin{eqnarray}
\sum _{i,j}\tilde{J}_{ij}d\theta^i d\theta^j
\label{eqn:berrycurv}
\end{eqnarray}

corresponds to the curvature form.

§ BERRY'S PHASE IN QUANTUM ESTIMATION THEORY

It must be noted that the curvature form is
deeply related to the multiplication of the imaginary unit ${\bf D}$.
Actually, The curvature form $(\ref{eqn:berrycurv})$
is identical to a map from
${\cal T}_{\theta}({\cal M})\times {\cal T}_{\theta}({\cal M})$
to $\R$ such that

\begin{eqnarray}
\ds{\sum_{ij}}\tilde{J}_{ij}d\theta^id\theta^j\: :\:
\left\lgl \partial_i,\frac{1}{2}{\bf D}\partial_j \right\rgl.
\nonumber
\end{eqnarray}

Hence, the eigenvalues of ${\bf D}$
can be interpreted in terms of Berry's phase.
Concretely speaking,
taking the coordinate system which is orthonormal at $\theta$
in terms of the metric $\lgl*,*\rgl_{\theta}$,
they are the half of the Berry's phase obtained when the state
goes around the infinitesimal loop $(\ref{loop})$.
Especially, when the model is only with two  parameters,
the eigenvalues of ${\bf D}$ are the half of the Berry's phase per
unit area, where unit of the area is naturally induced from
the metric $\lgl *,* \rgl_{\theta}$.

Therefore, we can roughly say that
the more the Berry's phase for  the loop $(\ref{loop})$,
the harder it is to estimate $\theta^i$ and $\theta^j$ simultaneously.
Namely, $\theta^i$ and $\theta^j$ are informationally independent
iff the Berry's phase for  the loop $(\ref{loop}g)$ vanishes and
$\lgl \partial_i, \partial_j\rgl_{\theta}=0$ holds.
On the other hand, iff the Berry's phase for  the loop $(\ref{loop})$
is maximal and $\lgl \partial_i, \partial_j\rgl_{\theta}=0$ holds,
i.e., $\beta=1$, the two parameters
are informationally exclusive.

This discussion is parallel to that
in the section $\ref{sec:estphase}$,
which was about relations between Uhlmann's parallelism and
the noncommutative nature of
the quantum estimation theory of the faithful model.
Because Berry's phase is nothing but the restriction of
Uhlmann's RPF to the pure state model, this parallelism is

What about the models with arbitrary number of parameters?
By virtue of theorem $\ref{theorem:purelsls0}$,
if Berry's phase for any closed loop vanishes,
the model is localy quasi-classical.
For general pure state models, we have the following theorem.

For any pure state model,

\begin{eqnarray}
&=&\Tr\{{\rm Re}(I_m+i J^{S-1/2}\tilde{J}J^{S-1/2})^{1/2}\}^{-2}\nonumber\\
&=&\sum_{\alpha \in\{\mbox{eigenvalues of ${\bf D}$}\}}
\frac{2}{1+(1-|\alpha|^2)^{1/2}}.
\nonumber
\end{eqnarray}

The estimation theoretical siginificance of $\CR(J^S)$
is hard to verify. However, this value remains invariant under
any transform of the coordinate in the model ${\cal M}$,
and can be an good index of distance between ${\cal V}$ and $J^{S-1}$.

is invariant by any affine coordinate transform in the model ${\cal M}$,
we choose a coordinate in which
$J^S$ writes $I_m$ and $\tilde{J}$ writes

\begin{eqnarray}
\tilde{J}=\left[
\begin{array}{ccccccccc}
0& -\beta_1& 		0&\cdots&0&0&0&\cdots&0\\
\beta_1&0 & 		0&\cdots&0&0&0&\cdots&0\\
0& 0& 		\ddots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\
\vdots&\vdots&\ddots&\ddots&0&0&0&\cdots&0\\
0&0&		\cdots &0&0& -\beta_{l}&0&\cdots&0\\
0&0&		\cdots &0&\beta_{l}&0 &0&\cdots&0\\
0&0&		\cdots &0&    0    &0&0&\ddots&\vdots\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\ddots&\vdots\\
0&0&		\cdots &0&    0    &0 &\cdots&\cdots&0
\end{array}
\right].
\nonumber
\end{eqnarray}

Then, The model  ${\cal M}$ is decomposed into
the direct sum of the submodels one or two parameter ${\cal M}_{\kappa}$,

\begin{eqnarray}
{\cal M}=\bigoplus_{\kappa} {\cal M}_{\kappa},
\nonumber
\end{eqnarray}

any two submodels  ${\cal M}_{\kappa}$ and ${\cal M}_{\kappa'}$
are informationally independent, and
$\tilde{J}$ of a two parameter submodel ${\cal M}_{\kappa}$

\begin{eqnarray}
\left[
\begin{array}{cc}
0& -\beta_{\kappa} \\
\beta_{\kappa}&0
\end{array}
\right].
\nonumber
\end{eqnarray}

Because the weight matrix $J^S=I_m$ writes in the form of
direct some of the weight matrix on the model ${\cal M}_{\kappa}$,
by virtue of lemma $\ref{lemma:independent}$
and the equation $(\ref{eqn:minvv})$,
we have the theorem.

§ BERRY'S PHASE IN THE GLOBAL THEORY OF QUANTUM ESTIMATION

In this section, we present a geometrical sufficient condition
for the pure state model ${\cal M}$ to be quasi-classical
in the sense of section $\ref{sec:questf}$.

For simplicity,
we say that the manifold ${\cal N}$ in ${\cal P}_1$
is a horizontal lift of the model ${\cal M}$ if

\begin{eqnarray}
\pi({\cal N})&=&{\cal M},\\
\forall |\phi(\theta)\rgl\in {\cal N},& &
\frac{1}{2}\frac{\partial}{\partial \theta^i}|\phi(\theta)\rgl
\in{\cal LS}_{|\phi(\theta)\rgl}.
\nonumber
\end{eqnarray}

The horizontal lift ${\cal N}$ exists iff ${\cal M}$
is quasi-classical.

If  the model ${\cal M}$ is parallel, that model is
quasi- classical
in the sense of section $\ref{sec:questf}$.

First, apply  Schmidt's orthonormalization to
the horizontal lift ${\cal N}$ of ${\cal M}$,
to obtain the orthonomal basis ${\bf B}=\{ |e_i\rgl\,|\, i=1, 2, ...\}$
such that ${\cal N}$ is a subset of the real span of ${\bf B}$.
We immerse Hilbert space ${\cal H}$ into $L^2(\R,\,\C)$

\begin{eqnarray}
=\sum_i a_i|e_i\rgl\mapsto {\bf \iota}\sum_i a_i(\theta)\psi_i(x),
\nonumber
\end{eqnarray}

where $\{\ psi_i(x)\,|\, i=1, 2, ...\}$ is an orthonormal basis
in $L^2(\R,\,\C)$. Then, letting $E(dx)=|x\rgl\lgl x|dx$,
the triplet $(\hat\theta(\theta),\, E\, , \R)$ is
one of the best estimators.
This assertion is easily proved by caluculating
the Fisher information matrix of the family,

\begin{eqnarray}
\{ p(x\, |\, \theta)=\sum_i (a_i(\theta))^2 |\psi_i(x)|^2\}
\end{eqnarray}

of probability distributions.

The converse of the latter theorem is, however, not true, because
the following counter-examples exist.

We consider the position shifted model which is defined by

\begin{eqnarray}
&&{\cal M}_x=\pi({\cal N}_x)\nonumber\\
&&{\cal N}_x=\{ |\phi(\theta)\rgl\: |\:
\mbox{\it const.}\times(x-\theta)^2 e^{-(x-\theta)^2+ig(x-\theta)},\:
\theta\in\R \},
\nonumber
\end{eqnarray}

where $c$ is a normalizing constant,
$g$ the function such that

\begin{eqnarray}
\begin{array}{cc}
0 & (x\geq 0),\\
\alpha & (x< 0).
\end{array}
\right.
\nonumber
\end{eqnarray}

Then, as easily checked,
${\cal N}_x$ is a horizontal lift of the model ${\cal M}_x$,
$\lgl\phi(\theta)|\phi(\theta')\rgl$ is not real
unless $\alpha=n\pi\,(n=0,1,...)$.
However, SLD CR bound is uniformly attained by
the measurement obtained
by the spectral decomposition $E_x(dx)=|x\rgl\lgl x| dx$
of the position operator,where $|x_0\rgl=\delta(x-x_0)$.
as is  checked by
comparing SLD Fisher information of the model ${\cal M}_x$
and the classical Fisher information of
the probability distribution family

\begin{eqnarray}
\{ p(x|\theta)\,|\,
\theta\in\R\}.
\nonumber
\end{eqnarray}

Note that $|\phi(\theta)\rgl$ is
an eigenstate of the Hamiltonian

\begin{eqnarray}
+\frac{\hbar^2}{m}\left( 2(x-\theta)^2+\frac{1}{(x-\theta)^2}\right),
\nonumber
\end{eqnarray}

whose potential  has two wells
with infinite height of wall between them.

Let ${\cal H}$ be $L^2([0,\,2\pi],\,\C)$,
and  define a one parameter model ${\cal M}$ such that,

\begin{eqnarray}
&&{\cal M}=\pi({\cal N})\nonumber\\
&&{\cal N}=\{|\phi(\theta)\rgl\: |\:
|\phi(\theta)\rgl=\mbox{\it const.}\times(2-\cos\omega)\,
e^{i\alpha(f(\omega-\theta)+\theta)},\:(0\leq\omega, \theta< 2\pi) \},
\nonumber\\
\label{eqn:ring}
\end{eqnarray}

where $\alpha$ is a real number and
$f$ the function defined by

\begin{eqnarray}
\begin{array}{cc}
\omega-\theta & (\omega-\theta\geq 0)\\
\omega+2\pi-\theta &(\omega-\theta < 0)
\end{array}
\right. .
\nonumber
\end{eqnarray}

Physically, $(\ref{eqn:ring})$ is an eigenstate of the Hamiltonian $H$
such that,

\begin{eqnarray}
\nonumber
\end{eqnarray}

which characterize the dynamics of an electron
confined to the one-dimensional ring which encircles magnetic flax
$\Phi=2\pi\alpha c/e$, where $m$ is the mass of the electron,
$-e$ the charge of the electron, $c$ the velocity of light,
and $A$, $B$ the appropriately chosen constant.

It is easily checked that ${\cal N}$ is  a  horizontal lift of the model
${\cal M}$,
and that
the model ${\cal M}$ is not parallel unless $\alpha= n\pi\,(n=0,1,...)$.
However, consider
the projection valued measure $E_{\omega}$ such that

\begin{eqnarray}
E_{\omega}(d\omega)=|\omega\rgl\lgl\omega| d\omega,
\nonumber
\end{eqnarray}

where $|\omega_0\rgl=\delta(\omega-\omega_0)$.
Then, it is easily checked that
the classical Fisher information of
the probability distribution family

\begin{eqnarray}
\{ p(\omega|\theta)\,|\,
p(\omega|\theta)=|\lgl \phi(\theta)|\omega\rgl |^2,\,
0\leq\omega,\theta< 2\pi\}
\nonumber
\end{eqnarray}

is equal to the SLD Fisher information of ${\cal M}$.

§ ANTIUNITARY OPERATORS

The  transformation $A$

\begin{eqnarray}
|\tilde{a}\rgl=A|a\rgl,\:\:\:  |\tilde{b}\rgl=A|b\rgl
\nonumber
\end{eqnarray}

is said to be antiunitary iff

\begin{eqnarray}
\lgl \tilde{a} |\tilde{b}\rgl&=&\ol{\lgl a| b \rgl},\nonumber\\
\nonumber
\end{eqnarray}

where $\ol{*}$ means complex conjugate.

Fix  an orthonormal basis ${\sf B}=\{|i\rgl\,|\, i=1,2, ... ,d\}$,
and we can then define antiunitary operator $K_{\sf B}$ which
takes complex conjugate of any components in this basis,

\begin{eqnarray}
K_{\sf B}=\sum_i\alpha_i |i\rgl= \sum_i \ol{\alpha}_i|i\rgl.
\nonumber
\end{eqnarray}

For different basis ${\sf B}, {\sf B}'$,
we have

\begin{eqnarray}
K_{\sf B}=U K_{\sf B}'U^*,
\nonumber
\end{eqnarray}

where $U$ is a unitary operator corresponding
to the change of the basis.

Suppose that any member of the manifold
${\cal N}=\{|\phi \rgl\}$ in $\tilde{\cal H}$
is invariant by the antiunitary operator $A$,
and let $|\tilde{\phi}\rgl=A|\phi\rgl,\,|\tilde{\phi}'\rgl=A|\phi'\rgl$.
Then, we have

\begin{eqnarray}
\lgl \phi |\phi'\rgl=\lgl \tilde{\phi'}|\tilde{\phi}\rgl=\lgl \phi'|\phi\rgl
\in\R.
\nonumber
\end{eqnarray}

Conversely, if $\lgl \phi|\phi'\rgl$ is real for any
$|\phi\rgl,\,|\phi'\rgl\in{\cal N}$,
by Schmidt's orthonormalization, we can obtain the basis ${\sf B}$
such that  ${\cal N}$ is subset of the real span of ${\sf B}$,
which means any member of ${\cal N}$ is invariant by
the antiunitary operator $K_{\sf B}$.

Therefore, the premise of the statement of
theorems $\ref{th:p-plgl1}$-$\ref{th:p-plgl2}$
is satisfied iff the horizontal lift of the model is
invariant by some antiunitary operator.

§ TIME REVERSAL SYMMETRY

As an example of the antiunitary operator,
we discuss time reversal operator
(see Ref.[26], pp. 266-282).
The time reversal operator $T$ is an antiunitary operator
in $L^2(\R^3,\,\C)$ which transforms
the wave function $\psi(x)\in L^2(\R^3,\,\C)$ as:

\begin{eqnarray}
T \psi(x)=\ol{\psi(x)}=K_{\{|{\bf x}\rgl\}}\psi(x).
\nonumber
\end{eqnarray}

The term `time reversal' came from the fact
that if  $\psi({\bf x}, t)$ is a solution of
the Schödinger equation

\begin{eqnarray}
i\hbar\frac{\partial\psi}{\partial t}=
\left(-\frac{\hbar^2}{2m}\nabla^2+V\right)\psi,
\nonumber
\end{eqnarray}

then $\ol{\psi({\bf x}, -t)}$ is also its solution.

The operator $T$ is sometimes called motion reversal operator,
since it transforms the momentum eigenstate
$e^{i{\bf p}\cdot{\bf x}/\hbar}$ corresponding to eigenvalue ${\bf p}$
to the eigenstate  $e^{-i{\bf p}\cdot{\bf x}/\hbar}$
corresponding to eigenvalue $-{\bf p}$.

Define the position shifted model by

\begin{eqnarray}
{\cal M}_{\bf x}=
\{\rho(\theta)\,|\,
\rho(\theta)=\pi(\psi({\bf x}-{\bf x}_0)\,),\,{\bf x}_0 \in\R^3\},
\nonumber
\end{eqnarray}

and suppose that any member of
the horizontal lift ${\cal N}_{\bf x}$
of the model ${\cal M}_{\bf x}$
has time reversal symmetry.
Then, since time reversal operator $T$ is antiunitary,
the model  ${\cal M}_{\bf x}$
is quasi-classical in the wider sense.
The spectral decomposition of the position operator
gives optimal measurement.

Now, we discuss the generalization of time reversal operator.
The antiunitary transform

\begin{eqnarray}
T_{\alpha}\, : \, e^{i{\bf p}\cdot{\bf x}/\hbar}
\rightarrow
e^{i\alpha({\bf p})}\,e^{-i{\bf p}\cdot{\bf x}/\hbar}
\nonumber
\end{eqnarray}

is also called
motion reversal operator, or time reversal operator.

If any member $\psi({\bf x}-{\bf x}_0)$ of the horizontal lift
${\cal N}_{\bf x}$
of the position shifted model ${\cal M}_{\bf x}$
is invariant by the time reversal operator $T_{\alpha}$,

\begin{eqnarray}
\int_{\R^3}
\psi({\bf x}-{\bf x}_0)\,\ol{\psi({\bf x}-{\bf x}'_0)}\,
d{\bf x}
\in\R
\label{eqn:psipsi}
\end{eqnarray}

holds true for any ${\bf x}_0,\, {\bf x}'_0$,
which is equivalent to
the premise of theorems $\ref{th:p-plgl1}$-$\ref{th:p-plgl2}$.

Conversely, if $(\ref{eqn:psipsi})$ holds true,
Fourier transform of $(\ref{eqn:psipsi})$ leads to

\begin{eqnarray}
|\Psi({\bf p})|^2=|\Psi(-{\bf p})|^2,
\nonumber
\end{eqnarray}

\begin{eqnarray}
\Psi({\bf p})=
\frac{1}{\sqrt{2\pi}}
\int\psi({\bf x})e^{-i{\bf p}\cdot{\bf x}/\hbar} d{\bf x}.
\nonumber
\end{eqnarray}

any member of ${\cal N}_{\bf x}$ is transformed  to itself
by the time reversal operator $T_{\alpha}$
such that

\begin{eqnarray}
T_{\alpha}\, : \, e^{i{\bf p}\cdot{\bf x}/\hbar}
\rightarrow
e^{i(\beta({\bf p})+\beta(-{\bf p})\,)}\,e^{-i{\bf p}\cdot{\bf x}/\hbar},
\nonumber
\end{eqnarray}

\begin{eqnarray}
e^{i\beta({\bf p})}=\frac{\Psi({\bf p})}{|\Psi({\bf p})|}.
\nonumber
\end{eqnarray}

$(\ref{eqn:psipsi})$ is equivalent to the existence of
the time reversal operator which transforms
any member of the horizontal lift ${\cal N}_{\bf x}$ to itself.

CHAPTER: UNCERTAINTY PRINCIPLE IN VIEW OF QUANTUM ESTIMATION THEORY

§ THE POSITION-MOMENTUM SHIFT MODEL

In this section and the next, we examine the position-momentum uncertainty
in view of quantum estimation theory.

First, it must be emphasized that
so-called `Heisenberg's uncertainty',

\begin{eqnarray}
\lgl (\Delta X)^2\rgl\lgl (\Delta P)^2\rgl \ge \frac{\hbar^2}{4},
\label{eqn:robertson}
\end{eqnarray}

where $\lgl (\Delta X)^2\rgl$ stands for

\begin{eqnarray}
\lgl\phi|(X-\lgl\phi|X|\phi\rgl)^2|\phi\rgl,
\nonumber
\end{eqnarray}

has nothing to do with the Heisenberg's gedanken experiment
which deals with the simultaneous measurement of the position
and the momentum.

$\lgl (\Delta X)^2\rgl$ (, or $\lgl (\Delta P)^2\rgl$)
in $(\ref{eqn:robertson})$
is the variance of the data when
only position (, or momentum) is measured.
$(\ref{eqn:robertson})$ corresponds to the
experiment where
the position is measured for the one of the group of identical particles
and the momentum for the other group of identical particles.
As a matter of fact,
the inequality $(\ref{eqn:robertson})$,
is  derived by H. P. Robertson $[25]$
and some careful researcher call
the inequality Robertson's uncertainty
(Heisenberg himself had nothing to do with the inequality).

The purpose of this  chapter is
to examine the simultaneous measurement of the position
and the momentum from the estimation theoretical viewpoint.
However, since
the measurement obtained by the spectral decomposition of
the position operator
differs from that of the momentum operator,
the simultaneous measurement in exact sense is impossible.
Here, we formulate the problem as a estimation of the shift parameters
$x_0$ and $p_0$,
in the position-momentum shifted model

\begin{eqnarray}
{\cal M}_{xp}=\{\rho(\theta)\: |\:
\rho(x_0,p_0)=\pi(D(x_0,p_0)|\phi_0\rgl),\,
\theta=(x_0,p_0)\in\R^2 \},
\nonumber
\end{eqnarray}

\begin{eqnarray}
D(x_0,p_0)=\exp \frac{i}{\hbar}(p_0 X- x_0 P)
\label{eqn:weilop}
\end{eqnarray}

and $|\phi_0\rgl=\phi_0(x)$ is a member of $L^2(\R,\C)$ such that

\begin{eqnarray}
\lgl\phi_0|X|\phi_0\rgl=\lgl\phi_0|P|\phi_0\rgl=0.
\nonumber
\end{eqnarray}

§ THE ESTIMATION OF THE SHIFT PARAMETERS

Our purpose is to examine how efficiently we can estimate
the shift parameters $\theta=(x_0,p_0)$.

The  horizontal lifts of
$\partial/\partial x_0$, $\partial/\partial p_0$ are

\begin{eqnarray}
\hlift_{|\phi(\theta)\rgl}
\left(\frac{\partial}{\partial x_0}\right)
&=&-\frac{2i}{\hbar}\Delta P_{\theta}|\phi(\theta)\rgl,\nonumber\\
\hlift_{|\phi(\theta)\rgl}
\left(\frac{\partial}{\partial p_0}\right)
&=&\frac{2i}{\hbar}\Delta X_{\theta}|\phi(\theta)\rgl,
\nonumber
\end{eqnarray}

\begin{eqnarray}
\lgl  A \rgl_{\theta}&\equiv& \lgl\phi(\theta)| A |\phi(\theta)\rgl,\nonumber\\
\Delta A_{\theta} &\equiv& A -\lgl  A \rgl_{\theta}
\nonumber
\end{eqnarray}

and the SLD Fisher information matrix $J^S(\theta)$ is,

\begin{eqnarray}
=\frac{4}{\hbar^2}\lgl (\Delta P_{\theta})^2 \rgl,\nonumber\\
&&=\frac{1}{\hbar^2}\lgl (\Delta P_{(0,0)})^2 \rgl,\nonumber\\
\frac{4}{\hbar^2}\lgl (\Delta X_{(0,0)})^2 \rgl, \nonumber\\
\frac{4}{\hbar^2} Cov(X,P)_{(0,0)},
\nonumber
\end{eqnarray}

\frac{1}{2}
\lgl (\Delta X_{\theta}\Delta P_{\theta}
+\Delta P_{\theta}\Delta X_{\theta}) $.
The absolute value $\beta(\theta)$
of the eigenvalue of ${\bf D}$ at $\theta$ is calculated as,

\begin{eqnarray}
\beta(\theta)&=&
\frac{1}{2i}\frac{\lgl [P,X]\rgl_{(0,0)}}
{\sqrt{\lgl(\Delta X_{(0,0)})^2 \rgl\lgl(\Delta P_{(0,0)})^2 \rgl
\nonumber\\
{2\sqrt{\lgl(\Delta X_{\theta})^2\rgl\lgl (\Delta P_{\theta})^2 \rgl
\label{eqn:sftb}
\end{eqnarray}

Notice that $J^S(\theta)$ and $\beta(\theta)$  are independent of
the true value of parameters.

We are interested in
the attainable CR type bound
and in the index $\beta$ of noncommutative nature of the model.

As for $\beta$, $(\ref{eqn:sftb})$ indicates that
the larger the formal `covariance matrix'
[ Note that this formal `covariance matrix' is not
equal to
the covariance matrix of any measurement related to position or momentum ]
of $X$ and $P$

\begin{eqnarray}
\left[
\begin{array}{cc}
\lgl (\Delta P_{(0,0)})^2\rgl & Cov(X,P)_{(0,0)}\\
Cov(X,P)_{(0,0)} & \lgl (\Delta X_{(0,0)})^2\rgl
\end{array}
\right]
\nonumber
\end{eqnarray}

the smaller the noncommutative nature between $x_0$ and $p_0$.

As for the attainable CR type bound,
the SLD Fisher information matrix is proportional to the
formal `covariance matrix' and $\beta$ decreases as
the determinant of the `covariance matrix' increases,
we can metaphorically say that
the larger the `covariance matrix' implies the possibility
of more efficient estimate of $x_0$ and $p_0$,
which is seemingly paradoxical.

We examine these points in  the shifted harmonic oscillator  model
${\cal M}_{xp,n}$, which is defined to be
the position-momentum shifted model in which
$\phi_0(x)$ is equal to
the $n$th eigenstate $|n\rgl$ of the harmonic oscillator.
For ${\cal M}_{xp,n}$, we have

\begin{eqnarray}
&[J^S(\theta)]_{x_0,x_0}&=\frac{4}{\hbar} \left(n+\frac{1}{2}\right),
\nonumber\\
\nonumber\\
\nonumber
\end{eqnarray}

\begin{eqnarray}
\beta(\theta)=\frac{-1}{2(n+1/2)}.
\label{eqn:sfnb}
\nonumber
\end{eqnarray}

Hence, if $n$ is large,
`noncommutative nature' of the parameters is small.

What about the efficiency of the estimate ?
We define ${\cal M}'_{xp,n}$ by
normalizing the parameters in ${\cal M}_{xp,n}$ as

\begin{eqnarray}
\theta=(x_0, p_0)\longrightarrow
\theta=(n+1/2)^{-1/2}x_0,(n+1/2)^{-1/2} p_0).
\nonumber
\end{eqnarray}

Then, directly from the definition,

\begin{eqnarray}
\CR (G,\theta,{\cal M}_{xp,n})=\frac{1}{n+1/2}\CR(G,\theta,{\cal M}'_{xp,n}),
\label{eqn:crmcrm}
\end{eqnarray}

and the SLD Fisher information matrix of  ${\cal M}'_{xp,n}$
is equal to $\frac{4}{\hbar}I_m$ for any $n$.

Because the index $\beta$ is unchanged by the change of the parameter,

\begin{eqnarray}
\CR (G,\theta,{\cal M}'_{xp,0})\ge
\CR (G,\theta,{\cal M}'_{xp,1})\ge
... \ge \CR (G,\theta,{\cal M}'_{xp,n})\ge ...,
\nonumber
\end{eqnarray}

which, combined with $(\ref{eqn:crmcrm})$ leads to

\begin{eqnarray}
\CR (G,\theta,{\cal M}_{xp,0})\ge
\CR (G,\theta,{\cal M}_{xp,1})\ge
... \ge \CR (G,\theta,{\cal M}_{xp,n})\ge ...,
\nonumber
\end{eqnarray}

for any $\theta$ and any  $G$.
if the `covariance matrix' larger,
the more efficient estimate of the parameter is possible.
Especially, when $n=0$, or in the case of the so-called
`minimum uncertainty state', the efficiency of the estimation is
the lowest.

Especially, when $n=0$, or in the case of the so-called
`minimum uncertain state',
the position parameter $x_0$ and the momentum parameter $p_0$
are maximally `noncommutative' in the sense $\beta$ is larger than
that of any other ${\cal M}_{xp,n}\,(n\neq 0)$.
It is easily shown that $\beta$ is maximal, or coherent,
iff $|\phi_0\rgl$ is in the squeezed state, or
where $S(\xi)$ is
the operator defined by $(\ref{eqn:defsqz})$.
In addition, the efficiency of the estimation is
lower than any other ${\cal M}_{xp,n}\,(n\neq 0)$.

If $n$ is very  large, how efficiently can we estimate?
Given $N$ particles, we divide them into two groups,
to one of which we apply the best measurement for
$x_0$ and to the other of which we apply the best measurement for $p_0$ .
The parameter $x_0$ and $p_0$ is estimated only from the data from the
first group and the second group, respectively.
Then, the attained efficiency of the estimation of
$x_0$ is

\begin{eqnarray}
\frac{\hbar}{(N/2)\,4(n+1/2)}
\nonumber
\end{eqnarray}

and the efficiency of the estimation of $p_0$ is

\begin{eqnarray}
\frac{\hbar}{(N/2)\,4(n+1/2)}
\nonumber
\end{eqnarray}

which are combined to yield the efficiency
of this estimate par sample,

\begin{eqnarray}
g_{x_0} [V[M]]_{x_0,x_0}+g_{p_0} [V[M]]_{p_0,p_0}
=\frac{\hbar g_{x_0}}{2(n+1/2)}
+\frac{\hbar g_{p_0}}{2(n+1/2)},
\label{eqn:effmle}
\end{eqnarray}