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\author{Armin Uhlmann,}
\title{Density Operators as an Arena for Differential Geometry}
\date{University of Leipzig}
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\begin{document}
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\maketitle
What I am going to describe may be called an interplay between
concepts of differential geometry and the superposition principle of
quantum physics. In particular it concerns a metrical distance
introduced by Bures \cite{Bu69} as a non- commutative version of a
construction of Kakutani \cite{Ka48}
on the one hand, and on the other hand the
purifications of mixed states in physically larger systems, including the
problem of geometric phases associated with a distinguished class of
such extensions. The Bures distance and the general transition
probability \cite{Ca75}, \cite{U76a} are discussed in \cite{Ar72},
\cite{AR82}, \cite{U87a}, and further papers.
For the sake of clarity, and to avoid technicalities, I will be concerned
with finite dimensional objects.
Let
${\cal H}$ denote an Hilbert space with complex dimension $n$.
The set of density operators defined on it is
$$ \Omega \, = \{ \varrho \geq 0, \quad \hbox{trace} \, \varrho =1 \}
\eqno(1) $$
which in turn containes the pure states as its extremal set.
This extremal set is isomorphic to the complex projective
space ${\bf P}({\cal H})$, the points of which are conveniently described
by the projection operators of rank one:
$$ {\bf P}({\cal H}) \, = \, \{ P \in \Omega \, : \, P^2 = P \, \}
\eqno(2) $$
There are natural mappings
$$ \psi \, \mapsto \, {\psi \over \sqrt{<\psi. \psi>}} \, \mapsto
\, P_{\psi} := {|\psi><\psi| \over <\psi, \psi>},
\quad \psi \not\equiv 0 \eqno(3) $$
from $({\cal H}) - \{0\}$ onto the unit sphere of the Hilbert space
$$ {\bf S}({\cal H}) \, = \, \{ \psi \in {\cal H} \, : \, <\psi. \psi>
= 1 \} \eqno(4) $$
and from that unit sphere onto
${\bf P}({\cal H})$. \hfill \\
Further, referring to an arbitrarily choosen orthogonal base,
the sets (1) and (2) may be considered as sets of hermitian matrices.
This gives embeddings
$$ {\bf P}({\cal H}) \, \subset \, \Omega \, \subset \,
{\cal R}^{n^2} \eqno(5) $$
\subsection*{Pure States}
This section is a short review of some selected geometrical properties
of ${\cal H}$, ${\bf S}{\cal H}$, and ${\bf P}{\cal H}$. Their study
goes back to the work on adiabatic approximations of Born, Oppenheimer,
and Fock \cite{born} in the late twenties (see also \cite{Fo28}, appendix).
In the last decade the interest on this
topic has been triggered by the ideas of Berry \cite{Be84}, of
Simon \cite{Si83}, and many others, in particular \cite{WZ84},
\cite{AA87}, \cite{AA90}, \cite{An90}. See also \cite{books1}.
The following consideration should prepare the study of similar
problems for density operators. Let
$$ t \mapsto \psi_t \, \in \, {\bf S}({\cal H}),
\quad 0 \leq t \leq 1 \eqno(6) $$
be a path on the unit sphere of ${\cal H}$. Its length is given by
$$ \int_0^1 \, \sqrt{<\dot \psi, \dot \psi>} \, dt \eqno(7) $$
where here and later the dot notation indicates differentiation.
Hence $\dot \psi$ is an element of the appropriate tangent space at
$\psi$.
The length (7) does {\it not} remain invariant under phase changes
$$ \psi_t \, \mapsto \, \epsilon(t) \psi_t, \quad | \epsilon | = 1
\eqno(8) $$
which can be understood as a $U(1)$ gauge transformation of the path
(6). This opens the possibility of a canonical gauge fixing by
demanding a minimal length of the path: The gauge is fixed if
a replacement (8) never can result in a shorter path.
What can be achieved by the
desired gauge fixing is easily seen by the help of the inequality
$$ <\dot \psi, \dot \psi> \, \geq \, <\dot \psi, \dot \psi> -
<\psi, \dot \psi><\dot \psi, \psi> \, \geq 0 \eqno(9) $$
showing that minimality of the length is reached if and only if
$$ <\psi, \dot \psi> \, = \, 0, \eqno(10) $$
i.e. if the Berry - Simon parallel transport condition is fulfilled.
The gauge is fixed by (10) up to a parameter independent gauge (a
global gauge). Therefore, if (10) is fulfilled for the path (6)
$$ |\psi_0><\psi_1| \quad {\rm and} \quad
A \mapsto <\psi_0, A \, \psi_1> \eqno(11) $$
are gauge invariant quantities. They depend only on the image
in ${\bf P}{\cal H}$ of
the curve (6) under the map (3). If this image is a closed curve then
$\psi_1 = \epsilon \psi_0$, and $\epsilon$ is Berry's
geometric phase factor.
It is sometimes necessary not to restrict oneself to the unit sphere
and to allow more general gauges
$$ \psi_t \, \mapsto \, \lambda(t) \psi_t, \quad \lambda \not= 0,
\, {\rm complex} \eqno(12) $$
then the inequaltiy (9) is usefully rewritten as
$$ {<\dot \psi, \dot \psi> \over <\psi, \psi>} \, \geq \,
{<\dot \psi, \dot \psi> \over <\psi, \psi>} -
{<\psi, \dot \psi><\dot \psi, \psi> \over <\psi, \psi>^2} =
\bigl( {ds \over dt} \bigr)_{SF}^2 \eqno(13) $$
where the right hand side is the Study Fubini metric which is invariant
with respect of (12). Hence we can consider this line element as
a lift of a metric of ${\bf P}({\cal H})$. Indeed, because of (3)
a short calculation yields
$$ \bigl( {ds \over dt} \bigr)_{SF}^2 = {1 \over 2} \,
{\rm tr} \, ( \dot P_{\psi} )^2 \eqno(14) $$
Further: In leaving the unit sphere one has to replace the crucial
condition (10) by
$$ {1 \over 2} \, {<\psi, \dot \psi> -
<\dot \psi, \psi> \over <\psi, \psi> } \, = \, 0, \eqno(15) $$
where the left hand side is a $U(1)$-connection form which remains
unchanged under a rescaling of the vector length. (Remark that its
curvature defines the K\"ahler Hodge 2-form of ${\bf P}({\cal H})$ which
also defines the Chern character of the bundle ${\bf S}({\cal H})$
over ${\bf P}({\cal H})$. )
A solution of
(15) is given by
$$ \dot \psi = G \psi, \qquad G = \dot P + \bigl( {d \over dt}
\ln \sqrt{<\psi, \psi>} \bigr) \, P, \eqno(16) $$
so that $G = \dot P$ at the unit sphere. Because $G$
is hermitian (15) is fulfilled.
\subsection*{Real Planes and Circles.}
For later use I need some elementary geometric facts.
A real 2-dimensional subspace of a Hilbert space is called a real plane.
Every such plane can be given as
$$ \{ \psi \, : \, \, \psi = \lambda_1 \psi_1 +
\lambda_2 \psi_2, \quad \lambda_j \quad \hbox{reell } \}
\eqno(17) $$
Every pair of vectors that generates the plane as in (17) is called
a base of the plane. The intersection of a real plane with the
unit sphere ${\bf S}({\cal H})$ is a large circle. Its length in
the Hilbert space metric equals $2 \pi$. Every geodesic closes on the
unit sphere. Every closed geodesic on the
sphere is a large circle and vice versa.
Let us now assume that in (17)
$$ <\psi_1, \psi_1> = <\psi_2, \psi_2> = 1, \quad <\psi_1, \psi_2> =
a + i b \eqno(18) $$
with reals $a, b$. Then the intersection of the plane (17) with the
sphere is parametrized by an angle $\alpha$ as
$$ \lambda_1 = \cos \alpha - \frac{a}{\sqrt{1 - a^2}} \,
\sin \alpha, \qquad
\lambda_2 = \frac{\sin \alpha}{\sqrt{1 - a^2}} \eqno(19) $$
This implies $<\dot \psi, \dot \psi> = 1$, and the differential
$d \alpha$ is the line element on the circle. The (oriented) arc length
$\alpha_{1,2}$ between $\psi_1$ and $\psi_2$ is given by
$\cos \alpha_{1,2} = a$.
If one varies the relative phases of the two
vectors, the arc length becomes minimal iff $b = 0$. Then
the transition amplitude becomes real, and {\it every}
curve on the plane satisfies the parallel transport condition (15).
On a general circle it follows from (18) and (19)
$$ < \dot \psi, \psi> = - i {b \over \sqrt{1 - a^2}}, \eqno(20) $$
and the invariant (11) is given by $\epsilon \, |\psi_2><\psi_1|$
with $\epsilon = \exp < \dot \psi, \psi> \alpha_{1,2}$.
Further, introducing (20) into the Fubini Study metric (13) one
gets along the circle the line element
$$ ds = {1 \over 2} \sqrt{{1 - ( a^2 + b^2 ) \over 1 - a^2 }} \,
d \check \alpha, \qquad \check \alpha = 2 \alpha \eqno(21) $$
The manipulation with the factor $2$ has been done because after
projecting down the Hilbert space circle according to (3) into
${\bf P}({\cal H})$,
it appears as a double covering of the resulting
base curve. This is a reminiscent of the Hopf bifurcation: $\psi$
changes it sign if the circle is rotated by $\pi$.
Therefore, the factor
before $d \check \alpha$ becomes the radius of a circle if
${\bf P}({\cal H})$ is embedded into an Euclidean space (5) with
a suitable choosen Euclidean metric. This radius becomes minimal,
namely ${1 \over 2}$, iff $b=0$. On a circle with this
condition the equality sign holds in (13). (14) then shows (3)
to be locally isometric. Thus a geodesic on the unit sphere gives
a geodesic on the projective space of pure states iff $b=0$ is
fulfillled. As a further consequence, every geodesic closes in
${\bf P}({\cal H})$ with length $\pi$.
It has been shown above that a unit circle in a plane with
$b=0$ contains with $\psi$ also $\dot \psi$, and both vectors
form an orthoframe. The arc connecting both vectors is of arc
length ${\pi / 2}$.
It has been already remarked that on a real plane with $b=0$
{\it every} curve fulfills the parallell transport condition
(15). Let us call such a plane {\it horizontal}.
\subsection*{Density Operators}
While in the pure state case the starting point is a Hilbert space
${\cal H}$ which is projected down to ${\bf P}({\cal H})$
(with the exception of its zero vector), the situation with the
density operators is somewhat reversed. We have to start with the
space of density operators, $\Omega$, and to look for a covering
by an Hilbert space, say ${\cal H}^{\rm ext}$, which carries a
representation of the operators acting on the Hilbert space
${\cal H}$ which defines $\Omega$. One may use any "large enough"
representation, the results are the same. Physically, the
construction of ${\cal H}^{\rm ext}$ is an embedding of the original
physical system into a larger one so that it becomes a subsystem.
(This procedure is also heavily used in the so-called thermo field theory.)
The vectors of the extended system, i.e.~the pure states of
the extended system, can be reduced to the subsystem. The result of the
reduction is a density operator (a mixed state). The extension should
be large enough, so that every
density operator should be reachable by reducing a vector state
of ${\cal H}^{\rm ext}$.
If a density operator $\varrho$ can be
gained by reducing a vector $W$ of ${\cal H}^{\rm ext}$ then
$W$ is called a {\it purification} of $\varrho$. While a
reduction gives a unique result, there is some arbitrariness in
the procedure of purification: There are many vectors in the
extended systems which purify a given density operator.
\smallskip
A good choice to realize ${\cal H}^{\rm ext}$ is by the set of all
those operators
$W$ acting on ${\cal H}$ for which $W W^* \,$ (and hence $W^* W$)
has finite trace, i.e. we consider Hilbert -
Schmidt operators. Of course, the trace condition is trivially
fulfilled for finite
dimensional Hilbert spaces with which we are dealing with.
The scalar product of ${\cal H}^{\rm ext}$ is given by
$$ (W_1, W_2) \, = \hbox{tr } \, W_1^* W_2
= \hbox{tr } \, W_2 W_1^* \eqno(22) $$
The mapping that replaces (3) is
$$ W \, \mapsto \, W W^* \, \mapsto
\, \varrho_{w} := {W W^* \over \hbox{tr } \, W W^* },
\quad W \not\equiv 0 \eqno(23) $$
It goes from $({\cal H}^{\rm ext}) - \{0\}$ onto the unit sphere of the
extended Hilbert space, ${\bf S}({\cal H})^{\rm ext}$,
and from that unit sphere onto $\Omega$. \hfill \\
If $\varrho = W W^{*}$ then $W$ is called
a {\it standard purification} of $\varrho$.
Let $\mu_1, \mu_2, \dots$ be the set of non-zero eigenvectors,
and $\psi_1, \psi_2, \dots$ an orthoframe of corresponding
eigenvectors of $\varrho$.
If now
$W$ is a standard purification of $\varrho$,
there exists a unique orthoframe,
$\varphi_1, \varphi_2, \dots$, of the same length such that
$$ W \, = \, \sum \, \mu_j^{1/2} \, |\psi_j><\varphi_j| \eqno(24) $$
Clearly, every $W$, given by an expression (24), is a purification of
$\varrho$. Given $\varrho$ the fibre (or leaf) of all purifications
sitting on the unit
sphere of the extended Hilbert space is the Stiefel manifold of
orthoframes of $\cal H$ of rank $k$, $k = {\rm rank(\varrho)}$.
The fibre admits the unitaries of ${\cal H}$ as right multipliers,
$$ W \mapsto W U, \qquad U \, {\rm unitary} \eqno(25) $$
If the rank, $k$, equals $n = {\rm dim} \, {\cal H}$, then this action is
free. Otherwise there is a stable subgroup ${\cal U}(n-k)$.
If $A$ is an operator acting on ${\cal H}$ then one defines
$$ A \mapsto A W := L_A \, W \quad {\rm and} \quad
A \mapsto W A := R_A \, W. \eqno(26) $$
The first expression defines the representation of the
operators of ${\cal H}$ as operators acting on the Hilbert
space ${\cal H}^{\rm ext}$. The second expression defines its
commutant. Hence the {\it gauge group} given by (25) is build from
the unitaries of the commutant.
After these preliminaries it is possible to mimic the pure state
case in the domain of density operators.
Starting with a curve of density operators and one of its purifications,
$$ t \mapsto \varrho_t, \quad t \mapsto W_t , \qquad
\varrho_t = W_t W_t^*, \eqno(27) $$
the length of the latter in the extended Hilbert space is by no means
invariant against gauge transformations (25). To get a gauge fixing, and
at the same time a length to the original curve of density operators,
we consider the variational problem \cite{U87a}, \cite{U91c}
$$ {\rm length}(\, t \mapsto \varrho_t \,) =
\inf \, \int \sqrt{(\dot W, \dot W)} \, dt \eqno(28) $$
where the infimum is running over {\it all} purifications, or, what
is the same, over all curves $t \mapsto W_t U_t$ with unitaries $U_t$.
The Euler equations of (28) read
$$ \dot{W}^* \, W \, = \, W^* \, \dot{W}, \eqno(29) $$
a set of equations which I call the {\it (extended) parallel condition},
see \cite{U86a}. It is a family of Berry conditions which is stable
under the commutant of the representation $A \mapsto L_A$. Indeed,
(29) is equivalent with
$$ (R_A \dot W, R_A W) = (R_A W, R_A \dot W)
\quad \hbox{for all} \quad A \eqno(30) $$
If $W_1$ is the endpoint and $W_0$ the beginning of a parallel lift
fullfilling (29), $W_0 W_1^*$ is important for the definition of the
geometric phase attached to curves of density operators. It
generalizes the invariant (11). However, in this paper I shall not go into
this branch of the game, which can be successfully considered
even in von Neumann and C$^*$-algebras \cite{Al92a}, \cite{Al92b}.
The parallel condition can be solved by an ansatz \cite{DJ89}, \cite{U89b}
$$ \dot W = G W, \qquad G = G^{*} \eqno(31) $$
After inserting this into the differentiated relation $\varrho = W W^*$
one gets $G$ as the solution of a Bloch like equation
$$ \dot \varrho = G \varrho + \varrho G \qquad {\rm or} \qquad
d \varrho = \varrho {\bf G} + {\bf G} \varrho \eqno(32) $$
The last relation defines a matrix-valued 1-form, ${\bf G}$, the
restriction of which to a given curve of density operators yields $G$.
(31), (32) generalize (16). If in (32) $\varrho$ and $\dot \varrho$
commute, then $G$ is the logarithmic derivative of $\sqrt{\varrho}$.
If no eigenvalue of $\varrho$ equals zero, (32) has exactly one solution.
It is easy to insert $\dot W = G W$ into the expression for
the Hilbert space metric, and one gets, using (32),
$$ (\dot W, \dot W) = (GW, GW) =
{\rm tr} \, G^2 \varrho =
{1 \over 2} {\rm tr} \, G \, \dot \varrho
\eqno(33) $$
Now the minimal lenght (28) can be calculated as well on the unit sphere
of the extended Hilbert space as on $\Omega$.
One can get rid of
the norm condition and extend these expressions to
${\cal H}^{\rm ext} - \{ 0 \}$ and to the cone of all positive
(trace class) operators to obtain the analogue to (14), which may
be called the projective Bures metric:
$$ { (W, G^2 W) \over (W, W) } - { (W, G W)^2 \over (W, W)^2 } \, = \,
{1 \over 2} { {\rm tr} \, G \, \dot \varrho \over {\rm tr} \, \varrho }
- {1 \over 4}
\bigl( { {\rm tr} \, \dot \varrho \over {\rm tr} \,\varrho } \bigr)^2
\eqno(34) $$
This metric is scale invariant and gauge invariant. \smallskip
The next goal is to ask for the appropriate connection form
in the extended Hilbert space. To this end let us use the differential
1-form ${\bf G}$ which is a gauge invariant. Hence ${\bf G}W$
behaves under gauge transformations (25) like $W$. If one therefore
defines the 1-form ${\bf A}$ by
$$ {\rm d}W - W {\bf A} = {\bf G} W \eqno(35) $$
the result is an operator valued connection form:
$$ {\bf A} \mapsto \tilde {\bf A} =: U^* {\bf A} U +
U^* {\rm d} U \eqno(36) $$
Reinserting (35) into (32) one finds compatibility iff
$$ {\bf A} + {\bf A} ^* = 0 \eqno(37) $$
Finally, calculating the left hend side of the following equation
by the help of the relations above, one easily finds
$$ W^* {\rm d} W - ({\rm d} W^*) W = W^*W \cdot {\bf A} +
{\bf A} \cdot W^*W \eqno(38) $$
Originally, \cite{U91a}, I guessed this as the
relevant definition of ${\bf A}$.
Unfortunately it is difficult to obtain explicite expressions for
${\bf G}$ and ${\bf A}$ with the exception of density operators of
rank two, where various results have been obtained: \cite{Ru90},
\cite{DG90}, \cite{Hu92a}, \cite{Hu92b}, \cite{U92d}.
See also \cite{DR92a}, \cite{DR92b} and a forthcoming paper of
J.~Dittmann.
\subsection*{$\Omega$- Horizontality, Geodesics}
To get some insight into the geometric meaning of the Bures metric,
we play again the game with real planes and circles, but this time within
the extendet Hilbert space. In accordance with (17) we consider
real planes
$$ \{ W \, : \, \, W = \lambda_1 W_1 +
\lambda_2 W_2, \quad \lambda_j \quad \hbox{reell},
\quad W_1, W_2 \in {\cal H}^{\rm ext} \} \eqno(39) $$
Let us assume, the plane contains an invertible element. Then
one may choose as a base two elements, $W_1, \, W_2$,
which are both invertible and
of Hilbert norm one. The determinant of $W_1 + \lambda W_2$ can
be zero for at most $n$ real values of $\lambda$. From (39) one sees
$\lambda = \lambda_2 \lambda_1^{-1}$, and with $W$ always $-W$ sits
on the circle. Hence the number k
of points on the unit circle of the plane which are not invertible
is even and restricted by $k \leq 2n$. After
projecting down the circle to $\Omega$ according to (23), these
points and no others become boundary points of $\Omega$.
If there is one turn of the $W$-circle the projection on $\Omega$
runs through two turns.
Parametrizing the circle as indicated in (19), we may call its
non-invertible $W$'s and the corresponding boundary points
$$ W(\alpha_1), W(\alpha_2), \dots, W(\alpha_k) \quad {\rm and}
\quad \varrho(\alpha_1), \varrho(\alpha_2), \dots, \varrho(\alpha_k)
\eqno(40) $$
ordered with increasing $\alpha$-parameters.
(Every $\varrho$ appears twice.)
If $k = 0$ the projected curve remains in the interior of $\Omega$.
If $k = 2$, the $\varrho$-curve starts at a boundary point
and return to it.
Generally, the projected curve may be started at $\varrho(\alpha_1)$,
to go through inner points to $\varrho(\alpha_2)$.
It then starts from this point to go to $\varrho(\alpha_3)$ through
the interior of $\Omega$, and so on. At last it goes from
the projection $\varrho(\alpha_k)$ of $W(\alpha_k)$
to that of $W(\alpha_1)$, i.e. to $\varrho(\alpha_1)$.
This shows that the projected curve generally goes through the
interior of the set of density operators, and if it hits a boundary
point, it will be reflected to start again into the interior region.
We may image this by a polygon with $k/2$ vertices which
are sitting on the boundary of $\Omega$.
\smallskip
A plane (39) is called {\it $\Omega$-horizontal} iff
$$ (W_1)^{*} W_2 = (W_2)^{*} W_1 \eqno(41) $$
is valid for one of its bases, or, what is the same, is valid
for every base of the plane. It is also evident that
a plane is $\Omega$-horizontal iff every curve on it fulfills
the extended parallelity condition (29).
A circle of an $\Omega$-horizontal plane consists of peaces, which,
after projecting them down according to (23) to $\Omega$,
become geodesics with respect to the metric of Bures.
Returning to the "generic" case where there are inverible elements
within in plane,
let us choose a base of the plane consisting of normed elements
$W_1, \, W_2$ which
are sufficiently near neighbours. Then $W_1^* W_2$ is not only hermitian
but also positve definite. In moving these two points continuously,
the positive definiteness can be lost only if on eigenvalue becomes
zero, and, consequently, the determinant of one of the two points
becomes zero. This means at least one of the $\varrho$'s goes into
the boundary of $\Omega$. The argument shows
$$ W^*(\gamma) W(\beta) \, > \, 0 \qquad \beta, \gamma \in
\{ \alpha \, : \, \, \alpha_j < \alpha < \alpha_{j+1} \} \eqno(42) $$
where $\alpha_{k+1} = \alpha_1$ is to be understood.
$k$ is twice the number of real zeros of the
determinant of $W_2 + \lambda W_1$
if $\lambda$ varies. Multiplying by the non-singular $W^*_1$ and
exploiting $\Omega$-horizontality one gets a characteristic equation
of the form $\det (A + \lambda B) = 0$ with positive definite matrices
$A$ and $B$. Such an equation allows only for real, negative
solutions. If a root is degenerated the corresponding
density operator posesses as many
zero eigenvalues as the degree of the relevant root. This means
$$ \sum \, \hbox{number of zeros of} \, \varrho(\alpha_j) \, =
\, n \, = \, {\rm dim}({\cal H}), \quad
0 \leq \alpha_j < \pi \eqno(43) $$
\smallskip
>From this and other observations I wonder (as a {\it working hypothesis})
wether there is a Riemann manifold of dimension $n^2 - 1$ which
allows for a decomposition into $n$ cells. The interior of every cell
should be isometric to the interior of $\Omega$, equipped with the metric
of Bures. Let me call that hypothetical Riemann manifold
$\Omega^{\rm fancy}$. A more stringent hypothesis for $\Omega^{\rm fancy}$
is by demanding additional: A density operator $\varrho$ is covered by
$\Omega^{\rm fancy}$ exactly rank($\varrho$) times. The hypothesis is
true if $n = 2$ where
the wanted manifold is a 3-sphere with radius $1/2$.
\smallskip
$\Omega$-horizontality obviously imlies horizontality as defined in
the case of pure states. Consequently, the intersection of
such a plan with the unit sphere ${\bf S}({\cal H})$ is a large
circle of this sphere, and at the same time one of its geodesics.
If $\alpha \mapsto W(\alpha)$ parametrizes this circle as indicated
in (19) then $W_, \, \dot W$ is an orthonormal base of the plane for
every $W = W(\alpha)$, and it is $\pm \dot W = W(\alpha \mp \pi / 2)$.
There is an intersting observation. Let $W_1, W_2$ denote a base
of an $\Omega$-horizontal plane (39), and $Y$ an invertible operator.
Then
$$ {\tilde W}_1 = {Y^* W_1 \over (W_1, Y Y^* W_1)}, \qquad
{\tilde W}_2 = {Y^{-1} W_2 \over (W_2, (Y Y^*)^{-1} W_2) }
\eqno(44) $$
generate an $\Omega$-horizontal plane. Indeed, (41) remains valid
after such a replacement. Inserting (44) into (22) alllows for the
corollary:
If $W_1, W_2$ is an orthonormal base, so it is $\tilde W_1, \tilde W_2$.
Hence (44) induces, if applied to an orthonormal base, an isometric
map from one $\Omega$-horizontal plane onto another one. Remark,
however, that the positions of the singular points (40) are changed
by that transformation, and the same is with the regions (42).
As a next step we choose an element $W \in {\bf S}({\cal H}^{\rm ext})$
which is not singular and call ${\bf h}_p(W)$ the union of all those
$\Omega$-horzontal planes which contain $W$. The intersection of two
different planes contributing to ${\bf h}_p(W)$ consists of $W$ only.
(Otherwise they would be identical.) ${\bf h}_p(W)$ is an Euclidean
submanifold of ${\cal H}^{\rm ext}$. Proof: In order that $X$
belongs to ${\bf h}_p(W)$ the expression $H = W^* X$
has to be hermitian. Thus ${\bf h}_p(W)$ is parametrized by
$H \mapsto X = (W^*)^{-1} H$.
Two such spaces, ${\bf h}_p(W)$ and ${\bf h}_p(\tilde W)$,
are diffeomerphic, and, moreover, their constitutent planes are
isometric. To see this one sets $W_1 = W$, $\tilde W_1 = \tilde W$
in (44), and for $W_2$, $\tilde W_2$ their orthonormal partners.
By (23) ${\bf h}_p(W)$ is mapped onto the cone of positive
(trace class) operators
$$ X \in {\bf h}_p(W) \, \mapsto \, X X^{*} = H ( W W^* )^{-1} H \eqno(45) $$
while $H$ runs through the hermitian operators.
All this remains essentially valid if going from ${\bf h}_p(W)$
to ${\bf h}_c(W)$ which will be defined as the union of
all horizontal unit circles containing $W$.
${\bf h}_c(W)$ contains a cell ${\bf h}_c^0(W)$
diffeomorphic to $\Omega$. The diffeomorphism can be realized by (23).
The reason is, $W$ belongs to exactly one segment (42) of every
circle constituing the manifold ${\bf h}_c(W)$, and the
union of this segments is called ${\bf h}_c^0(W)$.
Along the intersection of ${\bf h}_c^0(W)$ with a horizontal
circle
the diffeomorphism is an isometry from the induced Hilbert
space geometry to the metric of Bures.
Its projections to $\Omega$ give all the Bures geodesics
of $\Omega$ which cross $\varrho_w$.
Transverse to the mentioned directions all this is not true.
\smallskip
How far away is now $\Omega^{\rm fancy}$ ? Let us equip
${\bf h}_c(W)$ with the lifted Bures metric (33) or (34).
Then, already for $n = 2$, there are points on ${\bf h}_c(W)$
where this metric degenerates.
Let $\varrho \in \Omega$. Because ${\bf h}_c(W)$ is mapped
continuously onto $\Omega$, the inverse mapping maps $\varrho$
onto a closed subset of ${\bf h}_c(W)$
with (eventually) several connected components. $\Omega^{\rm fancy}$
should be obtained after contracting these connected components
to points for every $\varrho$.
But is the result a manifold? As already said, presently this is only
known in the most simple case $\dim {\cal H} = 2$. It would be
certainly very interesting to get complete control for the
dimensions 3 and 4. And this seems to be not out of range.
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