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\line{{\bf PARALLEL LIFTS AND HOLONOMY } \hfill}
\smallskip
\line{{\bf ALONG DENSITY OPERATORS:} \hfill}
\smallskip
\line{{\bf COMPUTABLE EXAMPLES USING O(3)-ORBITS.} \hfill}
%\line{{\bf COMPUTABLE EXAMPLES USING O(3)-ORBITS.}\footnote{\dag}{dedicated
%to Lawrence C. Biedenharn on the occasion of his 70th birthday.} \hfill}
\vskip 1truecm
\hskip 3truecm Armin Uhlmann
\bigskip
\hskip 3truecm University of Leipzig
\hskip 3truecm Department of Physics
\hskip 3truecm Leipzig, Germany
\vskip 1truecm
{\bf PARALLEL TRANSPORT ALONG DENSITY OPERATORS}
\medskip
The parallel transport governing Berry's phase [1], [2], and
the Wilzcek and Zee [3] holonomy for degenerate states extends
naturally to {\it density operators} [4] (and, up to the pecularities
of infinite dimensional analysis, at least partly to state spaces of
certain $^*$-algebras [5].) Our approach is consistent,
with the point of view of [6], to which it
reduces by restricting to the pure states case.
Let
$$ t \mapsto \varrho_t \eqno(1) $$
be a curve of density operators in the set
$$ \Omega \, = \{ \varrho \geq 0, \quad \hbox{trace} \, \varrho =1 \}
\eqno(2) $$
of all density operators of an Hilbert space ${\cal H}$. \hfill
One considers now {\it extensions}
$$ {\cal H}^{\rm ext} = {\cal H} \otimes {\cal H}^{{\rm aux}} \eqno(3) $$
of ${\cal H}$ such that the original Hilbert space can be considered
as a {\it subsystem} of the extended one. Every unit vector of the
extension can be reduced to a density operator of ${\cal H}$.
The inverse operation is called {\it purification}. Therefore, a
curve of vectors
sitting in ${\cal H}^{\rm ext}$ is called a purification of
(1) if its reduction onto ${\cal H}$ coincides with the curve (1).
The task of purification is of course not a unique one.
However, in the set of all possible
purifications of a given curve of density operators (1) there are
exceptional ones. These exceptional ones are those
purifications for which the usual Berry transport condition
remains stable after applying operators out of the commutant
of ${\cal B}({\cal H})$ in ${\cal B}({\cal H}^{\rm ext})$.
(Remind that every imbedding (3) induces a well defined imbedding
of the operators acting on ${\cal H}$ into those acting on
${\cal H}^{\rm ext}$.)
Purifications of this type are called {\it parallel}, and the
condition just mentioned {\it (extended) parallel condition}.
Under certain continuity assumptions a parallel purification of a
given curve of density operators is
already determined by its initial value. Two parallel purifications
of the same curve of density operators do not intersect or they
are identical. Thus parallel purifications give rise to holonomy
invariants if the curve which is to be lifted is closed.
For our present purposes it is suffincient to consider only
standard purifications. This means to identify
${\cal H}^{{\rm aux}}$ with the dual of ${\cal H}$ in (3). Then
${\cal H}^{\rm ext}$ can be realized by the Hilbert space of the
Hilbert - Schmidt
operators defined on ${\cal H}$, i.e. by the linear space
of operators $W$ such that
$$ \hbox{trace} \, W W^* = \hbox{trace} \, W^* W < \infty
\eqno(4) $$
Its scalar product is given by
$$ (W_1, W_2) := \hbox{trace} \, W_1^* W_2 \eqno(5) $$
A {\it standard purification} of (1) is nothing but a curve
$$ t \mapsto W_t , \qquad W_t \quad \hbox{Hilbert - Schmidt}
\eqno(6) $$
such that
$$ \varrho_t = W_t W_t^* \quad \hbox{for all} \quad t \eqno(7) $$
The (extended) parallelity condition can now be expressed [4] as
$$ \dot{W}^* \, W \, = \, W^* \, \dot{W} \eqno(8) $$
Indeed, this condition is equivalent with the more general reciepe above,
as seen as follows.
If $A$ is an operator acting on ${\cal H}$ then
$$ A \mapsto A W := L_A \, W \eqno(9) $$
is the canonical embedding of the operators of ${\cal H}$ into
the operators acting on the Hilbert space ${\cal H}^{\rm ext}$
of Hilbert Schmidt operators. The commutant of this embedding
is given by the operations
$$ A \mapsto W A := R_A \, W \eqno(10) $$
Hence the stability of the Berry condition under the operators of
the commutant is expressed by
$$ (R_A \dot W, R_A W) = (R_A W, R_A \dot W)
\quad \hbox{for all} \quad A \eqno(11) $$
Now $R_A$ can be moved from the left hand side to the right hand side
in the scalar products, giving there an additional operator $R_{A^*}$.
Every operator $R_B$ can be represented by a linear combination of
operators of the form $R_{A^*} R_A$. As a consequence equation (11)
is equivalent with
$$ (\dot W, R_A W) = (W, R_A \dot W)
\quad \hbox{for all} \quad A \eqno(12) $$
Using (5) one immediately derives (8) from (12) and vice versa.
\smallskip
An important conclusion is: Let $t \mapsto W_t$ be a parallel
lift of $t \mapsto \varrho$ for $0 \leq t \leq \tau$ . Then
$$ t \mapsto W_t \, W_0^* , \qquad 0 \leq t \leq \tau \eqno(13) $$
depends only on $\varrho_s, \quad 0 \leq s \leq t $.
If the curve
of density operators closes for $t = \tau$ then $W_{\tau} W_0^*$
is an {\it holonomy invariant}. (It should be remarked that (13)
defines trace class operators, and hence a path of normal linear
functionals $A \mapsto$ trace$(AW_tW^*)$ on the bounded
operators of ${\cal H}$.)
\smallskip
The parallelity condition (8), (12) allows for links
to a pecular Riemannian
metric on $\Omega$, i.e. on the set of density operators, and to a
connection (a gauge field) on the unit sphere of
${\cal H}^{\rm ext}$.
One gets the first link by the observation, that the parallelity
condition (8) can be considered as the Euler equations of the
variational problem
$$ {\rm length}(\, t \mapsto \varrho_t \,) =
\inf \, \int \sqrt{(\dot W, \dot W)} \, dt \eqno(14) $$
if the infimum is running over all purifications of the curve
of density operator in question [7]. This extends nicely an idea of
Fock (see appendix of [8]) to fix the arbitrary phases of moving orthogonal
m-frames by requiring their minimal change during their movement.
Further aspects of this observations are discussed more recently
in [9].
The length obtained by (14) is measured equally well by Riemannian
metric on $\Omega$. This metric has been at first considered
in the context of $W^*$-algebra representations by Bures [10], who
defined it as a distance function for positive functionals of
W$^*$-algebras. On $\Omega$ the line element of the metric equals
$$ ds^2 := {1 \over 2} \, {\rm trace} \, {\bf G} \, d \varrho
\eqno(15) $$
where $d$ denotes the total differential. $\bf{G}$ is determined
by the Bloch-like equation [11], [12]
$$ d \varrho = \varrho {\bf G} + {\bf G} \varrho \eqno(16) $$
The metric (15) extends the Study Fubini metric from the
pure states to the mixed states.
\smallskip
To obtain the second link one is looking for a connection form
${\bf A}$ for the gauge transformations
$$ W \mapsto W U, \quad U \quad {\rm unitary} \eqno(17) $$
acting in the fibre of all $W$ purifying a given $\varrho$.
Again, within the many possibilities to satisfy this demand there
is a canonical one. It can be defined by [13]
$$ W^* {\rm d} W - ({\rm d} W^*) W = W^*W \cdot {\bf A} +
{\bf A} \cdot W^*W \eqno(18) $$
Denoting by $D$ the covariant differential that goes with the
connection form ${\bf A}$ one computes
$$ {\rm D} W := {\rm d}W - W {\bf A} = {\bf G} W \eqno(19) $$
and this is one way to see how strongly ${\bf A}$
and the gauge invariant ${\bf G}$
are bound together. And further:
\item{ } Because of (18) ${\bf A} = 0$ is a gauge fixing along a lift.
In a gauge with this property the parallelity condition is
satisfied. The gauge fixing is up to a global gauge
transformation, i.e. it is a local one.
\item{ } Accoding to (19) this gauge fixing is equivalent with
solving the system of partial differential equations
${\rm d}W - {\bf G} W = 0$ [11].
%\vfill \eject
\medskip
{\bf LIFTS OF HAMILTONIAN MOTION}
\medskip
If the eigenvalues of the density operators of a sufficiently regular
curve (1) remain unchanged, this curve is a solution of a
von Neumann - Liouville eqations with a time-dependent Hamiltonian.
$$ i \, \dot \varrho = [ H(t), \varrho ] \eqno(20) $$
If (1) is a solution of (20), and if $W = W_t$ is an arbitrary lift
then one may write with a suitable $\tilde H(t)$
$$ i \dot W = H(t) W - W \tilde H(t) \eqno(21) $$
This relation can be considered as a Schr\"odinger equation
$$ i \dot W = H^{{\rm ext}}(t) W \quad \hbox{with} \quad
H^{{\rm ext}}(t) := ( L_H - R_{\tilde H} ) \eqno(22) $$
in ${\cal H}^{\rm ext}$. In [13] it is shown how to choose
$\tilde H(t)$ for parallel purifications.
\smallskip
In the following the Hamiltonian in (20) is always assumed to be
independent of time. The formal solution of this equation reads
$$ t \mapsto \varrho_t := U(t) \varrho_0 U(-t),
\quad U(t) = e^{-it H} \eqno(23) $$
The corresponding solution for (21) with $\tilde H$ independent of
time too is given by
$$ W(t) = U(t) \varrho_0^{1/2} V(t),
\qquad V(t) = e^{i t \tilde H} \eqno(24) $$
where the initial value for $t = 0$ is the positive square root of
$\varrho_0$. Assuming that this is already a parallel lift, the
gauge invariant (13) can be written as
$$ U(t) \varrho_0^{1/2} V(t) \varrho_0^{1/2} $$
If now (1) is a loop, and the curve of densitiy operator closes for
$t = \tau$, then $U(\tau)$ commutes with $\varrho_0$. Thus the
associated holonomy invariant reads
$$ \varrho_0^{1/2} \, U(\tau) \, V(\tau) \, \varrho_0^{1/2}
\quad \hbox{with} \quad \varrho_{\tau} = \varrho_0 \eqno(25) $$
The parallelity condition demands the hermiticity of
$$ W^* \, {d W \over dt} = V^* \varrho_0^{1/2} [ - i H ]
\varrho_0^{1/2} V + V^* \varrho_0 [ i \tilde H ] V \eqno(26) $$
resulting in
$$ 2 \varrho_0^{1/2} H \varrho_0^{1/2} =
\varrho_0 \tilde{H} + \tilde{H} \varrho_0 \eqno(27) $$
If $\varrho$ is faithful this equation defines $\tilde H$.
Otherwise, in order to determine $\tilde H$ uniquely, one may
require
$$ <\psi, \tilde H \psi> = 0 \quad {\rm if} \quad
\varrho_0 |\psi> = 0 $$
However, in calculating the gauge invariants (13), the holonomy
invariant (25), or the line element of the Bures metric the
ambiguitity coming from (27) is cancelled automatically.
\smallskip
Let us assume for simplicity the faithfulness of $\varrho_0$.
Then (27) defines a linear map
$$ H \, \mapsto \, \tilde H \eqno(28) $$
in the linear space of Hamiltonians.
This map satisfies the following \hfill
\item{} a) If $H \varrho = \varrho H$ then $H = \tilde H$.
\item{} b) If $H \varrho = \varrho H$ then $[H, H'] = [H, \tilde H']$
for all $\tilde H'$.
\item{} c) The map (28) is trace preserving for ${\rm dim} \cal{H} <
\infty$.
\item{} d) It is trace$\varrho H$ = trace$\varrho \tilde H$
\smallskip
\noindent Denoting the eigenvalues and eigenvectors of
$\varrho$ by $\lambda_m$ and $\psi_m$ respectively, one writes
$$ \varrho_0 = \sum \, \lambda_m \, |\psi_m><\psi_m| \eqno(29) $$
Inserting this into (27) results in
$$ \tilde H = \sum \, {2 \sqrt{\lambda_m \lambda_n}
\over \lambda_m + \lambda_n} \, |\psi_m> <\psi_m, H \psi_n>
<\psi_n| \eqno(30) $$
The coefficient in (30) is the quotient of the geometric mean by the
arithmetic mean, and hence a number between 0 and 1. (30) can be
rewritten as
$$ \tilde H = H - \sum \, {(\lambda_n^{1 \over 2} - \lambda_m^{1 \over 2})^2
\over \lambda_m + \lambda_n} \,
|\psi_m> <\psi_m, H \psi_n> <\psi_n| \eqno(31) $$
Another useful relation is
$$ \eqalign{(\dot W, \dot W) &=
\hbox{trace} \varrho_0^2 \{ H^2 - \tilde H^2 \} \cr
&= {1 \over 2} \sum { (\lambda_m - \lambda_n)^2
\over \lambda_m + \lambda_n} |<\psi_m, H \psi_n>|^2 \cr} \eqno(32) $$
The following section is devoted to Hamiltonians $H$ which are
generators of the rotation group. There eigenvalues are integers or
half integers, and (23) closes after a time
period of $2 \pi$. Thus
$$ W_{2 \pi} = (-1)^{2j} \, \varrho_0^{1/2} \exp{2 \pi i \, \tilde H},
\qquad j \, \, \hbox{denotes spin} \eqno(33) $$
and the holonomy invariant (25) of a loop will be
$$ W_{2 \pi} W_0^{*} = (-1)^{2j} \,
\varrho_0^{1/2} \, e^{2 \pi i \, \tilde H} \, \varrho_0^{1/2}
\eqno(34) $$
\medskip
{\bf ROTATIONALLY SYMMETRIC 2-SPHERES OF STATES }
\medskip
The unit vectors $\vec{n}$ in 3-space characterize the points
of a 2-sphere. By the help of
the generators of an irreducible representation of $SU(2)$,
$J_x, J_y, J_z$, one associate to it an 2-sphere of states
with monopol-like structure. To this end one considers the
vectors $|m, \vec{n}>$ of norm one satisfying
$$ \vec{n} \, \vec{J} \, |m, \vec{n}> = m \, |m, \vec{n}> \eqno(35) $$
If $\lambda_m > 0$ with $\sum \lambda_m = 1$ is given then
$$ \varrho = \sum \, \lambda_m \, |\psi_m><\psi_m| \quad {\rm where}
\quad \psi_m = |m, \vec{n}> \eqno(36) $$
is a density operator.
Varying the direction of the unit vector $\vec{n}$ one gets
a set of density operators which uniquely fill a 2-sphere
${\bf S} = {\bf S}_{j, \lambda}$.
It is determined by the given eigenvalues $\lambda_m$, and the
choosen irreducible representation labelled by $j$.
It is an obviously rotational invariant sphere.
On this sphere we shall consider curves which are (parts of) circles.
Their starting point $\varrho_0$ should be attached to the $z$-direction
in the sense of (36), while
$\vec{n} = \{0, \sin \theta, \cos \theta \}$ will be choosen as
rotational axis. The resulting curve is given by
$$ \phi \mapsto U(\phi) \varrho_0 U(-\phi), \qquad
U(\phi) = e^{- i \phi \, ( \sin \theta \, J_y + \cos \theta \, J_z )}
\eqno(37) $$
and the associated parallel lift of this curve
with initial value $\varrho_0^{1/2}$ by
$$ \phi \mapsto U(\phi) \varrho_0 V(\phi), \qquad
V(\phi) = e^{ i \phi \, ( \sin \theta \, \tilde J_y + \cos \theta \, J_z )}
\eqno(38) $$
in accordance with (23) and (24). The holonomy invariant (34) can now
be obtained by setting $\phi = 2 \pi$.
An obvious calculation shows for the linear map (28)
$$ \eqalign{\tilde J_z &= J_z, \cr
\tilde J^{+} &= \sum a_{m+1,m} \sqrt{j(j+1) - m(m+1)} \, |\psi_{m+1}>
<\psi_m|, \cr
\tilde J^{-} &= \sum a_{m-1,m} \sqrt{j(j+1) - m(m-1)} \, |\psi_{m-1}>
<\psi_m|, \cr } \eqno(39)
$$
where
$$ a_{m, m'} = {2 \sqrt{\lambda_m \lambda_{m'}} \over
\lambda_m + \lambda_{m'}} \eqno(40) $$
One remarks as a byproduct
$$ [ [ \tilde J_x, \tilde J_y ], J_z ] = 0, \qquad [ J_z, J^{\pm} ]
= \pm J^{\pm} \eqno(41) $$
and further
$$ (\dot W, \dot W) = {1 \over 4} \sin^2\theta \, \sum_m
{ (\lambda_{m+1} - \lambda_{m})^2
\over \lambda_{m+1} + \lambda_{m}} \, \{ j(j+1) - m(m+1) \} \eqno(42) $$
\smallskip
$$ $$
Things become simpler for suitable choosen eigenvalues of the $\varrho$.
At first, as a check, let ${\bf S}$ be a sphere of pure states, i.e.
it consists of 1-dimensional projectors. Thus let $\varrho_0$ project
onto a vector $|m>$ with $J_z$-eigenvalue $m$ in a spin $j$
representation. Then $\lambda_m = 1$, and all other eigenvalues are
zero. Conseqently the numbers (40) will be zero, and $\tilde J_y = 0$,
therefore. Our holonomy invariant (34) reads
$$ W_{2 \pi} W_0^{*} = (-1)^{2j} \,
\varrho_0^{1/2} \, e^{2 \pi i \, \cos \theta \, J_z} \, \varrho_0^{1/2}
= e^{-2 \pi i m \, ( 1 - \cos \theta )} \, |m>From the number (40) we only need
$$ a := a_{m, m+1} = {2 \, e^{\alpha \over 2} \over 1 + e^{\alpha}}
= {1 \over \cosh{\alpha \over 2}} \eqno(46) $$
It follows now (39) that $\tilde J^{\pm} = a J^{\pm}$, and hence
$$ \tilde H = \cos{\theta} \, J_z + a \,
\sin{\theta} \, J_y \eqno(47) $$
It is now possible to look at $V(\phi)$ as a rotation with angle
$$ \tilde \phi = \kappa \phi, \quad \kappa = \sqrt{ \cos^2{\theta} +
a^2 \sin^2{\theta}} \leq 1 \eqno(48) $$
and rotation axis
$$ \vec{\xi} = \{ 0, {\sin \theta \over \kappa},
{a \, \cos \theta \over \kappa } \} \eqno(49) $$
The holonomy invariant can be written as
$$ (-1^{2j}) \varrho_0^{1/2} \, e^{2 \pi i (\cos{\theta} \, J_z + a \,
\sin{\theta} \, J_y )} \varrho_0^{1/2} =
(-1^{2j}) \varrho_0^{1/2} \, e^{2 \pi i \kappa \vec{\xi} \vec{J}} \,
\varrho_0^{1/2} \eqno(50) $$
See also [14], [15], and [16] for $j = {1 \over 2}$.
The Bures radius of the considered circle of density operators
is easily computed from (32) because $H$ and $\tilde H$ are known
as linear combinations of $J_y$ and $J_z$, and because $\varrho_0$
commutes with $J_z$. The result is
$$ \hbox{radius of the circle: } \, \sin \theta \,
\sqrt{ (1 - a^2) {\rm trace} \, \varrho_0 J_y^2} :=
\sin \theta \, r_j^{\alpha} \eqno(51) $$
where $r_j^{\alpha}$ is the radius of the sphere
${\bf S}_j^{\alpha}$ as determined by the Bures metric.
As a cross check one may consider the high and low temperature limites
$$ \alpha \mapsto \pm \infty, \quad \varrho_ \mapsto |m>