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\centerline{{\bf The Metric of Bures and the Geometric Phase.}}
\medskip
\bigskip
\centerline{Armin Uhlmann}
\medskip
\centerline{University of Leipzig, Dept. of Physics}
\bigskip
\bigskip
After the appearance of the papers of Berry [1], Simon [2], and
of Wilczek and Zee [3], I tried to understand [4], whether there is a
reasonable extension of the {\it geometric phase} - or, more accurately, of
the accompanying phase factor - for general (mixed) states. A known
recipe for such exercises is to use {\it purifications}: One looks for
larger, possibly fictitious, quantum systems from which the original
mixed states are seen as reductions of pure states. For density
operators there is a standard way to do so by the use of Hilbert Schmidt
operators (or by Hilbert Schmidt maps from an auxiliary Hilbert space
into the original one).
Thus let
$$ {\bf c} \, : \quad t \mapsto \varrho_t ,
\qquad 0 \leq t \leq 1 \eqno(1) $$
be a path of density operators. A {\it standard purification} of (1)
is a path
$$ t \mapsto W_t , \qquad \varrho_t = W_t W_t^* \eqno(2) $$
sitting in the Hilbert space of Hilbert Schmidt operators with
scalar product
$$ := \, {\rm tr} \, W_1^* \, W_2 \eqno(3) $$
The construction of standard purifications is by no means unique.
Indeed, not only (2) but every {\it gauged} path
$$ W_t \to W_t U_t , \qquad U_t \quad {\rm unitary}, \eqno(4) $$
is a purification of the same path of density operators.
\smallskip
The problem is, therefore, to distinguish within all purifications
of the curve (1) of mixed states exceptional ones. In [4] this
has been achieved as following. Let $W_1, ...., W_m$ be a
{\it subdivision} of (2), i.e. a time-ordered subset of operators
(2). These operators are of norm one since the density operators have
trace one. Now the expression
$$ \xi = ... \eqno(5) $$
will be considered according to (4) for all gauges
$$ \xi \mapsto \tilde \xi \quad {\rm by} \quad
W_j \mapsto W_j U_j , \qquad U_j \quad {\rm unitary}, \eqno(6) $$
and it will be looked within the set of gauged $\tilde \xi$ for choices with
$$ | \tilde \xi | \, = \, {\rm maximum !} \eqno(7) $$
The necessary and sufficient condition for (7) reads [5], [6]:
$$ | <\tilde W_{j+1}, \tilde W_j> | = {\rm tr} \, \bigl( \varrho_j^{1/2}
\varrho_{j+1} \varrho_j^{1/2} \bigr)^{1/2} \quad {\rm for}
\quad j = 1, \dots , m-1 \eqno(8) $$
It should be remarked that
$$ p(\varrho_1, \varrho_2) := \bigl( {\rm tr} \, \bigl( \varrho_1^{1/2}
\varrho_2 \varrho_1^{1/2} \bigr)^{1/2} \bigr) \eqno $$
is called {\it transition probability} of the pair $\varrho_1, \varrho_2$.
If (8) and hence (7) is fulfilled, the remaining arbitrariness
is in a regauging $\tilde W_j \to \epsilon_j U \tilde W_j$ of
the subdivision by numbers of modulus one and by an independent of
$j$ unitary $U$ - provided the rank of the density operators (1) remains
constant.
This, however, means the gauge invariance of the quantity
$$ X \mapsto \nu_{{\bf c}}^{subdivision}(X) = \xi \,
<\tilde W_1, X \tilde W_m> \eqno $$
and it depends therefore only on the ordered set of the density
operators $\varrho_k = W_k W_k^*$ .
In the limit of finer and finer subdivisions,
$$ X \mapsto \nu_{{\bf c}}(X) := \lim \, \nu_{{\bf c}}^{subdivision}(X)
, \eqno(11) $$
one obtains a gauge invariant linear form depending only on the original
path (1). For closed loops of pure states the number
$\nu_{{\bf c}}({\bf 1})$ is exactly Berry's phase factor.
\smallskip
(11) defines a certain noncommutative product integral. For curves
of faithful density operators it can be conveniently expressed by
the help of the geometric (quadratic) mean
$$ a \# b := a^{1 \over 2} \, ( a^{-{1 \over 2}} b
a^{-{1 \over 2}} )^{1 \over 2} \, a^{1 \over 2} \eqno(12) $$
of two positive operators [8], [9].
To this end one introduces the {\it holonomy} $V({\bf c})$
of ${\bf c}$ by
$$ \nu_{{\bf c}}(X) = {\rm tr } V({\bf c}) \varrho_0 X \eqno(13) $$
to find [20] ( - in [20] the exponents are not correctly assigned - )
$$ V({\bf c}) = \lim_{subdivisions} \,
( \varrho_m \# \varrho_{m-1}^{-1} )
( \varrho_{m-1} \# \varrho_{m-2}^{-1} )
\cdots ( \varrho_2 \# \varrho_1^{-1} )
\eqno(14) $$
\smallskip
My next aim is to obtain expressions of the above procedure which are
more manageable. One idea is to use an infinitesimal variant of (8).
Indeed one may sharpen (8) by adding the requirement
$$ \tilde W_{j+1}^* \, \tilde W_j \,\, \geq \, 0 \eqno(15) $$
which in turn implies (8) for faithful density operators.
Going to finer and finer subdivisions - and removing the tilde -
(15) results in ($\dot W$ denotes the t-derivation of $W$)
$$ W^* \, \dot W = \dot W^* \, W , \eqno(16) $$
the so-called {\it parallelity condition} [4] :
A lift (2) of (1) fulfilling
(16) is called a (standard) {\it parallel purification}
or a {\it parallel lift}.
Thus choosing a parallel purification of (1), it is
$$ \nu_{{\bf c}}(X) \, = \, \eqno(17) $$
where $W_0$ and $W_1$ are the starting and the end point of a
parallel lift.
Though the word {\it parallel} points to a parallel
transport governed by a connection form (described later on),
a more elementary explanation
is possible. The scaler products of the subdivision attain their
maximal possible value if (8) is true. The vectors $W_j$ have norm
one and hence the scalar product is the cosine between neighbouring
vectors. Therefore (8) indicates that the angles between neighbouring
vectors is as small as possible. Hence for infinitesimal neighbouring
they are parallelly directed.
Note that from (16) it follows for parallel lifts
$$ W^* \, \ddot W = \ddot W^* \, W , \eqno(b) $$
\smallskip
Another idea is already indicated in a paper of Fock [7], who
tried to minimize the arbitrariness in the transport of phases of
degenerate eigenstates of Hamiltonians.
The observation [10] is as following:
After choosing appropriate phases in (5) the scalar products
$$ can be made real and positive. But then $\xi$ in (7)
attains its maximum if and only if
$$ \parallel W_m - W_{m-1} \parallel + ... +
\parallel W_3 - W_2 \parallel + \parallel W_2 - W_1 \parallel
\eqno(19) $$
attains its minimum. On the other hand, in going to finer and finer
subdivisions, (19) tends to the length of the curve (2) in the
metric given by (3). Therefore a purification (2) is a parallel one
iff it solves the variational problem
$$ \int \sqrt{ <\dot W, \dot W>} {\rm d}t =
\hbox{Min !} \eqno(20) $$
However, the Euler equations of this variational problem
are nothing else than the parallelity condition (16) !
\smallskip
One can calculate the minimal length (20), which, indeed, is the
{\it Bures length} [9] of the path (1) of density operators. To do so
one has to solve the parallelity condition. According to Dabrowski and
Jadczyk [12], and to [13], this is done by an ansatz
$$ \dot W = \, G \, W , \qquad G^* = G \eqno(21) $$
which gives easily the equation
$$ \dot \varrho = G \varrho + \varrho G \eqno(22) $$
for the unknown $G$. $G$ is gauge invariant, and
depends only on the pair $\{ \varrho, \dot \varrho \}$. This
reflects the fact that the Bures length of the path (1) can be
expressed without using lifts (2) : Inserting (21) into (20)
one gets
$$ L^{\rm Bures}({\bf c}) \, = \, \int \sqrt{ } {\rm d}t
\eqno(23) $$
and a straightforward calculation shows
$$ dt^2_{\rm Bures} = = \, {\rm tr} \, \varrho G^2 = \,
{1 \over 2} {\rm tr} \, G \dot \varrho \eqno(24) $$
There is a formal solution of (22) which reads for faithful
density operators
$$ G = \int_0^{\infty} (\exp -s \varrho) \dot \varrho
(\exp -s \varrho) \, ds \eqno(25) $$
and which implies for the metric form (24) the expression
$$ {1 \over 2} {\rm tr} \,
\int_0^{\infty} (\exp -s \varrho) \dot \varrho
(\exp -s \varrho) \, \dot \varrho \, ds \eqno(26) $$
Now, switching to density operators of finite dimension $n$, one
may choose a base $E_k$ , where $k = 1, \dots , n^2 - 1$, of
traceless hermitian matrices, and write
$$ \varrho \, = \, {1 \over n} {\bf 1} + \sum x^k E_k \eqno(27) $$
to get from (26)
$$ {1 \over 2} {\rm tr} \, \dot \varrho G = \sum g_{jk} \dot x^j \dot x^k
\quad {\rm with} \quad g_{jk} = {1 \over 2} {\rm tr} \,
\int_0^{\infty} (\exp -s \varrho) E_j
(\exp -s \varrho) E_k \, ds \eqno(28) $$
Therefore one has for the "moments conjugate to the coordinates", $x_k$,
$$ p_k = 2 \sum g_{kj} \dot x^j = {\rm tr} \, G E_k \eqno(29) $$
\smallskip
{\it Example 1.}
Here I show the simplest possible case, the Bures metric for $n = 2$.
That this case can be solved is due to the following:
Let $\delta$ be a derivation, $X > 0, Y$, 2-by-2 matrices, then
$$ \delta X = Y X + X Y \eqno(30) $$
is solved by
$$ Y \, {\rm tr} X = \, \delta X + {1 \over 2} X^{-1} \delta \det X
- {\bf 1} \, {1 \over 2} {\rm tr} X \eqno(31) $$
which is easily derived by $\delta$-differentiating the characteristic
equation of $X$. Describing now the density operators by
$$ \varrho = {1 \over 2} \bigl( {\bf 1} +
x_1 \sigma_1 + x_2 \sigma_2 + x_3 \sigma_3 \bigr) \eqno(32) $$
which is a variant of (27), the metric space
$$ \{ \varrho > 0, \quad {\rm tr} \varrho = 1, \quad dt^2_{\rm Bures}
\} \eqno(33) $$
can be isometrically imbedded into a sphere ${\cal S}^3$ given by
$$ 1 = x_1^2 + x_2^2 + x_3^2 + x_4^2 \eqno(34) $$
where $x_4$ is defined by
$$ x_4 \geq 0, \qquad x_4^2 = 4 \det \varrho \eqno(35) $$
and which is equipped with the metric
$$ {1 \over 4} \bigl( {\rm d} x_1^2 + {\rm d} x_2^2 + {\rm d} x_3^2
+ {\rm d} x_4^2 \bigr) \eqno(36) $$
This example shows that the Bures metric
turns the set of all 2-by-2 density matrices into a piece of a symmetric
space, i.e. into half of a 3-sphere, see also [14], showing a hidden
O(4)-symmetry. Further, let
$$ \omega = {1 \over 2} \bigl( {\bf 1} +
y_1 \sigma_1 + y_2 \sigma_2 + y_3 \sigma_3 \bigr) \eqno(c) $$
be another 2-by-2 density operator. Then one calculates
$$ p(\varrho, \omega) = {1 \over 2} ( x_1 y_1 + x_2 y_2 +
x_3 y_3 + x_4 y_4 + 1 ) \eqno(d) $$
showing that for $n = 2$ the transition probability charcterize the
relative position of $\varrho$ and $\omega$ up to an O(4)-rotation.
\smallskip
{\it Example 2.}
Here the restriction of the Bures metric to maximal commutative submanifolds
will be described. In a suitable base such a submanifold can be given
by diagonal density matrices.
$$ \varrho = \bigl( \, \lambda_j \, \delta_{jk} \, \bigr) , \qquad
G = \bigl( \, g_j \, \delta_{jk} \, \bigr) \eqno(39) $$
Now (22) yields
$$ g_j = \frac{\dot \lambda_j}{2 \lambda_j} \eqno(40) $$
Introducing the new variables
$$ \lambda_j = y_j^2 \eqno(41) $$
the metric of Bures reads
$$ dt^2_{\rm Bures} = \sum \dot y_j^2 \eqno(42) $$
Hence the restriction on a maximal commutative subset of the
Bures metric is isometrically isomorph to a piece of a sphere, i.e.
of a symmetric space. \smallskip
The set of density operators, equipped with the Bures metric,
is metrically incomplete. One may ask whether there is a completion
in which all geodesics close for $\dim > 2$. To support this question
let us consider
\smallskip
{\it Example 3.}
The geodesic connecting two faithful density operators, $\varrho_j,
\, j=1,2$, within the space of density operators can be described as
follows. Let $\varrho_j = W_j W_j^*$ . Then the geodesic in the $W$-space
connecting $W_1$ with $W_2$ is part of a large circle of the unit
sphere. Its equation is
$$ W = \lambda_1 W_1 + \lambda_2 W_2,
\qquad = 1 \eqno(43) $$
where
$$ a := \hbox{Real } \eqno(44) $$
$$ \lambda_1 = \cos \vartheta - \frac{a}{\sqrt{1 - a^2}} \, \sin \vartheta
\eqno(45) $$
$$ \lambda_2 = \frac{\sin \vartheta}{\sqrt{1 - a^2}} \eqno(46) $$
The (oriented) length is hence the arc $\vartheta_0$ given by
$$ \cos \vartheta_0 = {\rm Real } \quad
{\rm with } \quad -{\pi \over 2} < \vartheta_0 < {\pi \over 2} \eqno(47) $$
Clearly, the length attains its minimum if we choose lifts such that
$a$ is of maximal value. This can be achieved if (15), and hence (8),
is valid for $j = 1$.
Thus the Bures length $\vartheta_0$ of the geodesic joining the two
density operators is given by
$$ \cos \vartheta_0 = {\rm tr} \, \bigl( \varrho_1^{1/2}
\varrho_2 \varrho_1^{1/2} \bigr)^{1/2} , \quad
{\rm with } \quad 0 < \vartheta_0 < {\pi \over 2} \eqno(48) $$
One easily constructs pairs $W_1, W_2$ for which (43) is a parallel
lift [4]. Expressing with them the holonomy (14)
and the linear form (13) results in
$$ V( {\rm geodesic} ) = \varrho_2 \# \varrho_1^{-1}, \qquad
\nu_{{\rm geodesic}} (X) = {\rm tr } X (\varrho_2 \# \varrho_1^{-1})
\varrho_1 \eqno(e) $$
Comparing (48) with (d) of example 1 yields in the $n = 2$ case
$$ x_1 y_1 + x_2 y_2 + x_3 y_3 + x_4 y_4 = \cos 2 \theta_0
\eqno(f) $$
\smallskip
The metric on the unit sphere of the Hilbert Schmidt W-space can be
decomposed into a horizontal and a vertical part by an ansatz
$$ <\dot W, \dot W> = \, + ,
\qquad A^* = - A \eqno(51) $$
Then $A$ can be defined equally well by [15]
$$ W^* \dot W - \dot W^* W = \, A \, W^* W + W^* W \, A \eqno(52) $$
This can be seen as follows. Going into (22) with an ansatz [15]
$$ \dot W - W A = G W \eqno(53) $$
and with $\varrho = W W^*$ , it follows that $A$ is antihermitian.
Knowing this and the hermiticity of $G$ one easily recovers (51).
On the other hand, substituting (53) into the left side of
(52), one arrives at the right side of this equation.
$A$ is the restriction on the given lift of a connection 1-form,
${\bf A}$, for the gauge transformations (4), and one has
$$ W^* {\rm d} W - {\rm d} W^* \, W =
\, {\bf A} \, W^* W + W^* W \, {\bf A} \eqno(54) $$
Introducing the ${\bf A}$-covariant derivation of an expression $X$
transforming as $W$ by
$$ {\rm D} X = {\rm d} X - X {\bf A} \eqno(55) $$
another form of (53) is
$$ {\rm D} \, W \, = \, {\bf G} \, W \eqno(56) $$
I now rewrite (54) in a form similar to (30).
As a complex linear space defines a complex analytic
structure, the total differential is decomposed naturally into
$d = \partial + \bar \partial $. Using this one may rewrite (54) as
$$ ( \partial - \bar \partial ) \, ( W^* W ) =
\, {\bf A} \, W^* W + W^* W \, {\bf A} \eqno(57) $$
This may be contrasted to
$$ {\rm d} ( W W^* ) = ( \partial + \bar \partial ) ( W W^* )
= W W^* {\bf G} + {\bf G} W W^* \eqno(58) $$
Thus we have
$$ W^* {\rm d} W =
\, {\bf A}^{1,0} \, W^* W + W^* W \, {\bf A}^{1,0} \eqno(59) $$
$$ {\rm d} W \, W^*
= W W^* {\bf G}^{1,0} + {\bf G}^{1,0} W W^* \eqno(60) $$
Remark: In the case of 2-by-2 density operators (51) can be solved
effectively by (31) using $\delta = \partial - \bar \partial$ ,
$X = W^* W$ , and $Y = {\bf A}$.
The first explicit expression for ${\bf A}$ was obtained in [16],
see also [17].
\medskip
For rank$(\varrho) = 1$ one falls back to the Berry case, and
${\bf A}$ describes the monopole structure.
For rank$(\varrho) = 2$ one gets instanton structures [18].
It is unknown what is with rank$(\varrho) > 2$ .
$$ * * * $$
Note added in proof: In a recent preprint [19] some of the
constructions are generalized and examined for C$^*$-algebras. It
is further indicated how possibly to proceed if the states
(or density operators) have mutually inequivalent supports.
\smallskip
\subsection*{References}
\begin{trivlist}
\item[ 1) ] M. V. Berry, Proc. Royal. Soc. Lond. A 392 (1984) 45
\item[ 2) ] B. Simon, Phys. Rev. Lett. 51 (1983) 2167
\item[ 3) ] F. Wilczek, A. Zee, Phys. Rev. Lett. 52 (1984) 2111
\item[ 4) ] A. Uhlmann, Rep. Math. Phys. {\bf 24}, 229, 1986
\item[ 5) ] H. Araki, RIMS-151, Kyoto 1973
\item[ 6) ] A. Uhlmann, Rep. Math. Phys. {\bf 9}, 273, 1976
\item[ 7) ] V. Fock, Z. Phys. {\bf 49} (1928) 323
\item[ 8) ] Pusz, W., Woronowicz, L., Rep. Math. Phys. {\bf 8} (1975) 159
\item[ 9) ] Ando, T., Linear Algebra Appl. 26 (1979) 203
\item[10) ] A. Uhlmann, Parallel Transport and Holonomy along Density
Operators. In: "Differential Geometric Methods in Theoretical
Physics", (Proc. of the XV DGM conference), H. D. Doebner and
J. D. Hennig (ed.), World Sci. Publ., Singapore 1987, p. 246 - 254
\item[11) ] D. J. C. Bures, Trans Amer. Math. Soc. {\bf 135}, 119, 1969
\item[12) ] L. Dabrowski and A. Jadczyk, Quantum Statistical Holonomy.
preprint, Trieste 1988
\item[13) ] A. Uhlmann, Ann. Phys. (Leipzig) {\bf 46}, 63, 1989
\item[14) ] M. H\"ubner: Explicit Computation of the Bures distance
for Density Matrices. NTZ-preprint 21/91, Leipzig 1991
\item[15) ] A. Uhlmann, Lett. Math. Phys. {\bf 21}, 229, 1991
\item[16) ] G. Rudolph: A connection form governing parallel transport
along $2 \times 2$ density matrices. Leipzig - Wroclaw
- Seminar, Leipzig 1990.
\item[17) ] J. Dittmann, G. Rudolph: A class of connections governing
parallel transport along density matrices.
Leipzig, NTZ-preprint 21/1991.
\item[18) ] J. Dittmann, G. Rudolph: On a connection governing parallel
transport along 2 x 2 -density matrices. To appear.
\item[19) ] P. M. Alberti: A study of pairs of positive linear forms,
algebraic transition probabilitiy, and geometric phase
over noncommutative operator algebras.
Leipzig, NTZ preprint 29/1991
\item[20) ] A. Uhlmann: Parallel Transport of Phases, in:
Differential Geometry, Group Representation, and
Quantization. (Hennig, Tolar, L\"ucke, editors),
Lecture Notes in Physics, p. 55-72, Springer 1991
\end{trivlist}
\end{document}