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\centerline{{\bf An Energy Dispersion Estimate.}}
\medskip
\bigskip
\centerline{Armin Uhlmann}
\medskip
\centerline{University of Leipzig, Dept. of Physics}
\bigskip
Given a curve of density operators $t \mapsto \varrho = \varrho(t)$
which is a solution of
$$ i \hbar \dot \varrho = [ H, \varrho ] , \qquad H = H(t) \eqno(1) $$
it is assumed $\varrho_j = \varrho(t_j)$ for $j = 1,2$ with $t_1 < t_2$.
Using an idea of [1] it is my aim to derive the apriori inequality
$$ \int_{t_1}^{t_2} \, \sqrt{ {\rm tr }\varrho H^2 - ({\rm tr }\varrho H)^2}
\, dt \, \geq \, \hbar \, \gamma_{1 2} \eqno(2) $$
where
$$ 0 \leq \gamma_{1 2} \leq {\pi \over 2}, \qquad \cos \gamma_{1 2}
= \tau_{1 2} :=
{\rm tr} \, \sqrt{ \varrho_1^{1 \over 2} \varrho_2 \varrho_1^{1 \over 2}}
\eqno(3) $$
The physical meaning of the quantity
$ \tau_{1 2}$
is as follows [2]: Let be
$\psi_1, \psi_2$ two pure vector states of a (possibly fictitious) larger
quantum system and let their reduced density operators coincide with
$\varrho_1, \varrho_2$. Then $|<\psi_1, \psi_2>|$ should be not bigger than
$\tau_{1 2}$, and $\tau_{1 2}$ is the smallest number with that
property. In short, $\tau_{1 2}$ is the supremum of $|<\psi_1, \psi_2>|$
if the pair of unit vectors $\psi_1, \psi_2$ run through all possible
simultaneous purifications of the pair $\varrho_1, \varrho_2$.\medskip
In case the Hamiltonian is time independent, the expectation values
$\bar{E}, \bar{E^2}$ of $H$ and $H^2$ are constants of motion and
inequality (2) simplifies to
$$ \Delta t \, \Delta E \, \geq \, \hbar \, \arccos \tau_{1 2} \quad
{\rm with} \quad \Delta t = t_2 - t_1, \,\, \Delta E =
\sqrt{ \bar{E^2} - (\bar E)^2} \eqno(4) $$
A further important special case appears if $\varrho(t) =
|\varphi(t)><\varphi(t)|$
describes a curve of pure states which is a solution of a
Schr\"odinger equation $i \hbar \dot \varphi = H \varphi$.
Now equation (4) looks like
$$ \Delta t \, \Delta E \, \geq \, \hbar \, \arccos |<\varphi(t_1),
\varphi(t_2)>| \eqno(5) $$
Finely, if $\varphi(t_1)$ and $\varphi(t_2)$ are orthogonal
then the right hand side of (5) takes its maximal value $h/4$.
This is the result of Anandan and Aharonov [1].
\medskip
The {\it proof} of (2), (3) is in two steps, and will be done for
non-singular density operators. The general case follows by continuity.
The first step starts by lifting the curve of density operators into
the Hilbert space of Hilbert Schmidt operators, $W$, with scalar
product
$$ \, = \, {\rm tr} \, W' W^* \eqno(6) $$
i.e. we purify this curve by an ansatz
$$ t \mapsto W = W(t) \quad {\rm with} \quad
\varrho(t) = W W^* \eqno(7) $$
There is a gauge freedom $W(t) \to W(t) U(t)$ with arbitrary unitaries
$U(t)$ for these purifications. The freedom can be diminished by
choosing a curve (7) which is as short as possible in the metric given
by the scalar product (6). This variational demand produces the
parallelity condition [3, 4, 5]
$$ W^* \, \dot W \, = \, \dot W^* \, W \eqno(8) $$
(8) can be satisfied, as shown in [5], by a Schr\"odinger motion
$$ i \hbar \dot W = H W - W \tilde W, \qquad
\tilde H = \tilde H^* \eqno(9) $$
where
$$ \tilde H = 2 W^* Y W \quad \hbox{with } \quad
H = \varrho Y + Y \varrho \eqno(10) $$
and the last equation uniquely determines an operator $Y = Y(t)$.
Inserting (9) into (8) yields
$$ 2 W^* H W = W^* W \tilde H + \tilde H W^* W \eqno(11) $$
As is easily seen (9) is compatible with (1) and implies in
addition
$$ i \hbar {\partial \over \partial t} \tilde \varrho =
[ \tilde H , \tilde \varrho ] \quad
\hbox{with} \quad \tilde \varrho = W^* W \eqno(12) $$
Now inserting (9) into $<\dot W, \dot W>$ and using (11) gives
$$ \hbar^2 \, <\dot W, \dot W> = {\rm tr }\varrho H^2 -
{\rm tr }\tilde \varrho {\tilde H}^2 \eqno(13) $$
Taking the trace of (11) yields
$$ {\rm tr }\tilde \varrho {\tilde H} = {\rm tr } \varrho H
\eqno(14) $$
Taking this into account one gets by the help of Schwarz
inequality from (13) the relation
$$ {\rm tr }\varrho H^2 - ( {\rm tr } \varrho H )^2
\geq \hbar^2 \, <\dot W, \dot W> \eqno(15) $$
with which step one of the proof is done. \medskip
The final step is in proving
$$ \int_{t_1}^{t_2} \sqrt{ <\dot W, \dot W> } dt \, \geq \,
\gamma_{12} \eqno(16) $$
which gives, together with (15), the desired inequality (2).
This will be done by the aid of the Bures metrics [6]. To calculate
the Bures distance between $\varrho_1$ and $\varrho_2$ one considers
a general simultaneous purification
$$ W_1 = W(t_1) \, U_1, \quad W_2 = W(t_2) \, U_2,
\qquad \varrho_j = W_j \, W_j^* \eqno(17) $$
in order to get
$$ \parallel \varrho_2 - \varrho_1 \parallel_{\rm Bures} =
\inf \sqrt{ } = \sqrt{2 - 2 \tau_{12}}
\eqno(18) $$
where the infimum runs through the unitaries $U_1, U_2$. Then $\tau_{12}$
is given by (3). The proof of (18) can be found in [7, 2].
Now let be $W_1, W_2$ a pair of points on the unit sphere
${\rm tr }W W^* = 1$. Its
Hilbert space distance $d$ equals $d = 2 \sin(\gamma /2)$
where $\gamma$ denotes the length of a geodesic arc
on the unit sphere connecting $W_1, W_2$. This follows because
the geodesic arc is part of a large circle on the unit sphere crossing
$W_1, W_2$. But the length of any curve on the unit sphere of
the $W$-space connecting $W_1$ and $W_2$ cannot be shorter than
$\gamma$. Therefore, having in mind (18) and (3), the left hand side of
(16) cannot be smaller than $\gamma_{12}$.
However, $\gamma_{12}$ fulfills
$$ 2 \sin{\gamma_{12} \over 2} = \sqrt{2 - 2 \tau_{12}} =
\parallel \varrho_2 - \varrho_1 \parallel_{\rm Bures} \eqno(19) $$
and this relation implies (3). This finishes the proof. \medskip
{\it Remarks:} 1) Appropriate changes of the operator
$H$ and the parameter $t$ produce mathematically
equivalent but physically different inequalities.
As an example one may choose instead of $H$ the projection of angular
momentum onto an axis and the rotation angle instead of the time parameter.
2) Under the condition (8) the right hand side of (15) is
equal to the line element of the Bures metric (up to the square of
Planck's constant and restricted to the density operators).
Defining $G$ by
$$ \dot \varrho = \varrho G + G \varrho \eqno(20) $$
according to [4, 8] one gets $\dot W = G W$ for curves satisfying the
parallelity condition (8), and this gives
$$ \left( {ds \over dt} \right)^2_{\rm Bures} =
{\rm tr} \, \varrho G^2 = {1 \over 2} {\rm tr} \, G \dot \varrho
\eqno(21) $$
Thus the inequality (15) can be rewritten as
$$ {\rm tr }\varrho H^2 - ( {\rm tr } \varrho H )^2
\geq \hbar^2 \, {1 \over 2} {\rm tr} \, G \dot \varrho \eqno(22) $$
3) Restricting on pure states there is no need in step one of the
given proof. Furthermore, for pure states the metric of Bures is
nothing than the Fubini-Studi metric.
Based on that, the inequality (5) - also in the form including time
dependent Hamiltonians - has been derived already in [12].
4) The square of $\tau_{12}$ as given by (3) has been called
{\it (generalized) transition probability} in [2]. Using results
of [9] it could be shown in [10] that the same expression
results from a quite different definition given in [11].
For useful discussions I like to thank {\it P. M. Alberti, J. Anandan,
B. Crell,} \\
{\it M. H\"ubner,} and {\it G. Rudolph}.\bigskip
\subsection*{References}
\begin{trivlist}
\item[ 1) ] J. Anandan and Y. Aharonov, Phys. Rev. Lett. {\bf 65 }, 1697, 1991
\item[ 2) ] A. Uhlmann, Rep. Math. Phys. {\bf 9}, 273, 1976
\item[ 3) ] A. Uhlmann, Rep. Math. Phys. {\bf 24}, 229, 1986
\item[ 4) ] L. Dabrowski and A. Jadczyk, Quantum Statistical Holonomy.
preprint, Trieste 1988
\item[ 5) ] A. Uhlmann, Lett. Math. Phys. {\bf 21}, 229, 1991
\item[ 6) ] D. J. C. Bures, Trans Amer. Math. Soc. {\bf 135}, 119, 1969
\item[ 7) ] H. Araki, RIMS-151, Kyoto 1973
\item[ 8) ] A. Uhlmann, Ann. Phys. (Leipzig) {\bf 46}, 63, 1989
\item[ 9) ] H. Araki and G. A. Raggio, Lett. Math. Phys. {\bf 6}, 237, 1982
\item[10) ] P. M. Alberti and A. Uhlmann, Lett. Math. Phys. {\bf 7}, 163, 1983
\item[11) ] V. Cantoni, Comm. Math. Phys. {\bf 44}, 125, 1975
\item[12) ] J. S. Anandan, A Geometric View of Quantum Mechanics.
In: Quantum Coherence, J. S. Anandan (ed.), World Scientific,
1990.
\end{trivlist}
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