  We examine the geodesic between two mixed states of arbitrary dimension by
means of their geometric mean operator. We utilize the fiber bundle approach by
which the distance between two mixed state density operators $\rho_1$ and
$\rho_2$ in the base space $M$ is given by the shortest distance in the
(Hilbert Schmidt) bundle space $E$ of their purifications. The latter is
well-known to be given by the Bures distance along the horizontal lift in $E$
of the geodesic between the $\rho_1$ and $\rho_2$ in $M$. The horizontal lift
is that unique curve in $E$ that orthogonally traverses the fibers $F\subset E$
above the curve in $M$, and projects down onto it. We briefly review this
formalism and show how it can be used to construct the intermediate mixed
quantum states $\rho(s)$ along the base space geodesic parameterized by affine
parameter $s$ between the initial $\rho_1$ and final $\rho_2$ states. We
emphasize the role played by geometric mean operator $M(s) = \rho_1^{-1/2}\,
\sqrt{\rho_1^{1/2}\rho(s)\rho_1^{1/2}}\,\rho_1^{-1/2}$, where the Uhlmann root
fidelity between $\rho_1$ and $\rho(s)$ is given by $\sqrt{F}(\rho_1,\rho(s)) =
Tr[M(s)\,\rho_1] = Tr[\sqrt{\rho_1^{1/2}\rho(s)\rho_1^{1/2}}]$, and $\rho(s) =
M(s)\,\rho_1\,M(s)$. We give examples for the geodesic between the maximally
mixed state and a pure state in arbitrary dimensions, as well as for the
geodesic between Werner states $\rho(p) = (1-p) I/N + p\,|\Psi\rangle\langle
\Psi|$ with $|\Psi\rangle = \{|GHZ\rangle, |W\rangle\}$ in dimension $N=2^3$.
For the latter, we compare expressions in the limit $p\to1$ to the infinite
number of possible geodesics between the orthogonal pure states $|GHZ\rangle$
and $|W\rangle$. Lastly, we compute the analytic form for the density matrices
along the geodesic that connects two arbitrary endpoint qubit density matrices
within the Bloch ball for dimension $N=2$.
