  We consider quantum fidelity between two states $\rho$ and $\sigma$, where we
fix $\rho$ and allow $\sigma$ to be sent through a quantum channel. We
determine the minimal fidelity where one minimizes over (a) all unital
channels, (b) all mixed unitary channels, and (c) arbitrary channels. We derive
results involving the minimal eigenvalue of $\rho$, which we can interpret as a
convex combination coefficient. As a consequence, we give a new geometric
interpretation of the minimal fidelity with respect to the closed, convex set
of density matrices and with respect to the closed, convex set of quantum
channels. We further investigate the geometric nature of fidelity by
considering density matrices arising as normalized projections onto subspaces;
in this way, fidelity can be viewed as a geometric measure of distance between
two spaces. We give a connection between fidelity and the canonical (principal)
angles between the subspaces.
