  We investigate two special classes of two-mode Gaussian states of light that
are important from both the experimental and theoretical points of view: the
mode-mixed thermal states and the squeezed thermal ones. Aiming to a parallel
study, we write the Uhlmann fidelity between pairs of states belonging to each
class in terms of their defining parameters. The quantum Fisher information
matrices on the corresponding four-dimensional manifolds are diagonal and allow
insightful parameter estimation. The scalar curvatures of the Bures metric on
both Riemannian manifolds of special two-mode Gaussian states are evaluated and
discussed. They are functions of two variables, namely, the mean numbers of
photons in the incident thermal modes. Our comparative analysis opens the door
to further investigation of the interplay between geometry and statistics for
Gaussian states produced in simple optical devices.
