  The space of quantum states can be endowed with a metric structure using the
second order derivatives of the relative entropy, giving rise to the so-called
Kubo-Mori-Bogoliubov inner product. We explore its geometric properties on the
submanifold of faithful, zero-displacement Gaussian states parameterised by
their covariance matrices, deriving expressions for the geodesic equations,
curvature tensors and scalar curvature. Our analysis suggests that the
curvature of the manifold is strictly monotonic with respect to the von Neumann
entropy, and thus can be interpreted as a measure of state uncertainty. This
provides supporting evidence for the Petz conjecture in continuous variable
systems.
