  Information geometry promotes an investigation of the geometric structure of
statistical manifolds, providing a series of elucidations in various areas of
scientific knowledge. In the physical sciences, especially in quantum theory,
this geometric method has an incredible parallel with the distinguishability of
states, an ability of great value for determining the effectiveness in
implementing physical processes. This gives us the context for this work. Here
we will approach a problem of uniparametric statistical inference from an
information-geometric perspective. We will obtain the generalised Bhattacharyya
higher-order corrections for the Cram\'{e}r-Rao bound, where the statistics is
given by a mixed quantum state. Using an unbiased estimator $T$, canonically
conjugated to the Hamiltonian $H$ that generates the dynamics, we find these
corrections independent of the specific choice of estimator. This procedure is
performed using information-geometric techniques, establishing connections with
corrections to the pure states case.
