Information geometry of quantum systems

This thesis presents an introductory exposition on the information geometry of quantum mechanics. The treatment begins with classical information geometry, defining statistical manifolds and the Fisher information metric, and finally demonstrating the Cramér-Rao inequality. Subsequently, the geometric formulation of quantum mechanics is introduced: starting from the postulates in their vectorial formulation, density operators are defined, and it is shown how the inner product of the Hilbert space induces the Fubini-Study metric on the projective Hilbert space. In the last part, the two geometric descriptions are connected by defining Fisher quantum information as a generalization of the classical one and finding that it coincides, up to a constant factor, with the Fubini-Study metric. The conclusion employs this description to demonstrate the quantum version of the Cramér-Rao inequality, a central result of quantum estimation theory (QET).