  Geometric quantum mechanics, through its differential-geometric underpinning,
provides additional tools of analysis and interpretation that bring quantum
mechanics closer to classical mechanics: state spaces in both are equipped with
symplectic geometry. This opens the door to revisiting foundational questions
and issues, such as the nature of quantum entropy, from a geometric
perspective. Central to this is the concept of geometric quantum state -- the
probability measure on a system's space of pure states. This space's continuity
leads us to introduce two analysis tools, inspired by Renyi's information
theory, to characterize and quantify fundamental properties of geometric
quantum states: the quantum information dimension that is the rate of geometric
quantum state compression and the dimensional geometric entropy that monitors
information stored in quantum states. We recount their classical definitions,
information-theoretic meanings, and physical interpretations, and adapt them to
quantum systems via the geometric approach. We then explicitly compute them in
various examples and classes of quantum system. We conclude commenting on
future directions for information in geometric quantum mechanics.
