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.

翻译：几何量子力学通过其微分几何基础，提供了额外的分析与解释工具，使量子力学更接近经典力学：两者中的状态空间均配备了辛几何结构。这为从几何视角重新审视基础问题（如量子熵的本质）开辟了道路。其核心概念是几何量子态——系统纯态空间上的概率测度。该空间的连续性促使我们引入两种受Renyi信息论启发的分析工具，以刻画并量化几何量子态的基本性质：量子信息维度（几何量子态压缩的速率）和维度几何熵（监测存储于量子态中的信息）。我们回顾了它们的经典定义、信息论含义及物理解释，并通过几何方法将其适配至量子系统。随后，我们在各类量子系统实例中显式计算了这些量，并最后展望了几何量子力学中信息研究的未来方向。