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,通过其差分几何基础,提供了更多的分析和解释工具,使量子力学更接近经典力学:国家空间都配备了共振几何学。这打开了从几何角度重新审视基本问题和议题的大门,例如从几何角度看量子酶的性质。关键在于几何量子状态概念 -- -- 系统纯国家空间的概率测量。这一空间的连续性导致我们引入两个分析工具,在Renyi信息理论的启发下,确定和量化几何量子状态的基本特性:量子信息层面,即几何量子质子压缩率和监测量子状态中储存的信息的维度几何几何几何进方。我们用几何方法来描述其古典定义、信息理论含义和物理解释,并把它们调整到量子系统。然后我们明确将其纳入各种实例和量子系统类别。我们最后评论几何量力量力力力学未来的信息方向。