AI 中文总结
该研究针对开放量子系统，采用几何框架提出两种互补诊断，结合量子踢顶模型揭示了相互作用强度、宇称对称性对量子动力学复杂性的影响，为量化量子区域动力学复杂性提供了几何途径。
AI 中文摘要
我们研究有限尺寸量子系统在动力学从封闭相干演化过渡到相互作用主导的有效开放行为时复杂性的涌现。采用基于状态的几何框架，我们将混合量子态表示为复射影希尔伯特空间上的概率测度，该表示可追踪相互作用如何重塑底层纯态几何。我们引入两种互补诊断：一是基于混合态概率测度表示间Wasserstein距离的可区分性度量，用于量化对初始态的敏感性；二是测量子系统态空间长时间探索程度的态空间覆盖指数。这些诊断为量子动力学复杂性的涌现与演化提供了几何视角。将其应用于量子踢顶模型时，两种诊断通常随相互作用强度增大而升高；它们对环境尺寸的依赖由宇称对称性决定，整数自旋系统常表现出比半整数自旋系统更高的敏感性和态空间覆盖度。这些结果凸显了有限尺寸量子效应，并提供了一种量化量子区域深处动力学复杂性的几何方法。
英文摘要
We study the emergence of complexity in finite-size quantum systems as their dynamics transition from closed and coherent evolution to interacting and effectively open behavior. Using a state-based geometric framework, we represent mixed quantum states as probability measures on complex projective Hilbert space. This representation allows us to track how interactions reshape the underlying pure-state geometry. We introduce two complementary diagnostics: a distinguishability measure, based on the Wasserstein distance between probability-measure representations of mixed states, that quantifies sensitivity to initial states, and a state-space coverage index that measures long-time exploration of the subsystem state space. These diagnostics provide a geometric perspective on the emergence and evolution of quantum dynamical complexity. When applied to the quantum kicked top, both diagnostics generally increase with interaction strength. Their dependence on environment size is structured by parity symmetry, with integer-spin systems often exhibiting greater sensitivity and state-space coverage than half-integer-spin systems. These results highlight finite-size quantum effects and provide a geometric approach to quantifying dynamical complexity deep in the quantum regime
Comments30 pages, 18 figures, 6 appendices; http://csc.ucdavis.edu/~cmg/compmech/pubs/QuantumStateComplexity.htm
