量子混沌与量子最优传输
Quantum Chaos and Quantum Optimal Transport
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中文总结 AI 辅助
本文利用量子最优传输理论建立量子混沌的统一数学基础，定义了避免发散的量子李雅普诺夫指数，其在半经典极限下可恢复经典结果，还阐明了时序无序关联函数的诊断作用。
中文摘要 AI 辅助
经典系统中的混沌可由李雅普诺夫指数表征，该指数用于测量相邻轨迹的指数发散，但将这一框架直接推广至量子力学长期以来一直是一项挑战。量子态的类波性质以及量子相空间的非交换几何阻碍了经典混沌理论的直接推广。本文中，我们利用量子最优传输理论开发了一种严谨的量子混沌研究方法，该理论为测量扩展量子分布间的距离提供了缺失的几何基础。我们定义了量子李雅普诺夫指数，该指数可自然避免在朴素推广经典指数时遇到的发散问题，并证明在半经典极限下，这些指数可恢复经典全局膨胀率，因此当它们重合时可得到通常的最大李雅普诺夫指数。我们的框架在经典轨迹的发散与半经典相空间演化之间建立了紧密联系，此外还阐明了时序无序关联函数作为量子混沌诊断工具的作用。这些结果确立了量子最优传输作为量子混沌理论的统一数学基础，为表征从简单少体模型到复杂多体系统的全范围量子动力学行为提供了新工具。
英文摘要
Chaos in classical systems can be characterized by Lyapunov exponents that measure the exponential divergence of nearby trajectories, but directly extending this framework to quantum mechanics has been a persistent challenge. The wavelike nature of quantum states and the non-commutative geometry of quantum phase space obstruct a straightforward generalization of classical chaos theory. Here we develop a rigorous approach to quantum chaos by leveraging quantum optimal transport theory, which provides the missing geometric foundation for measuring distances between extended quantum distributions. We define quantum Lyapunov exponents that naturally avoid divergences encountered when naïvely generalizing classical exponents, and show that in the semiclassical limit they recover the classical global expansion rate, and hence the usual maximal Lyapunov exponent when they coincide. Our framework provides a tight connection between the divergence of classical trajectories and semiclassical phase space evolution, and additionally clarifies the role of out-of-time-order correlators as diagnostics of quantum chaos. These results establish quantum optimal transport as a unifying mathematical foundation for quantum chaos theory, providing new tools to characterize dynamical behavior across the full range of quantum dynamics from simple few-body models to complex many-body systems.
