来自洛施密特振幅的克雷洛夫复杂度
Krylov Complexity from Loschmidt Amplitude
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中文总结 AI 辅助
研究基于洛施密特振幅，利用克雷洛夫复杂度算符代数性质，将克雷洛夫复杂度表示为特定洛施密特振幅导数，定义谱传播子，研究二维量子几何中克雷洛夫复杂度的性质，推导相关量之间的关系，为其动力学提供分类方案。
中文摘要 AI 辅助
克雷洛夫复杂度是量子动力学的有力诊断工具，与量子混沌和算符增长的其他度量有明确联系。洛施密特振幅是其中一种度量，定义为在两个略有不同的哈密顿量下演化的初始相同状态的重叠。其在某些系统中的衰减由经典李雅普诺夫指数控制。利用克雷洛夫复杂度算符的代数性质，我们将克雷洛夫复杂度表示为洛施密特振幅的导数，该振幅的微扰由角变量$\phi$参数化。此公式使我们能定义一个编码整个复杂度分布的谱传播子，并针对特定类型系统进行表征。我们研究了原始轨迹和变形轨迹所在的由时间和$\phi$跨越的二维量子几何，证明克雷洛夫复杂度受其体积的上限约束。我们还根据一个不同但相关的洛施密特振幅来表示克雷洛夫复杂度的时间导数。根据兰索斯系数的增长情况，该振幅中的微扰项可以截断。我们提出这种微扰的强度为克雷洛夫复杂度动力学提供了一种分类方案，并将其与谱传播子的$\phi$导数联系起来。利用这个分析框架，我们推导出生存振幅的时间依赖性与克雷洛夫空间度量之间的一般关系。
英文摘要
Krylov complexity is a powerful diagnostic of quantum dynamics, with clear connections to other measures of quantum chaos and operator growth. One such measure is the Loschmidt amplitude, defined as the overlap of initially identical states evolved under two slightly different Hamiltonians. Its decay in certain systems is controlled by the classical Lyapunov exponent. Using the algebraic properties of the Krylov complexity operator, we express Krylov complexity as the derivative of a Loschmidt amplitude whose perturbation is parameterized by an angular variable $ϕ$. This formulation allows us to define a spectral propagator that encodes the entire complexity distribution, which we characterize for specific types of systems. We study the two-dimensional quantum geometry spanned by time and $ϕ$ where the original and deformed trajectories reside, demonstrating that Krylov complexity is upper-bounded by its volume. We also express the time derivative of Krylov complexity in terms of a distinct Loschmidt amplitude. Depending on the growth of the Lanczos coefficients, the perturbation term in this amplitude can be truncated. We propose that the strength of this perturbation provides a classification scheme for Krylov complexity dynamics and relate it to the $ϕ$-derivative of the spectral propagator. Using this analytical framework, we derive general relations between the time-dependence of the survival amplitude and Krylov space measures.
发表机构
- ICTP South American Institute for Fundamental Research（ICTP南美基础研究所）
- Instituto de Física Teórica, UNESP - Univ. Estadual Paulista（圣保罗州立大学理论物理研究所）
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