Krylov complexity measures the spread of the wavefunction in the Krylov basis, which is constructed using the Hamiltonian and an initial state. We investigate the evolution of the maximally entangled state in the Krylov basis for both chaotic and non-chaotic systems. For this purpose, we derive an Ehrenfest theorem for the Krylov complexity, which reveals its close relation to the spectrum. Our findings suggest that neither the linear growth nor the saturation of Krylov complexity is necessarily associated with chaos. However, for chaotic systems, we observe a universal rise-slope-ramp-plateau behavior in the transition probability from the initial state to one of the Krylov basis states. Moreover, a long ramp in the transition probability is a signal for spectral rigidity, characterizing quantum chaos. Also, this ramp is directly responsible for the late-time peak of Krylov complexity observed in the literature. On the other hand, for non-chaotic systems, this long ramp is absent. Therefore, our results help to clarify which features of the wave function time evolution in Krylov space characterize chaos. We exemplify this by considering the Sachdev-Ye-Kitaev model with two-body or four-body interactions.
Unfolded Krylov complexity: universal chaotic dynamics without false positives
Julius-Maximilians-Universität Würzburg（维尔茨堡大学）；Shanghai Jiao Tong University（上海交通大学）；Pohang University of Science and Technology（韩国科学技术院）；Instituto de Física Teórica UAM/CSIC（马德里自治大学/西班牙国家科学研究委员会理论物理研究所）
Quantum Chaos and Spread of States in Krylov Subspace: A Topical Review
Asia Pacific Center for Theoretical Physics（亚太理论物理中心）；Pohang University of Science and Technology（浦项科技大学）；Instituto de Física Teórica UAM/CSIC（马德里自治大学/西班牙国家科学研究委员会理论物理研究所）
Quantum chaos and late-time equipartition of symmetry-resolved Krylov complexity
Saha Institute of Nuclear Physics（萨哈核物理研究所）；Homi Bhabha National Institute（霍米·巴巴国立研究所）；Instituto de Física Teórica UAM/CSIC（马德里自治大学/西班牙国家科学研究委员会理论物理研究所）；Universidad Autónoma de Madrid（马德里自治大学）
Quantum Chaos Diagnostics for non-Hermitian Systems from Bi-Lanczos Krylov Dynamics
Shanghai Jiao Tong University（上海交通大学）；Shanghai Research Center for Quantum Sciences（上海量子科学研究中心）；Instituto de Física Teórica UAM/CSIC（UAM/CSIC理论物理研究所）；Asia Pacific Center for Theoretical Physics（亚太理论物理中心）；Pohang University of Science and Technology（韩国科学技术院）；Technical University of Munich（慕尼黑工业大学）；Ludwig-Maximilians-Universität München（慕尼黑大学）；Gwangju Institute of Science and Technology（光州科学技术研究院）
Probing the Chaos to Integrability Transition in Double-Scaled SYK
Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）；Julius-Maximilians-Universität Würzburg（维尔茨堡大学）；Weizmann Institute of Science（魏茨曼科学研究所）；Massachusetts Institute of Technology（麻省理工学院）；Freie Universität Berlin（柏林自由大学）
Towards a Refinement of Krylov Complexity: Scrambling, Classical Operator Growth and Replicas
National Taiwan University（国立台湾大学）；National Center for Theoretical Sciences（国家理论科学中心）；National Sun Yat-sen University（国立中山大学）；Gwangju Institute of Science and Technology（光州科学技术院）
Stochastic Krylov Dynamics: Revisiting Operator Growth in Open Quantum Systems
Indian Institute of Technology Gandhinagar（印度理工学院甘德纳加尔分校）；University of Cape Town（开普敦大学）；National Institute of Theoretical and Computational Sciences (NITheCS)（国家理论与计算科学研究所）
Cosmological brick walls & quantum chaotic dynamics of de Sitter horizons
Universidad de Oviedo（奥维耶多大学）；Instituto de Física Teórica UAM/CSIC（UAM/CSIC理论物理研究所）；Asia Pacific Center for Theoretical Physics（亚太理论物理中心）；Pohang University of Science and Technology（浦项科技大学）
Krylov complexity from a simple quantum mechanical model for a radiating black hole
Instituto de Física Teórica, UNESP-Universidade Estadual Paulista（圣保罗州立大学理论物理研究所）；The Laboratory for Quantum Gravity and Strings, Department of Mathematics and Applied Mathematics, University of Cape Town（开普敦大学数学与应用数学系量子引力与弦实验室）
