标题： 量子信息纠缠、混沌、敏感性和涌现态设计 显示英文标题

标题： Quantum Information Scrambling, Chaos, Sensitivity, and Emergent State Designs

摘要： 理解量子混沌具有深远的理论意义，并对各种应用具有重要的影响，从凝聚态物理到量子纠错。最近，时间有序关联函数（OTOCs）已成为量化量子混沌的强大工具。对于给定的量子系统，OTOCs 测量在海森堡绘景中演化的算符与未演化的算符之间的不相容性。在本论文的第一部分，我们利用 OTOCs 研究扰动非科莫戈罗夫-阿诺德-莫泽（non-KAM）系统在量子极限下的动态敏感性，当表征$\textit{resonance}$条件的参数缓慢变化时。为此，我们考虑一个在相空间中表现出随机网状结构的量子化踢谐振子（KHO）模型。随后对共振处的量子费舍尔信息（QFI）进行研究，并与非共振情况作比较。我们将展示在共振处 QFI 随时间的尺度增强，使得非 KAM 系统的动力学成为量子传感的良好候选。在下一章中，我们研究了踢耦合顶点的双部分系统中的 OTOCs，特别关注混合相空间 OTOC 动力学。论文的最后一部分致力于研究量子态设计的出现作为量子混沌的特征以及对称性在此现象中的作用。最近提出的投影系综框架利用量子混沌作为资源来构建近似高阶态设计。尽管普遍存在，对称性对量子态设计出现的影响仍缺乏深入研究。我们通过展示对称性和测量在构建近似态设计中的相互作用来全面研究这一问题。最后，我们概述了一些与当前论文相关的开放方向。 显示英文摘要

摘要： Understanding quantum chaos is of profound theoretical interest and carries significant implications for various applications, from condensed matter physics to quantum error correction. Recently, out-of-time ordered correlators (OTOCs) have emerged as a powerful tool to quantify quantum chaos. For a given quantum system, the OTOCs measure incompatibility between an operator evolved in the Heisenberg picture and an unevolved operator. In the first part of this thesis, we employ OTOCs to study the dynamical sensitivity of a perturbed non-Komogorov-Arnold-Moser (non-KAM) system in the quantum limit as the parameter that characterizes the $\textit{resonance}$ condition is slowly varied. For this purpose, we consider a quantized kicked harmonic oscillator (KHO) model that displays stochastic webs in the phase space. The OTOC analysis is followed by a study of quantum Fisher information (QFI) at the resonances and a comparison with the non-resonance cases. We shall show that scaling of the QFI in time is enhanced at the resonances, making the dynamics of the non-KAM systems good candidates for quantum sensing. In the following chapter, we study the OTOCs in a bipartite system of kicked coupled tops with a special focus on the mixed phase space OTOC dynamics. The last part of the thesis is devoted to the study of the emergence of quantum state designs as a signature of quantum chaos and the role of symmetries in this phenomenon. Recently proposed projected ensemble framework utilizes quantum chaos as a resource to construct approximate higher-order state designs. Despite being ubiquitous, the effects of symmetries on the emergence of quantum state designs remain under-explored. We thoroughly investigate this by demonstrating the interplay between symmetries and measurements in constructing approximate state designs. Finally, we outline a few open directions relevant to the current thesis.