This work develops tools to understand how quantum information spreads, scrambles, and is reshaped by measurements in many-body systems. First, I study scrambling and pseudorandomness in the Brownian Sachdev-Ye-Kitaev (SYK) model, quantifying pseudorandomness using unitary k-designs and frame potentials. Using Keldysh path integrals with replicas and disorder averaging, I obtain analytic control of the approach to randomness, identify collective modes that delay convergence to Haar-like behavior, and estimate design times as functions of model parameters, clarifying links between scrambling, complexity growth, and random-circuit phenomenology. Second, I construct a field theory for weakly measured SYK clusters. Starting from a system-ancilla description and a continuum monitoring limit, and using fermionic coherent states with replicas and disorder averaging, I derive a nonlinear sigma model that captures measurement back-action and the competition between interaction-induced scrambling and information extraction, predicting characteristic crossover scales and response signatures that distinguish weak monitoring from fully unitary evolution. Third, I develop a strong-disorder renormalization group for measurement-only SYK clusters, based on the SO(2n) replica algebra and Dasgupta-Ma decimation rules. The flow shows features reminiscent of infinite-randomness behavior, but an order-of-limits subtlety renders the leading recursions non-robust, so the analytic evidence for an infinite-randomness fixed point is inconclusive, even though the average second Renyi entropy displays logarithmic scaling. Together, these results provide a unified language to diagnose when many-body dynamics generate operational randomness and how measurements redirect that flow.
This work develops tools to understand how quantum information spreads, scrambles, and is reshaped by measurements in many-body systems. First, I study scrambling and pseudorandomness in the Brownian Sachdev-Ye-Kitaev (SYK) model, quantifying pseudorandomness using unitary k-designs and frame potentials. Using Keldysh path integrals with replicas and disorder averaging, I obtain analytic control of the approach to randomness, identify collective modes that delay convergence to Haar-like behavior, and estimate design times as functions of model parameters, clarifying links between scrambling, complexity growth, and random-circuit phenomenology. Second, I construct a field theory for weakly measured SYK clusters. Starting from a system-ancilla description and a continuum monitoring limit, and using fermionic coherent states with replicas and disorder averaging, I derive a nonlinear sigma model that captures measurement back-action and the competition between interaction-induced scrambling and information extraction, predicting characteristic crossover scales and response signatures that distinguish weak monitoring from fully unitary evolution. Third, I develop a strong-disorder renormalization group for measurement-only SYK clusters, based on the SO(2n) replica algebra and Dasgupta-Ma decimation rules. The flow shows features reminiscent of infinite-randomness behavior, but an order-of-limits subtlety renders the leading recursions non-robust, so the analytic evidence for an infinite-randomness fixed point is inconclusive, even though the average second Renyi entropy displays logarithmic scaling. Together, these results provide a unified language to diagnose when many-body dynamics generate operational randomness and how measurements redirect that flow.
Quantum chaos concerns the emergence of universal statistical and dynamical prop- erties in quantum systems whose classical counterparts exhibit chaotic behavior. Unlike classical chaos, which can be characterized by sensitivity to initial conditions and exponential divergence of trajectories, quantum chaos manifests itself through a variety of indirect diagnostics, including spectral correlations, operator growth, and information scrambling. Understanding how these signatures arise from microscopic dynamics, and how they can be accessed in realistic many-body systems, is a central challenge in modern physics. This thesis investigates quantum chaos from complementary theoretical and practical perspectives. On the theoretical side, it develops effective descriptions of chaotic dy- namics in many-body systems with approximate symmetries, focusing on how weak symmetry breaking controls the exploration of Hilbert space and the crossover to random-matrix universality. Using perturbative arguments, numerical simulations, and a nonlinear sigma-model formulation, we identify a regime of Hilbert-space dif- fusion in which chaotic dynamics unfolds through a slow, collective spreading across nearly decoupled symmetry sectors. On the practical side, the thesis addresses the question of how paradigmatic chaotic models can be realized and probed in controlled experimental platforms. In partic- ular, it explores quantum simulation schemes for Sachdev–Ye–Kitaev (SYK) models using cavity quantum electrodynamics. By employing time-dependent disorder and Trotterized evolution in single-mode optical cavities, we demonstrate how low-rank interactions can be systematically promoted to effective all-to-all randomness, re- producing the spectral and dynamical signatures of SYK physics. Extensions to open-system dynamics are discussed, including Lindbladian realizations and the role of dissipation. The thesis thus highlights the connections between effective descriptions of quantum chaos and their implementation in controlled quantum simulation settings.
Chaos, intrication et mesure : perspectives de théorie des champs sur la dynamique de l'information quantique. Cette thèse développe des outils de théorie des champs pour comprendre comment l'information quantique se propage, se brouille (scrambling) et est remodelée par les mesures dans des systèmes à nombreux corps. Elle s'articule autour de trois projets complémentaires. extbf{Projet 1 — Scrambling et pseudo-aléatoire dans le SYK brownien.} Je quantifie le caractère pseudo-aléatoire à l'aide des (k)-designs unitaires et du potentiel de trame (frame potential), en prenant le modèle SYK brownien comme banc d'essai à la fois fortement chaotique et traitable. En utilisant des intégrales de chemin de Keldysh, combinées à la méthode des répliques et à la moyenne sur le désordre, j'obtiens un contrôle analytique de l'approche temporelle vers l'aléa et j'identifie des modes collectifs qui retardent la convergence vers un comportement de type Haar. Cela fournit des estimations des temps de (k)-design en fonction des paramètres du modèle et clarifie les liens entre scrambling, croissance de la complexité et phénoménologie des circuits aléatoires. J'esquisse également des protocoles de mesures aléatoires (randomized measurements) permettant d'accéder à ces prédictions sur du matériel quantique actuel. extbf{Projet 2 — Théorie des champs pour des amas SYK sous mesures faibles.} Je construis une description de premiers principes, en théorie des champs, d'amas SYK interactifs soumis à des mesures douces et fréquentes. Partant d'un schéma système-ancilla, je passe à la limite de monitoring continu et représente la dynamique à l'aide d'états cohérents fermioniques. En recourant aux répliques et à la moyenne sur le désordre, je dérive un modèle sigma non linéaire qui capture la rétroaction de la mesure, la compétition entre le scrambling induit par les interactions et l'extraction d'information, ainsi que l'organisation des modes collectifs mous et massifs. Ce cadre prédit des échelles de croisement caractéristiques en temps et en longueur, des spectres de fluctuations et des signatures de réponse qui distinguent les régimes de monitoring faible d'une évolution pleinement unitaire. extbf{Projet 3 — Renormalisation à fort désordre pour des amas SYK à dynamique purement mesurée.} Je construis un groupe de renormalisation à fort désordre (SDRG) adapté aux dynamiques « measurement-only », en partant de l'algèbre de répliques (SO(2n)) et en dérivant des règles de décimation à la Dasgupta-Ma. Si le flot présente des traits rappelant un comportement à désordre infini, une subtilité d'ordre des limites dans le traitement par répliques ((n! o!1) vs. (Gamma'! o!infty)) engendre un pôle simple dans le couplage induit et rend non robustes les récurrences au premier ordre. Par conséquent, les indices analytiques en faveur d'un point fixe à désordre infini (IRFP) dans le cadre monitoré sont emph{non conclusifs} et nécessitent une analyse plus poussée (p.,ex. une formulation stable en répliques ou sans répliques). Numériquement, l'emph{entropie de Rényi d'ordre 2 moyenne} suit l'échelle logarithmique prédite. À travers ces trois projets, la thèse propose un langage unifié — diagnostics par potentiel de trame et (k)-designs, techniques de Keldysh/répliques avec modèle sigma non linéaire, et RG fondé sur le désordre — pour décider quand l'évolution à nombreux corps génère un aléa opérationnel et comment les mesures réorientent ce flux. Les résultats suggèrent des signatures concrètes et testables pour des simulateurs quantiques à court terme (qubits supraconducteurs, atomes neutres, ions piégés).
Information scrambling, the process by which quantum information spreads and becomes effectively inaccessible, is central to modern quantum statistical physics and quantum chaos. These lecture notes provide an introduction to information scrambling from both static and dynamical perspectives. The spectral properties of reduced density matrices arising from Haar-random states are developed through the geometry of the unitary group and the universal results of random matrix theory. This geometric framework yields universal, model-independent predictions for entanglement and spectral statistics, capturing generic features of quantum chaos without reference to microscopic details. Dynamical diagnostics such as the spectral form factor and out-of-time-ordered correlators further reveal the onset of chaos in time-dependent evolution.
The notes are aimed at advanced undergraduate and graduate students in physics, mathematics, and computer science who are interested in the connections between quantum chaos, information dynamics, and quantum computing. Concepts such as entanglement growth, Haar randomness, random-matrix statistics, and unitary t-designs are introduced through their realization in random quantum circuits and circuit complexity. The same mathematical framework also underpins modern quantum-device benchmarking, where approximate unitary designs and random circuits translate geometric ideas into quantitative tools for assessing fidelity, noise, and scrambling efficiency in real quantum processors.
Chaos in many-body quantum systems is of great importance to both many-body physics as well as black hole physics. In the field of many-body physics, interactions and disorder in the system can lead to various dynamic phenomena, be it thermalisation, many-body localisation, chaotic behavior and scrambling of quantum information. In the field of high energy physics, the holographic principle connects chaos and scrambling in many-body quantum systems with the information-theoretic properties of black holes. Theoretical quantum physics provides the framework for the models, methods, and results in this thesis, while black hole physics partly provides some motivation and inspiration. First, we introduce a new model for a many-body quantum system based on random quantum circuits. These are a popular framework for theoretic study of disordered spin chains. By drawing the random unitaries in the circuit from different ensembles, we can adjust the disorder strength in the interactions, which in turn leads to a thermal/many-body localisation phase transition. Next, we study the Brownian SYK model, a disordered model of Majorana fermions with all-to-all interactions, motivated by its link to the holographic principle. We develop a new numerical method based on an effective permutational symmetry to reduce computational cost from exponential to linear or quadratic in system size N. As a consequence, we can compute scrambling quantifiers in detail, and find a log N scrambling time, as conjectured in the context of fast scrambling for black holes. Finally, we develop a model based on the continuous-time limit of a random quantum circuit. It serves as a microscopic toy model for the evaporation of a black hole. With a similar method as developed for the Brownian SYK model, we can analyse its information theoretic properties. In particular, we follow established protocols for information retrieval from the Hawking radiation. We find a separation of time scales for entanglement growth and information retrieval, related to the intrinsic black hole dynamics (∝log N) and the coupling to the environment (∝N).
Information scrambling is a hallmark of quantum chaos and thermalization in isolated quantum many-body systems. We investigate scrambling dynamics in the all-to-all interacting spin Sachdev-Ye-Kitaev (SYK)-q model using both pure- and mixed-state entanglement measures. We show that von-Neumann and Rényi entropies exhibit rapid growth followed by saturation near Haar-random values, signaling efficient scrambling. The scrambling rate reveals a nontrivial dependence on the interaction order, system size, and Hamiltonian scaling. We further employ mixed-state entanglement as a powerful probe of information scrambling. We numerically find a universal relation between the Rényi-1/2 mutual information and entanglement negativity for minimal interaction order in the early growth regime. Furthermore, entanglement negativity displays a Page-curve-like behavior under unequal subsystem partitioning, characterized by the birth, spread, and eventual death of quantum correlations. Our results provide a generic description of information scrambling using entanglement dynamics in all-to-all interacting spin systems with multi-body interactions.
A bstract We investigate minimal two-body Hamiltonians with random interactions that generate spectra resembling those of Gaussian random matrices, a phenomenon we term quadratic quantum chaos . Unlike integrable two-body fermionic systems, the corresponding hard-core boson models exhibit genuinely chaotic dynamics, closely paralleling the Sachdev-Ye-Kitaev (SYK) model in its spin representation. This chaotic behavior is diagnosed through spectral statistics and measures of operator growth, including Krylov complexity and the late-time decay of higher-order out-of-time-ordered correlators (OTOCs); the latter reveals the emergence of freeness in the sense of free probability. Moreover, the fractal dimension and Stabilizer Rényi entropy of a representative mid-spectrum eigenstate show finite-size deviations yet converge toward Haar-randomness as the system size increases. This convergence, constrained by local interactions, highlights the “weakly chaotic” character of these eigenstates. Owing to their simplicity and bosonic nature, these minimal models may constitute promising and resource-efficient candidates for probing quantum chaos and information scrambling on near-term quantum devices.