Chaos, Entanglement and Measurement: Field-Theoretic Perspectives on Quantum Information Dynamics

TL;DR

This thesis develops a field-theoretic framework to understand how quantum information scrambles and is reshaped by measurements in many-body systems. It combines three complementary projects: (i) a frame-potential analysis of the Brownian SYK model to diagnose emergent randomness and design formation via Keldysh replicas, (ii) a first-principles NLσM description of interacting SYK clusters under weak continuous monitoring, and (iii) a strong-disorder RG treatment of measurement-only SYK dynamics based on the SO(2n) replica algebra. The work reveals an operational distinction between scrambling-driven Haar-like randomness (with distinct Majorana/Dirac and Gaussian/non-Gaussian regimes) and measurement-dominated dynamics, including measurement-induced crossovers and potential infinite-randomness behavior. It provides concrete, testable predictions for near-term quantum simulators and establishes a unified language—frame potentials, NLσMs, and SDRG—for diagnosing when many-body dynamics generate randomness and how measurements redirect that flow. Overall, the results chart a coherent program connecting randomness, information scrambling, and measurement backaction in strongly interacting fermionic systems, with implications for quantum information processing and quantum simulation experiments.

Abstract

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.

Figures (16)

• Figure 1: Geometry controls mixing times. Left: nearest-neighbor brickwork with a Lieb–Robinson light cone. Right: all-to-all pairings (fast scrambler).
• Figure 2: Thermal average and its TFD representation. (a) Euclidean “thermal circle’’ of circumference $\beta$ with seam identifying $\tau{=}0\sim\beta$; an operator $O$ is inserted at Euclidean time $\tau$. (b) Two copies $L,R$ prepared in the thermofield–double state $|\mathrm{TFD}\rangle$; the top/bottom endcaps correspond to $\tau{=}0$ and $\tau{=}\beta$ (front rim dashed, back rim solid). The TFD matrix element reproduces the thermal trace, $\langle O\rangle_\beta = Z(\beta)^{-1}\,\mathrm{Tr}\!(e^{-\beta H} O) = \langle \mathrm{TFD}|\, O_L(\tau)\, |\mathrm{TFD}\rangle$ with $Z(\beta)=\mathrm{Tr}\!(e^{-\beta H})$.
• Figure 3: Left: One monitored micro-step of duration $\Delta t$: the blue block $U_{\rm sys}$ acts on the system, then an ancilla–system interaction $U_{\rm int}$ is applied and the ancilla is measured. A dot on the ancilla line marks that a fresh ancilla is prepared each step (see App. \ref{['App:SSE']}). Right: Full evolution shown as the repetition of blocks acting on system$+$ancilla.
• Figure 4: Entanglement structure at low ($p<p_c$) and high ($p>p_c$) measurement rates. Blue links dominate at low $p$ (volume law), while magenta measurements suppress long links at high $p$ (area law).
• Figure 5: Rescaled state-dependent frame potential $F^{(k)}(T)$ for Brownian SYK with $q=2$, plotted on a logarithmic (vertical) scale. Solid curves are exact numerics at $L=120$: green for $k=2$, brown for $k=3$. Dotted lines show the early-time Keldysh prediction; dashed lines show the late-time (sector-Haar) prediction. The saturation time $T_{\mathrm{sat}}$ is estimated by the intersection of the dotted and dashed analytics, up to finite-size corrections (larger for higher $k$). For $q=2$, $T^{q=2}_{\rm Dirac}=4\!\left(2\log 2-\tfrac{k}{L}\log L\right)+O(L^{-1})$ and $T^{q=2}_{\rm Majorana}=4\!\left(2\log 2-\tfrac{k-1}{L}\log L\right)+O(L^{-1})$; for $q>2$, $T^{q>2}_{\rm Dirac}=T^{q>2}_{\rm Majorana}=q^{2}\cdot 2\log 2+O(L^{-1})$.