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

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.