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

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.

💡 Research Summary

This doctoral thesis develops a suite of field-theoretic tools to dissect how quantum information propagates, scrambles, and is fundamentally reshaped by measurements in many-body quantum systems. It is structured around three complementary research projects that collectively build a unified framework for diagnosing “operational randomness” in quantum dynamics.

The first project tackles the quantification of scrambling and pseudorandomness in the strongly chaotic yet analytically tractable Brownian Sachdev-Ye-Kitaev (SYK) model. The author employs the concepts of unitary k-designs and frame potentials as operational measures of how closely a time-evolving unitary ensemble approximates the fully random Haar measure. By formulating the problem using Keldysh path integrals combined with replica techniques and disorder averaging, the work achieves analytic control over the time-dependent approach to randomness. It identifies collective modes that delay convergence to Haar-like behavior and provides explicit estimates for the “design time” (the time to approximate a k-design) as a function of model parameters like the number of fermions and interaction order. This clarifies the deep links between scrambling, quantum complexity growth, and the phenomenology observed in random quantum circuits.

The second project constructs a first-principles field theory for clusters of SYK models subjected to weak, continuous measurements. Starting from a microscopic system-ancilla description and taking a continuum monitoring limit, the author derives a stochastic Schrödinger equation governing the system’s conditional evolution. To analyze entanglement and purification, the replica trick is implemented by vectorizing the density matrix, leading to an effective Hamiltonian in a replicated Hilbert space governed by an SO(2n) algebra. Using a generalization of fermionic coherent states for n replicas, the author performs a symmetry-based reduction to derive a nonlinear sigma model (NLSM). This effective field theory captures the essential competition between Hamiltonian-induced scrambling (which spreads information) and measurement back-action (which extracts and localizes information). It predicts characteristic crossover scales in time and space, fluctuation spectra, and dynamical responses that distinguish weakly monitored phases from fully unitary or fully measured dynamics. A separate field theory is also derived for the measurement-only limit.

The third project designs a strong-disorder renormalization group (SDRG) approach tailored for measurement-only dynamics in a chain of fermionic clusters. Mapping the problem, via the replica trick, onto an effective chain of SO(2n) “spins” representing measurement operators, the author adapts the Dasgupta-Ma decimation procedure. The derived RG flow equations show features reminiscent of an infinite-randomness fixed point, where the distribution of measurement strengths becomes increasingly broad. However, a subtle order-of-limits issue—between taking the replica limit n→1 and the coupling strength limit—renders the leading-order recursion relations non-robust. Therefore, while the average second Rényi entropy exhibits the logarithmic scaling expected in such phases, the analytic evidence for a stable infinite-randomness fixed point in this monitored setting is deemed inconclusive, pointing to the need for a more stable replica-symmetric or replica-free formulation.

In synthesis, the thesis provides a coherent set of advanced field-theoretic methodologies—frame potentials and k-design diagnostics, Keldysh/replica techniques culminating in NLSMs, and disorder-based RG—to address when and how many-body evolution generates effective randomness and how the pervasive presence of measurements redirects that flow of information. The results offer concrete, testable predictions for near-term quantum simulation platforms like superconducting qubits, neutral atoms, and trapped ions.