Constructing an ergodic theory of quantum information dynamics

Abstract

The ergodic theory of classical dynamical systems, originating in Boltzmann's ergodic hypothesis, provides an idealized description of how the flow of information within energy surfaces of a classical phase space justifies the use of equilibrium statistical mechanics. While it is an extremely successful mathematical theory that establishes rigorous foundations for classical chaos and thermalization, its basic assumptions do not directly generalize to quantum mechanics. Consequently, previous approaches to quantum ergodicity have generally been limited to model-specific studies of thermalization, or well-motivated but imprecise general conjectures.

In this Dissertation, we develop a general theoretical framework for understanding how the energy levels of a quantum system drive the flow of quantum information and constrain the applicability of statistical mechanics, guided by two prominent conjectures. The first of these, the Quantum Chaos Conjecture (QCC), aims to characterize which quantum systems may thermalize, by postulating a connection between ergodicity or chaos and the statistical properties of random matrices. The second, the Fast Scrambling Conjecture (FSC), is concerned with how fast a quantum system may thermalize, and posits a maximum speed of thermalization in a sufficiently “local” many-body system.

This Dissertation is divided into three main parts. In the first part, Theory of Quantum Dynamics and the Energy Spectrum, we tackle these conjectures for a general isolated quantum system through results that may be understood as new formulations of the energy-time uncertainty principle. For QCC, we introduce precise quantum dynamical concepts of ergodicity and quantitatively establish their connections to the statistics of energy levels, deriving random matrix statistics as a special consequence of these dynamical notions. We subsequently build on one of these connections to derive an energy-time uncertainty principle that accounts for the full structure of the spectrum, introducing sufficient sensitivity for many-body systems. The resulting quantum speed limit allows us to prove a precise formulation of FSC from the mathematical properties of the energy spectrum. In doing so, we generalize QCC beyond the statistics of random matrices alone, and FSC beyond requirements of locality, establishing precise versions of these statements for the most general quantum mechanical Hamiltonian.

In the second part, Quantum Systems Beyond the Chaotic-Integrable Dichotomy, we demonstrate the need for the aforementioned precise formulations of these conjectures, by showing that looser formulations can be readily violated in “maximally” chaotic or integrable systems that would be most expected to satisfy them. Finally, in the third part, Experimental Probes of Many-Body Quantum Ergodicity, we develop tools to experimentally probe the structure of energy levels associated with ergodic dynamics, and demonstrate a generalization of these probes to open systems in an experiment with trapped ions.