Abstract: In this talk, we will describe a 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 some 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. To address QCC, we introduce precise quantum dynamical concepts of ergodicity and quantitatively establish their connection to the statistics of energy levels, deriving random matrix statistics as a special consequence of these dynamical notions. Building on these notions, we derive a tighter state-independent formulation of the energy-time uncertainty principle that accounts for the full structure of the energy spectrum, introducing sufficient sensitivity for many-body systems. The resulting quantum speed limit allows us to prove a quantitative formulation of FSC from the mathematical properties of the 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. Along the way, we will discuss some additional implications for “many-body quantum chaos” and quantum gravity.
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