Amit Vikram, University of Colorado Boulder

The energy-time uncertainty principle states that quantum dynamics can occur over a time no faster than the inverse energy spread of an initial state. However, this bound is often trivial for questions of many-body dynamics, as the relevant timescales grow with the number of particles. For example, the fast scrambling conjecture (partly motivated from models of black holes and quantum circuits) suggests that the scrambling of quantum information between N interacting particles at temperature T requires a time that is at least (log N)/T. Though such statements have been challenging to prove, they are of fundamental importance for the emergence of quantum statistical mechanics. In this talk, we will describe a formulation of the energy-time uncertainty principle that gives nontrivial speed limits for many-body systems, by taking into account the full structure of the energy spectrum. This structure is accessed through the spectral form factor, a time-domain quantity that can be efficiently probed in experiments, including present-day quantum simulation platforms. Applying this relation to a quantum system coupled to a thermal bath, we will formulate and prove a rigorous fast scrambling statement in terms of the entanglement entropy of scrambled states, showing this bound to be a fundamental property of quantum mechanics without any model-specific assumptions. We will also show that certain “maximally chaotic” systems exhibit slow scrambling behaviors due to sharp edges in their energy spectrum. These results build towards an “ergodic theory” of quantum information dynamics, establishing how energy spectra determine universal dynamical features of quantum systems.