In this talk I will discuss the projected ensemble, the collection of local post-measurement wavefunctions of a quantum many-body state. I will focus on states arising from evolution under generic isolated quantum dynamics, and describe the universal limiting distributions the projected ensemble attains, which depend on the interplay of conserved quantities and information obtained from the measurements.This amounts to a more refined version of quantum equilibration beyond standard thermalization of local observables, which has been dubbed “deep thermalization”. I will further explain how these limiting distributions are predicted by generalized maximum entropy principles rooted in quantum information theory. At the same time, I will also present a class of quantum dynamics where depending on the choice of measurements, the projected ensemble undergoes a phase transition from a maximally entropic one to a minimally entropic one, signatures of which are not detectable at the level of the density matrix. This constitutes a novel form of ergodicity-breaking, characterized not by the failure of the system to regularly thermalize, but rather by its failure to deep thermalize.