  We present exact results on a novel kind of emergent random matrix
universality that quantum many-body systems at infinite temperature can
exhibit. Specifically, we consider an ensemble of pure states supported on a
small subsystem, generated from projective measurements of the remainder of the
system in a local basis. We rigorously show that the ensemble, derived for a
class of quantum chaotic systems undergoing quench dynamics, approaches a
universal form completely independent of system details: it becomes uniformly
distributed in Hilbert space. This goes beyond the standard paradigm of quantum
thermalization, which dictates that the subsystem relaxes to an ensemble of
quantum states that reproduces the expectation values of local observables in a
thermal mixed state. Our results imply more generally that the distribution of
quantum states themselves becomes indistinguishable from those of uniformly
random ones, i.e. the ensemble forms a quantum state-design in the parlance of
quantum information theory. Our work establishes bridges between quantum
many-body physics, quantum information and random matrix theory, by showing
that pseudo-random states can arise from isolated quantum dynamics, opening up
new ways to design applications for quantum state tomography and benchmarking.
