  Chaotic quantum many-body dynamics typically lead to relaxation of local
observables. In this process, known as quantum thermalization, a subregion
reaches a thermal state due to quantum correlations with the remainder of the
system, which acts as an intrinsic bath. While the bath is generally assumed to
be unobserved, modern quantum science experiments have the ability to track
both subsystem and bath at a microscopic level. Here, by utilizing this
ability, we discover that measurement results associated with small subsystems
exhibit universal random statistics following chaotic quantum many-body
dynamics, a phenomenon beyond the standard paradigm of quantum thermalization.
We explain these observations with an ensemble of pure states, defined via
correlations with the bath, that dynamically acquires a close to random
distribution. Such random ensembles play an important role in quantum
information science, associated with quantum supremacy tests and device
verification, but typically require highly-engineered, time-dependent control
for their preparation. In contrast, our approach uncovers random ensembles
naturally emerging from evolution with a time-independent Hamiltonian. As an
application of this emergent randomness, we develop a benchmarking protocol
which estimates the many-body fidelity during generic chaotic evolution and
demonstrate it using our Rydberg quantum simulator. Our work has wide ranging
implications for the understanding of quantum many-body chaos and
thermalization in terms of emergent randomness and at the same time paves the
way for applications of this concept in a much wider context.
