  Systems reaching thermal equilibrium are ubiquitous. For classical systems,
this phenomenon is typically understood statistically through ergodicity in
phase space, but translating this to quantum systems is a long-standing problem
of interest. Recently a quantum notion of ergodicity has been proposed, namely
that isolated, global quantum states uniformly explore their available state
space, dubbed Hilbert-space ergodicity. Here we observe signatures of this
process with an experimental Rydberg quantum simulator and various numerical
models, before generalizing to the case of a local quantum system interacting
with its environment. For a closed system, where the environment is a
complementary subsystem, we predict and observe a smooth quantum-to-classical
transition in that observables progress from large, quantum fluctuations to
small, Gaussian fluctuations as the bath size grows. This transition is
universal on a quantitative level amongst a wide range of systems, including
those at finite temperature, those with itinerant particles, and random
circuits. Then, we consider the case of an open system interacting noisily with
an external environment. We predict the statistics of observables under largely
arbitrary noise channels including those with correlated errors, allowing us to
discriminate candidate error models both for continuous Hamiltonian time
evolution and for digital random circuits. Ultimately our results clarify the
role of ergodicity in quantum dynamics, with fundamental and practical
consequences.
