This paper explores the statistical properties of ensembles of pure states in quantum many-body systems, focusing on two classes of ensembles: those formed by the temporal trajectory of a quantum state under unitary evolution and those formed by partial, local projective measurements on small subsystems. These ensembles, respectively, exemplify "Hilbert-space ergodicity" and "deep thermalization." The authors propose a generalized maximum entropy principle for these ensembles, which states that the distributions of pure states maximize entropy subject to constraints such as energy conservation and effective constraints imposed by thermalization. They present explicit formulae for the statistical moments of these ensembles, prove necessary and sufficient conditions for universality under widely accepted assumptions, and describe their measurable consequences in experiments. The work generalizes the notions of Hilbert-space ergodicity to time-independent Hamiltonian dynamics and deep thermalization from infinite to finite effective temperature. The findings have implications for understanding the dynamics of quantum systems, the nature of information scrambling, and the operational meaning of ensemble entropy. The paper also discusses the information-theoretic implications, including the maximally difficult tasks of information compression and classical information transmission for these ensembles.This paper explores the statistical properties of ensembles of pure states in quantum many-body systems, focusing on two classes of ensembles: those formed by the temporal trajectory of a quantum state under unitary evolution and those formed by partial, local projective measurements on small subsystems. These ensembles, respectively, exemplify "Hilbert-space ergodicity" and "deep thermalization." The authors propose a generalized maximum entropy principle for these ensembles, which states that the distributions of pure states maximize entropy subject to constraints such as energy conservation and effective constraints imposed by thermalization. They present explicit formulae for the statistical moments of these ensembles, prove necessary and sufficient conditions for universality under widely accepted assumptions, and describe their measurable consequences in experiments. The work generalizes the notions of Hilbert-space ergodicity to time-independent Hamiltonian dynamics and deep thermalization from infinite to finite effective temperature. The findings have implications for understanding the dynamics of quantum systems, the nature of information scrambling, and the operational meaning of ensemble entropy. The paper also discusses the information-theoretic implications, including the maximally difficult tasks of information compression and classical information transmission for these ensembles.