Emergent Quantum State Designs from Individual Many-Body Wave Functions

Quantum chaos in many-body systems provides a bridge between statistical and quantum physics with strong predictive power. This framework is valuable for analyzing properties of complex quantum systems such as energy spectra and the dynamics of thermalization. While contemporary methods in quantum chaos often rely on random ensembles of quantum states and Hamiltonians, this is not reflective of most real-world systems. In this paper, we introduce a new perspective: across a wide range of exam-ples, a single nonrandom quantum state is shown to encode universal and highly random quantum state ensembles. We characterize these ensembles using the notion of quantum state k-designs from quantum information theory and investigate their universality using a combination of analytic and numerical tech-niques. In particular, we establish that k-designs emerge naturally from generic states in a Hilbert space as well as physical states associated with strongly interacting Hamiltonian dynamics. Our results offer a new approach for studying quantum chaos and provide a practical method for sampling approximately uniformly random states; the latter has wide-ranging applications in quantum information science from tomography to benchmarking.