Emergent quantum state designs from individual many-body wavefunctions

TL;DR

This paper demonstrates that individual many-body quantum states can encode universal quantum state designs, providing new insights into quantum chaos and practical methods for sampling random states in quantum information science.

Contribution

It introduces the concept that single non-random states can generate universal quantum state ensembles, linking quantum chaos with quantum information theory.

Findings

Quantum state $k$-designs naturally emerge from generic and individual states.

Single states associated with Hamiltonian dynamics encode highly random ensembles.

Provides a practical method for sampling approximately uniform random states.

Abstract

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 examples, a single non-random 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 techniques. In particular, we establish that $k$-designs arise naturally from generic states as well as individual states associated with strongly interacting, time-independent 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.