Exact emergent quantum state designs from quantum chaotic dynamics

TL;DR

This paper demonstrates that quantum chaotic systems at infinite temperature generate ensembles of states that form exact quantum state-designs, revealing a new form of emergent randomness beyond thermalization.

Contribution

It provides rigorous proof that quantum chaotic dynamics produce state ensembles equivalent to uniformly random states, establishing a novel connection between quantum chaos and quantum information theory.

Findings

Quantum chaotic systems produce state ensembles approaching uniform Hilbert space distribution.

The resulting ensembles form exact quantum state-designs, not just approximate.

This bridges quantum many-body physics with quantum information and random matrix theory.

Abstract

We present exact results on a novel kind of emergent random matrix universality that quantum many-body systems at infinite temperature can exhibit. Specifically, we consider an ensemble of pure states supported on a small subsystem, generated from projective measurements of the remainder of the system in a local basis. We rigorously show that the ensemble, derived for a class of quantum chaotic systems undergoing quench dynamics, approaches a universal form completely independent of system details: it becomes uniformly distributed in Hilbert space. This goes beyond the standard paradigm of quantum thermalization, which dictates that the subsystem relaxes to an ensemble of quantum states that reproduces the expectation values of local observables in a thermal mixed state. Our results imply more generally that the distribution of quantum states themselves becomes indistinguishable from those of uniformly random ones, i.e. the ensemble forms a quantum state-design in the parlance of quantum information theory. Our work establishes bridges between quantum many-body physics, quantum information and random matrix theory, by showing that pseudo-random states can arise from isolated quantum dynamics, opening up new ways to design applications for quantum state tomography and benchmarking.