Designs via Free Probability

TL;DR

This paper bridges unitary designs, free probability, and quantum chaos, showing that generic Hamiltonian evolution leads to freeness for ETH-restricted operators, linking pseudorandomness with thermalization and quantum ergodicity.

Contribution

It introduces the concept of $k$-freeness as an alternative to $k$-designs and connects free probability to quantum dynamics and ETH.

Findings

Freeness emerges at long times for ETH-restricted operators under generic Hamiltonian evolution.

Free probability tools facilitate calculations of mixed moments and quantum channels.

The work establishes a link between unitary designs, quantum chaos, and thermalization.

Abstract

Unitary Designs have become a vital tool for investigating pseudorandomness since they approximate the statistics of the uniform Haar ensemble. Despite their central role in quantum information, their relation to quantum chaotic evolution and in particular to the Eigenstate Thermalization Hypothesis (ETH) are still largely debated issues. This work provides a bridge between the latter and $k$-designs through Free Probability theory. First, by introducing the more general notion of $k$-freeness, we show that it can be used as an alternative probe to designs. In turn, free probability theory comes with several tools, useful for instance for the calculation of mixed moments or the so-called $k$-fold quantum channels. Our second result is the connection to quantum dynamics. Quantum ergodicity, and correspondingly ETH, apply to a restricted class of physical observables, as already discussed in the literature. In this spirit, we show that unitary evolution with generic Hamiltonians always leads to freeness at sufficiently long times, but only when the operators considered are restricted within the ETH class. Our results provide a direct link between unitary designs, quantum chaos and the Eigenstate Thermalization Hypothesis, and shed new light on the universality of late-time quantum dynamics.