# Universal fluctuations around typicality for quantum ergodic systems

## TL;DR

This paper demonstrates that fluctuations around typical states in quantum ergodic systems are described by random matrix theory, specifically the GUE, and extends this to fluctuations around the canonical Gibbs state, linking to ETH.

## Contribution

It provides a detailed characterization of quantum fluctuations around typicality and canonical states using random matrix theory, extending previous understanding of quantum ergodicity.

## Key findings

- Fluctuations are captured by the Gaussian unitary ensemble (GUE).
- Fluctuations around the Gibbs state are described by a deformation of the GUE.
- The results connect quantum typicality with random matrix theory and ETH.

## Abstract

For a quantum system in a macroscopically large volume $V$, prepared in a pure state and subject to maximally noisy or ergodic unitary dynamics, the reduced density matrix of any sub-system $v\ll V$ is almost surely totally mixed. We show that the fluctuations around this limiting value, evaluated according to the invariant measure of these unitary flows, are captured by the Gaussian unitary ensemble (GUE) of random matrix theory. An extension of this statement, applicable when the unitary transformations conserve the energy but are maximally noisy or ergodic on any energy shell, allows to decipher the fluctuations around canonical typicality. According to typicality, if the large system is prepared in a generic pure state in a given energy shell, the reduced density matrix of the sub-system is almost surely the canonical Gibbs state of that sub-system. We show that the fluctuations around the Gibbs state are encoded in a deformation of the GUE whose covariance is specified by the Gibbs state. Contact with the eigenstate thermalisation hypothesis (ETH) is discussed.