# On the size of the space spanned by a nonequilibrium state in a quantum spin lattice system

## TL;DR

This paper analyzes the growth of the subspace size that captures the time evolution of a nonequilibrium quantum spin lattice state, providing a quantitative estimate based on energy variance and system parameters.

## Contribution

It introduces a method to estimate the subspace size needed for accurate representation of quantum dynamics in large spin lattices, linking it to energy cumulants and system size.

## Key findings

- Subspace size scales as $L^{d/2} t$ with system size and time.
- Derived an explicit formula involving energy variance and error tolerance.
- Provides insight into the complexity of quantum state evolution in large systems.

## Abstract

We consider the time evolution of a state in an isolated quantum spin lattice system with energy cumulants proportional to the number of the sites $L^d$. We compute the distribution of the eigenvalues of the time averaged state over a time window $[t_0,t_0+t]$ in the limit of large $L$. This allows us to infer the size of a subspace that captures time evolution in $[t_0,t_0+t]$ with an accuracy $1-\epsilon$. We estimate the size to be $ \frac{\sqrt{2\mathfrak{e}_2}}{\pi}\mathrm{erf}^{-1}(1-\epsilon) L^{\frac{d}{2}}t$, where $\mathfrak{e}_2$ is the energy variance per site, and $\mathrm{erf}^{-1}$ is the inverse error function.