# Entanglement subvolume law for 2D frustration-free spin systems

## TL;DR

This paper establishes a subvolume law for entanglement entropy in 2D frustration-free spin systems, showing it scales as a fractional power of boundary length, and improves understanding of entanglement in such quantum systems.

## Contribution

It introduces a novel AGSP construction based on polynomial approximation, leading to the first subvolume entanglement bounds in 2D frustration-free systems.

## Key findings

- Entanglement entropy scales as O(L^{5/3}) for vertical cuts.
- Provides a new area law with improved scaling in 1D.
- First subvolume entanglement bound for 2D frustration-free systems.

## Abstract

Let $H$ be a frustration-free Hamiltonian describing a 2D grid of qudits with local interactions, a unique ground state, and local spectral gap lower bounded by a positive constant. For any bipartition defined by a vertical cut of length $L$ running from top to bottom of the grid, we prove that the corresponding entanglement entropy of the ground state of $H$ is upper bounded by $\tilde{O}(L^{5/3})$. For the special case of a 1D chain, our result provides a new area law which improves upon prior work, in terms of the scaling with qudit dimension and spectral gap. In addition, for any bipartition of the grid into a rectangular region $A$ and its complement, we show that the entanglement entropy is upper bounded as $\tilde{O}(|\partial A|^{5/3})$ where $\partial A$ is the boundary of $A$. This represents the first subvolume bound on entanglement in frustration-free 2D systems. In contrast with previous work, our bounds depend on the local (rather than global) spectral gap of the Hamiltonian. We prove our results using a known method which bounds the entanglement entropy of the ground state in terms of certain properties of an approximate ground state projector (AGSP). To this end, we construct a new AGSP which is based on a robust polynomial approximation of the AND function and we show that it achieves an improved trade-off between approximation error and entanglement.