# Entanglement subvolume law 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 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 O(|\partial A|^{5/3}) where `\partial A' is the boundary of A. This represents a 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.

Joint work with Itai Arad and David Gosset.

**Series:**Institute for Quantum Information (IQI) Weekly Seminar Series