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 Õ(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 Õ(|∂ A|5/3) where ∂ 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.
We present a new proof for the 1D area law for frustration-free systems with a constant gap, which exponentially improves the entropy bound in Hastings\^a 1D area law and which is tight to within a polynomial factor. For particles of dimension
d, spectral gap $\ensuremath{\epsilon}>0$, and interaction strength at most
J, our entropy bound is ${S}_{1D}\ensuremath{\le}\mathcal{O}(1)\ifmmode\cdot\else\textperiodcentered\fi{}{X}^{3}{\mathrm{log}}^{8}X$, where $X\stackrel{\mathrm{def}}{=}(J\mathrm{log}d)/\ensuremath{\epsilon}$. Our proof is completely combinatorial, combining the detectability lemma with basic tools from approximation theory. In higher dimensions, when the bipartitioning area is $|\ensuremath{\partial}L|$, we use additional local structure in the proof and show that $S\ensuremath{\le}\mathcal{O}(1)\ifmmode\cdot\else\textperiodcentered\fi{}{|\ensuremath{\partial}L|}^{2}{\mathrm{log}}^{6}|\ensuremath{\partial}L|\ifmmode\cdot\else\textperiodcentered\fi{}{X}^{3}{\mathrm{log}}^{8}X$. This is at the cusp of being nontrivial in the 2D case, in the sense that any further improvement would yield a subvolume law.