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We prove that the entanglement entropy of the ground state of a locallygapped frustration-free 2D lattice spin system satisfies an area law withrespect to a vertical bipartition of the lattice into left and right regions.We first establish that the ground state projector of any locally gappedfrustration-free 1D spin system can be approximated to within error $\epsilon$by a degree $O(\sqrt{n\log(\epsilon^{-1})})$ multivariate polynomial in theinteraction terms of the Hamiltonian. This generalizes the optimal bound on theapproximate degree of the boolean AND function, which corresponds to thespecial case of commuting Hamiltonian terms. For 2D spin systems we thenconstruct an approximate ground state projector (AGSP) that employs the optimal1D approximation in the vicinity of the boundary of the bipartition ofinterest. This AGSP has sufficiently low entanglement and error to establishthe area law using a known technique.

An area law for 2D frustration-free spin systems