Approximating local properties by tensor network states with constant bond dimension

Suppose we would like to approximate all local properties of a quantum many-body state to accuracy $\delta$. In one dimension, we prove that an area law for the Renyi entanglement entropy $R_\alpha$ with index $\alpha<1$ implies a matrix product state representation with bond dimension $\mathrm{poly}(1/\delta)$. For (at most constant-fold degenerate) ground states of one-dimensional gapped Hamiltonians, it suffices that the bond dimension is almost linear in $1/\delta$. In two dimensions, an area law for $R_\alpha(\alpha<1)$ implies a projected entangled pair state representation with bond dimension $e^{O(1/\delta)}$. In the presence of logarithmic corrections to the area law, similar results are obtained in both one and two dimensions.