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