tripartite entanglement and tensor network representations in two dimensions

The area law of entanglement entropy is a fundamental property of ground states of local Hamiltonians. In one dimension, the area law implies a faithful matrix product state representation, enabling systemic theoretical and numerical study of ground states of one-dimensional Hamiltonians. For two-dimensional states, there is strong evidence that the area law alone does not guarantee a tensor-network representation. In this work, we show that trivial tripartite entanglement and area-law bipartite entanglement is a sufficient condition for tensor-network representation. We use the entanglement of purification and the Markov gap to quantify tripartite entanglement. Our definition of trivial tripartite entanglement includes, for example, the cluster state and string-net states in 2D.