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Cutting an open-boundary matrix product state across one virtual bond of dimension $\chi$ gives Schmidt rank at most $\chi$, so its entanglement entropy obeysA periodic interval cuts two bonds and obeys $S(A)\le 2\log \chi$. Since the boundary of a one-dimensional interval has a constant number of points, this is an area law.

For a projected entangled pair state, a bipartition crossing $n_{\partial }$ virtual bonds has Schmidt rank at most ${\chi }^{n_{\partial }}$. The tensor-network area-law bound is thereforeBecause $n_{\partial }$ is proportional to the lattice boundary area, every fixed-bond-dimension PEPS satisfies an area law.

Cutting an open-boundary matrix product state crosses one virtual bond of dimension $\chi$. Its Schmidt rank is at most $\chi$, soA periodic interval crosses two bonds and obeys $S(A)\le 2\log \chi$. The one-dimensional boundary has a constant number of points, so this is an area law.

For a projected entangled pair state, cut every virtual bond crossing the boundary of a region $A$. If $n_{\partial }$ bonds are cut, the Schmidt rank is at most ${\chi }^{n_{\partial }}$, and thereforeSince $n_{\partial }$ is proportional to the lattice boundary area, every fixed-bond-dimension PEPS satisfies an area law.

A projected entangled pair state generalizes an MPS to higher-dimensional lattices. Cutting a region intersects one virtual bond per boundary edge.

A tensor network state represents many-body amplitudes by contracting auxiliary indices of smaller tensors. For two physical indices, a simple example is ${\psi }_{ij}={\sum }_{a=1}^{\chi }{A}_{ia}{B}_{aj}$: the auxiliary index joins the two local tensors and its dimension limits the matrix rank.

Larger graphs join many such tensors, making entanglement constraints visible through the bonds crossing a partition. A matrix product state uses a chain, while a projected entangled pair state permits a higher-dimensional graph. These representations can efficiently describe structured low-entanglement states even though a generic state needs exponentially many amplitudes. Small local tensors do not automatically make every network contraction or every encoded physical calculation efficient; graph geometry and bond dimensions matter.