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An entanglement area law bounds a region's entanglement entropy by its boundary size rather than its volume, in a specified family of states and with a stated short-distance convention. For a lattice region $A$, a typical form is $S(A)\le c|\partial A|$ with a constant independent of the region's linear size.

The physical intuition is that correlations contributing most strongly across the cut can be concentrated near its boundary. A generic highly entangled state need not satisfy this: independent Bell pairs joining many interior sites to the exterior can produce volume-scaling entropy. Tensor-network area-law bounds make the boundary counting explicit through the number of cut bonds. Area laws are therefore useful structural constraints, not automatic consequences of calling a state quantum or a Hamiltonian local.

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.