Proposition: PEPS cut bound

For a nonzero pure PEPS with edge dimensions $D_e$, its normalized state satisfies

In particular, a maximum bond dimension $D$ gives $S(A)\le |{\partial }_{\Lambda }A|\log D$.

Proof. Cut all virtual edges crossing the partition. Expanding their indices writes the physical vector as a sum of at most ${\prod }_{e\in {\partial }_{\Lambda }A}{D}_e$ product vectors across the cut. Its Schmidt rank is no larger, and the entropy is at most the logarithm of that rank.