= Tensor network state
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.
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