Normal projected entangled pair states generating the same state

Tensor networks (TNs) are generated by a set of small rank tensors and define many-body quantum states in a succinct form. The corresponding map is not one-to-one: different sets of tensors may generate the very same state. A fundamental question in the study of TNs naturally arises: what is then the relation between those sets? The answer to this question in one-dimensional setups has found several applications, like the characterization of local and global symmetries, the classification of phases of matter and unitary evolutions, or the determination of the fixed points of renormalization procedures. Here we answer this question for projected entangled pair states in any dimension and lattice geometry (including, for example, the Kagome lattice, hyperbolic lattices, or tree tensor networks), as long as the tensors generating the states are normal, which constitute an important and generic class.