On the closedness and geometry of tensor network state sets

Tensor network states (TNS) are a powerful approach for the study of strongly correlated quantum matter. The curse dimensionality is addressed by parametrizing many-body state in terms partially contracted tensors. These tensors form substantially reduced set effective degrees freedom. In practical algorithms, functionals like energy expectation values or overlaps optimized over certain sets TNS. Concerning algorithmic stability, it important whether considered closed because, otherwise, algorithms may boundary point that outside TNS and tensor elements diverge. We discuss closedness geometries sets, we propose regularizations optimization problems on non-closed sets. show matrix product (MPS) with open conditions, tree states, multiscale entanglement renormalization ansatz always closed, whereas translation-invariant MPS periodic conditions (PBC), heterogeneous PBC, projected entangled pair generally not closed. latter done using explicit examples W state, call two-domain fine-grained versions thereof.