Dimension of Tensor Network varieties
Abstract
The tensor network variety is a variety of tensors associated to a graph and a set of positive integer weights on its edges, called bond dimensions. We determine an upper bound on the dimension of the tensor network variety. A refined upper bound is given in cases relevant for applications such as varieties of matrix product states and projected entangled pairs states. We provide a range (the "supercritical range") of the parameters where the upper bound is sharp.
 Publication:

arXiv eprints
 Pub Date:
 January 2021
 DOI:
 10.48550/arXiv.2101.03148
 arXiv:
 arXiv:2101.03148
 Bibcode:
 2021arXiv210103148B
 Keywords:

 Quantum Physics;
 Mathematics  Algebraic Geometry;
 15A69;
 81P45
 EPrint:
 27 pages, 3 figures. Final version, to appear in Communications in Contemporary Mathematics