Dimension of Tensor Network Varieties
Alessandra Bernardi, Claudia De Lazzari, and Fulvio Gesmundo
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Submission date: 11. Jan. 2021
MSC-Numbers: 15A69, 81P45
Keywords and phrases: tensor network, dimension, gauge, isotropy group
Link to arXiv:See the arXiv entry of this preprint.
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.