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MiS Preprint

Dimension of Tensor Network Varieties

Alessandra Bernardi, Claudia De Lazzari and Fulvio Gesmundo


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.

MSC Codes:
15A69, 81P45
tensor network, dimension, gauge, isotropy group

Related publications

2023 Repository Open Access
Alessandra Bernardi, Claudia De Lazzari and Fulvio Gesmundo

Dimension of tensor network varieties

In: Communications in contemporary mathematics, 25 (2023) 10, p. 2250059