Dimension of Tensor Network varieties

TL;DR

This paper investigates the dimension of tensor network varieties, providing upper bounds and identifying conditions where these bounds are tight, with implications for quantum physics and tensor network applications.

Contribution

It establishes an upper bound on tensor network variety dimensions and refines this bound for important cases like matrix product states.

Findings

Derived a general upper bound on tensor network variety dimensions.

Identified a supercritical range where the upper bound is exact.

Applied results to matrix product states and projected entangled pairs states.

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.