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.

27 pages, 3 figures. Final version, to appear in Communications in Contemporary Mathematics

References in corpus (6)

• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Geometry of variational methods: dynamics of closed quantum systems
• Geometric descriptions of entangled states by auxiliaries varieties
• Optimization at the boundary of the tensor network variety
• Maximum likelihood estimation for tensor normal models via castling transforms
• Border rank non-additivity for higher order tensors

Cited by in corpus (5)

• The resource theory of tensor networks
• The Linear Span of Uniform Matrix Product States
• Nonlinear Algebra and Applications
• Partial Degeneration of Tensors
• Collineation varieties of tensors