On the geometry of tensor network states
Abstract
We answer a question of L. Grasedyck that arose in quantum information theory, showing that the limit of tensors in a space of tensor network states need not be a tensor network state. We also give geometric descriptions of spaces of tensor networks states corresponding to trees and loops. Grasedyck's question has a surprising connection to the area of Geometric Complexity Theory, in that the result is equivalent to the statement that the boundary of the Mulmuley-Sohoni type variety associated to matrix multiplication is strictly larger than the projections and re-labelings of matrix multiplication. Tensor Network States are also related to graphical models in algebraic statistics.
Cite
@article{arxiv.1105.4449,
title = {On the geometry of tensor network states},
author = {J. M. Landsberg and Yang Qi and Ke Ye},
journal= {arXiv preprint arXiv:1105.4449},
year = {2012}
}
Comments
8 pages, to appear in QIC. Exposition improved to make paper more accessible to physicists