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.Comment: 8 pages, to appear in QIC. Exposition improved to make paper more
accessible to physicist