On the geometry of tensor network states

Understand

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.