ON THE GEOMETRY OF TENSOR NETWORK STATES Academic Article

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 of matrix multiplication (and re-expressions of matrix multiplication and its projections after changes of bases). Tensor Network States are also related to graphical models in algebraic statistics.

authors

• Landsberg, Joseph

published proceedings

• QUANTUM INFORMATION & COMPUTATION

author list (cited authors)

• Landsberg, J. M., Qi, Y., & Ye, K. e.

complete list of authors

• Landsberg, Joseph M||Qi, Yang||Ye, Ke

publication date

• March 2012

published in

• Quantum Information and Computation  Journal

keywords

• Finitely Correlated States
• Geometric Complexity Theory
• Matrix Multiplication
• Matrix Product States
• Tensor
• Valence Bond Solids

start page

• 346

end page

• 354

volume

• 12

issue

• 3-4