On the geometry of tensor network states

Authors:
Joseph M. Landsburg;Yang Qi;Ke Ye
Affiliations:
Department of Mathematics, Texas AM University, College Station, TX;Department of Mathematics, Texas AM University, College Station, TX;Department of Mathematics, Texas AM University, College Station, TX
Venue:
Quantum Information & Computation
Year:
2012

Citing 4
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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.