On the geometry of tensor network states

Details On the geometry of tensor network states On the geometry of tensor network states

2011-05-23

arxiv.org/abs/1105.4449v2

Mathematics

Algebraic Geometry

8 pages, to appear in QIC. Exposition improved to make paper more accessible to physicists

Scientific paper

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.

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