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On
the geometry of tensor network states
(pp0346-0354)
Joseph M. Landsburg, Yang Qi, and Ke Ye
doi:
https://doi.org/10.26421/QIC12.3-4-12
Abstracts:
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. Grasedycks 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.
Key words:
tensor, finitely correlated states, valence bond solids,
matrix product states, geometric complexity theory, matrix
multiplication |
