Tensor network representations from the geometry of entangled states

Abstract

Tensor network states provide successful descriptions of strongly correlated quantum systems with applications ranging from condensed matter physics to cosmology. Any family of tensor network states possesses an underlying entanglement structure given by a graph of maximally entangled states along the edges that identify the indices of the tensors to be contracted. Recently, more general tensor networks have been considered, where the maximally entangled states on edges are replaced by multipartite entangled states on plaquettes. Both the structure of the underlying graph and the dimensionality of the entangled states influence the computational cost of contracting these networks. Using the geometrical properties of entangled states, we provide a method to construct tensor network representations with smaller effective bond dimension. We illustrate our method with the resonating valence bond state on the kagome lattice.

arXiv comment35 pages, 9 figures

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• The minimal canonical form of a tensor network
• Dimension of Tensor Network varieties
• Halos and undecidability of tensor stable positive maps
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• Border Ranks of Positive and Invariant Tensor Decompositions: Applications to Correlations