## Tensor network representations from the geometry of entangled states

Matthias Christandl, Angelo Lucia, Péter Vrana, Albert H. Werner

SciPost Phys. 9, 042 (2020) · published 30 September 2020

- doi: 10.21468/SciPostPhys.9.3.042
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### 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.

### Cited by 6

### Authors / Affiliations: mappings to Contributors and Organizations

See all Organizations.-
^{1}Matthias Christandl, -
^{1}^{2}^{3}Angelo Lucia, -
^{1}^{4}^{5}Peter Vrana, -
^{1}^{3}Albert H. Werner

^{1}Københavns Universitet / University of Copenhagen [UCPH]^{2}Niels Bohr Institute [NBI]^{3}Walter Burke Institute for Theoretical Physics^{4}Budapesti Műszaki és Gazdaságtudományi Egyetem / Budapest University of Technology and Economics [BUTE]^{5}Magyar Tudományos Akadémia / Hungarian Academy of Sciences [MTA]

- Alexander von Humboldt-Stiftung / Alexander von Humboldt Foundation
- Det Frie Forskningsråd / Danish Council for Independent Research [DFF]
- European Research Council [ERC]
- National Science Foundation [NSF]
- Nemzeti Kutatási, Fejlesztési és Innovációs Hivatal / National Research, Development and Innovation Office [NKFIH]
- Villum Fonden / Velux Foundation
- Walter Burke Institute for Theoretical Physics