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# Entanglement spread area law in gapped ground states

## Abstract

Ground-state entanglement governs various properties of quantum many-body systems at low temperatures and is the key to understanding gapped quantum phases of matter. Here we identify a structural property of entanglement in the ground state of gapped local Hamiltonians. This property is captured using a quantum information quantity known as the entanglement spread, which measures the difference between Rényi entanglement entropies. Our main result shows that gapped ground states possess limited entanglement spread across any partition of the system, exhibiting an area-law scaling. Our result applies to systems with interactions described by any graph, but we obtain an improved bound for the special case of lattices. These interaction graphs include cases where entanglement entropy is known not to satisfy an area law. We achieve our results first by connecting the ground-state entanglement to the communication complexity of testing bipartite entangled states and then devising a communication scheme for testing ground states using recently developed quantum algorithms for Hamiltonian simulation.

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## Acknowledgements

A.W.H. thanks D. Aharonov for insightful discussions regarding this project and raising the question of connection between the communication complexity of measuring the ground state and ground-state entanglement. A.A. thanks D. Gosset for discussions on the applications of quantum algorithms to area laws. M.S. thanks Z. Landau, A. Natarajan and U. Vazirani for helpful discussions. We thank I. Arad for suggesting the connection between entanglement spread and tensor network contraction. A.A. acknowledges support through the NSF award QCIS-FF: Quantum Computing & Information Science Faculty Fellow at Harvard University (NSF 2013303). Part of the work was done when A.A. was affiliated to the Institute for Quantum Computing and the Department of Combinatorics and Optimization, University of Waterloo, and the Perimeter Institute for Theoretical Physics, where research was supported by the Canadian Institute for Advanced Research, through funding provided to the Institute for Quantum Computing by the Government of Canada and the Province of Ontario. Perimeter Institute is also supported in part by the Government of Canada and the Province of Ontario. A.W.H. was funded by NSF grants CCF-1452616, CCF-1729369 and PHY-1818914; the NSF QLCI program through grant number OMA-2016245; and a Samsung Advanced Institute of Technology Global Research Partnership. M.S. was funded by NSF grant CCF-1729369.

## Author information

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All the authors contributed equally to all aspects of this work. The authors are arranged alphabetically by last name.

### Corresponding author

Correspondence to Aram W. Harrow.

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## Supplementary information

### Supplementary Information

Supplementary Sections 1–3.

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Anshu, A., Harrow, A.W. & Soleimanifar, M. Entanglement spread area law in gapped ground states. Nat. Phys. (2022). https://doi.org/10.1038/s41567-022-01740-7

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