An area law for one-dimensional quantum systems
Open accessAbstract
We prove an area law for the entanglement entropy in gapped one-dimensional quantum systems. The bound on the entropy grows surprisingly rapidly with the correlation length; we discuss this in terms of properties of quantum expanders and present a conjecture on matrix product states which may provide an alternate way of arriving at an area law. We also show that, for gapped, local systems, the bound on Von Neumann entropy implies a bound on Rényi entropy for sufficiently large α<1 and implies the ability to approximate the ground state by a matrix product state.
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In open-boundary one-dimensional systems, ground states of gapped Hamiltonians of N qubits can be approximated to arbitrarily high precision with an MPS of bond dimension χ that does not grow with the size of the system.
“Such MPS states naturally encode the area law of entanglement 75 : ground states of gapped Hamiltonians of N qubits can be approximated to arbitrarily high precision with an MPS of bond dimension χ, which does not grow with the size of the system.”
From Beyond-classical computation in quantum simulation (2025) · cites this paper as [75]
The Matrix Product States section (p. 5) constructs an MPS approximation for a gapped ground state under finite-range and finite-strength assumptions. But the approximation estimate on p. 4 scales with chain length N at a given bond dimension, so the paper does not establish arbitrarily high precision at a bond dimension independent of N. Its theorem also assumes a unique ground state (p. 1). (LLM, conf=0.88)
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Self-citation
- Authors
- Hastings, M. B.
- Affiliations
- Los Alamos National Laboratory (US)
- Corresponding author
- Hastings, M B
- Venue
- Journal of Statistical Mechanics: Theory and Experiment
- Publisher
- IOP Publishing
- Published
- 2007-08-20
- Volume
- 2007
- Issue
- 08
- Pages
- P08024-P08024
- ISSN
- 1742-5468 (ISSN-L)
- OpenAlex
- W2019692968
- Cited by
- 1,131
- References
- 4
- Citations by year
- 2026: 60 · 2025: 131 · 2024: 95 · 2023: 89 · 2022: 87 · 2021: 66 · 2020: 84 · 2019: 85 · 2018: 53 · 2017: 69 · 2016: 53 · 2015: 59 · 2014: 39 · 2013: 50 · 2012: 34
- FWCI
- 11.21
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Journal of Statistical Mechanics Theory and Experiment (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- U.S. Department of Energy
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Condensed Matter Physics › Advanced Condensed Matter Physics
- Keywords
- area law, entanglement entropy, one-dimensional quantum systems, gapped systems, correlation length, matrix product states, von Neumann entropy, Rényi entropy, ground state approximation, local Hamiltonians
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