# Entanglement and its relation to energy variance for local one-dimensional Hamiltonians

## Abstract

We explore the relation between the entanglement of a pure state and its energy variance for a local one-dimensional Hamiltonian, as the system size increases. In particular, we introduce a construction which creates a matrix product state of arbitrarily small energy variance $δ^2$ for $$\textit{N}$$ spins, with bond dimension scaling as $$\sqrt{N}D^{1/δ}_0$$, where $$D_0$$ > 1 is a constant. This implies that a polynomially increasing bond dimension is enough to construct states with energy variance that vanishes with the inverse of the logarithm of the system size. We run numerical simulations to probe the construction on two different models and compare the local reduced density matrices of the resulting states to the corresponding thermal equilibrium. Our results suggest that the spatially homogeneous states with logarithmically decreasing variance, which can be constructed efficiently, do converge to the thermal equilibrium in the thermodynamic limit, while the same is not true if the variance remains constant.

- Authors:

- Publication Date:

- Research Org.:
- Princeton Univ., NJ (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC); German Research Foundation (DFG); European Research Council (ERC); National Science Foundation (NSF)

- OSTI Identifier:
- 1616366

- Alternate Identifier(s):
- OSTI ID: 1802887

- Grant/Contract Number:
- SC0016244; 742102; NSF PHY-1748958

- Resource Type:
- Published Article

- Journal Name:
- Physical Review B

- Additional Journal Information:
- Journal Name: Physical Review B Journal Volume: 101 Journal Issue: 14; Journal ID: ISSN 2469-9950

- Publisher:
- American Physical Society

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; Materials Science; Physics

### Citation Formats

```
Bañuls, Mari Carmen, Huse, David A., and Cirac, J. Ignacio. Entanglement and its relation to energy variance for local one-dimensional Hamiltonians. United States: N. p., 2020.
Web. doi:10.1103/PhysRevB.101.144305.
```

```
Bañuls, Mari Carmen, Huse, David A., & Cirac, J. Ignacio. Entanglement and its relation to energy variance for local one-dimensional Hamiltonians. United States. https://doi.org/10.1103/PhysRevB.101.144305
```

```
Bañuls, Mari Carmen, Huse, David A., and Cirac, J. Ignacio. Tue .
"Entanglement and its relation to energy variance for local one-dimensional Hamiltonians". United States. https://doi.org/10.1103/PhysRevB.101.144305.
```

```
@article{osti_1616366,
```

title = {Entanglement and its relation to energy variance for local one-dimensional Hamiltonians},

author = {Bañuls, Mari Carmen and Huse, David A. and Cirac, J. Ignacio},

abstractNote = {We explore the relation between the entanglement of a pure state and its energy variance for a local one-dimensional Hamiltonian, as the system size increases. In particular, we introduce a construction which creates a matrix product state of arbitrarily small energy variance $δ^2$ for $\textit{N}$ spins, with bond dimension scaling as $\sqrt{N}D^{1/δ}_0$, where $D_0$ > 1 is a constant. This implies that a polynomially increasing bond dimension is enough to construct states with energy variance that vanishes with the inverse of the logarithm of the system size. We run numerical simulations to probe the construction on two different models and compare the local reduced density matrices of the resulting states to the corresponding thermal equilibrium. Our results suggest that the spatially homogeneous states with logarithmically decreasing variance, which can be constructed efficiently, do converge to the thermal equilibrium in the thermodynamic limit, while the same is not true if the variance remains constant.},

doi = {10.1103/PhysRevB.101.144305},

journal = {Physical Review B},

number = 14,

volume = 101,

place = {United States},

year = {2020},

month = {4}

}

https://doi.org/10.1103/PhysRevB.101.144305

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