# Entanglement and quantum phase transitions in matrix-product spin-1 chains

## Abstract

We consider a one-parameter family of matrix-product states of spin-1 particles on a periodic chain and study in detail the entanglement properties of such a state. In particular, we calculate exactly the entanglement of one site with the rest of the chain, and the entanglement of two distant sites with each other, and show that the derivative of both these properties diverge when the parameter g of the states passes through a critical point. Such a point can be called a point of quantum phase transition, since at this point the character of the matrix-product state, which is the ground state of a Hamiltonian, changes discontinuously. We also study the finite size effects and show how the entanglement depends on the size of the chain. This later part is relevant to the field of quantum computation where the problem of initial state preparation in finite arrays of qubits or qutrits is important. It is also shown that the entanglement of two sites have scaling behavior near the critical point.

- Authors:

- Department of Physics, Iran University of Science and Technology, Narmak, P.O. Box 16765-163, Tehran (Iran, Islamic Republic of)
- Department of Physics, Sharif University of Technology, P.O. Box 11365-9161, Tehran (Iran, Islamic Republic of)

- Publication Date:

- OSTI Identifier:
- 20982491

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. A; Journal Volume: 75; Journal Issue: 5; Other Information: DOI: 10.1103/PhysRevA.75.052322; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; GROUND STATES; HAMILTONIANS; MATRICES; PERIODICITY; PHASE TRANSFORMATIONS; QUANTUM COMPUTERS; QUANTUM ENTANGLEMENT; QUANTUM INFORMATION; QUANTUM MECHANICS; QUBITS; SPIN

### Citation Formats

```
Alipour, S., Karimipour, V., and Memarzadeh, L..
```*Entanglement and quantum phase transitions in matrix-product spin-1 chains*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVA.75.052322.

```
Alipour, S., Karimipour, V., & Memarzadeh, L..
```*Entanglement and quantum phase transitions in matrix-product spin-1 chains*. United States. doi:10.1103/PHYSREVA.75.052322.

```
Alipour, S., Karimipour, V., and Memarzadeh, L.. Tue .
"Entanglement and quantum phase transitions in matrix-product spin-1 chains". United States.
doi:10.1103/PHYSREVA.75.052322.
```

```
@article{osti_20982491,
```

title = {Entanglement and quantum phase transitions in matrix-product spin-1 chains},

author = {Alipour, S. and Karimipour, V. and Memarzadeh, L.},

abstractNote = {We consider a one-parameter family of matrix-product states of spin-1 particles on a periodic chain and study in detail the entanglement properties of such a state. In particular, we calculate exactly the entanglement of one site with the rest of the chain, and the entanglement of two distant sites with each other, and show that the derivative of both these properties diverge when the parameter g of the states passes through a critical point. Such a point can be called a point of quantum phase transition, since at this point the character of the matrix-product state, which is the ground state of a Hamiltonian, changes discontinuously. We also study the finite size effects and show how the entanglement depends on the size of the chain. This later part is relevant to the field of quantum computation where the problem of initial state preparation in finite arrays of qubits or qutrits is important. It is also shown that the entanglement of two sites have scaling behavior near the critical point.},

doi = {10.1103/PHYSREVA.75.052322},

journal = {Physical Review. A},

number = 5,

volume = 75,

place = {United States},

year = {Tue May 15 00:00:00 EDT 2007},

month = {Tue May 15 00:00:00 EDT 2007}

}