# Quantum walk on the line: Entanglement and nonlocal initial conditions

## Abstract

The conditional shift in the evolution operator of a quantum walk generates entanglement between the coin and position degrees of freedom. This entanglement can be quantified by the von Neumman entropy of the reduced density operator (entropy of entanglement). We show analytically that for a Hadamard walk with local initial conditions the asymptotic entanglement is 0.872 for all initial coin states. When nonlocal initial conditions are considered, the asymptotic entanglement varies smoothly between almost complete entanglement and no entanglement (product state). An exact expression for the asymptotic (long-time) entanglement is obtained for initial conditions in the position subspace spanned by [{+-}1>.

- Authors:

- Instituto de Fisica, Universidad de la Republica, C.C. 30, C.P. 11000, Montevideo (Uruguay)

- Publication Date:

- OSTI Identifier:
- 20787075

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. A; Journal Volume: 73; Journal Issue: 4; Other Information: DOI: 10.1103/PhysRevA.73.042302; (c) 2006 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DEGREES OF FREEDOM; DENSITY; ENERGY LEVELS; ENTROPY; EVOLUTION; INFORMATION THEORY; QUANTUM ENTANGLEMENT; QUANTUM OPERATORS

### Citation Formats

```
Abal, G., Siri, R., Romanelli, A., and Donangelo, R..
```*Quantum walk on the line: Entanglement and nonlocal initial conditions*. United States: N. p., 2006.
Web. doi:10.1103/PHYSREVA.73.0.

```
Abal, G., Siri, R., Romanelli, A., & Donangelo, R..
```*Quantum walk on the line: Entanglement and nonlocal initial conditions*. United States. doi:10.1103/PHYSREVA.73.0.

```
Abal, G., Siri, R., Romanelli, A., and Donangelo, R.. Sat .
"Quantum walk on the line: Entanglement and nonlocal initial conditions". United States.
doi:10.1103/PHYSREVA.73.0.
```

```
@article{osti_20787075,
```

title = {Quantum walk on the line: Entanglement and nonlocal initial conditions},

author = {Abal, G. and Siri, R. and Romanelli, A. and Donangelo, R.},

abstractNote = {The conditional shift in the evolution operator of a quantum walk generates entanglement between the coin and position degrees of freedom. This entanglement can be quantified by the von Neumman entropy of the reduced density operator (entropy of entanglement). We show analytically that for a Hadamard walk with local initial conditions the asymptotic entanglement is 0.872 for all initial coin states. When nonlocal initial conditions are considered, the asymptotic entanglement varies smoothly between almost complete entanglement and no entanglement (product state). An exact expression for the asymptotic (long-time) entanglement is obtained for initial conditions in the position subspace spanned by [{+-}1>.},

doi = {10.1103/PHYSREVA.73.0},

journal = {Physical Review. A},

number = 4,

volume = 73,

place = {United States},

year = {Sat Apr 15 00:00:00 EDT 2006},

month = {Sat Apr 15 00:00:00 EDT 2006}

}

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