# Asymptotic entanglement transformation between W and GHZ states

## Abstract

We investigate entanglement transformations with stochastic local operations and classical communication in an asymptotic setting using the concepts of degeneration and border rank of tensors from algebraic complexity theory. Results well-known in that field imply that GHZ states can be transformed into W states at rate 1 for any number of parties. As a generalization, we find that the asymptotic conversion rate from GHZ states to Dicke states is bounded as the number of subsystems increases and the number of excitations is fixed. By generalizing constructions of Coppersmith and Winograd and by using monotones introduced by Strassen, we also compute the conversion rate from W to GHZ states.

- Authors:

- Institute for Theoretical Physics, ETH Zürich, Wolfgang-Pauli-Strasse 27, CH-8093 Zürich (Switzerland)
- (Hungary)
- (Denmark)

- Publication Date:

- OSTI Identifier:
- 22403103

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 2; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMPTOTIC SOLUTIONS; CONVERSION; EXCITATION; QUANTUM ENTANGLEMENT; STOCHASTIC PROCESSES; TENSORS; TRANSFORMATIONS

### Citation Formats

```
Vrana, Péter, Department of Geometry, Budapest University of Technology and Economics, Egry József u. 1., 1111 Budapest, Christandl, Matthias, and Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen.
```*Asymptotic entanglement transformation between W and GHZ states*. United States: N. p., 2015.
Web. doi:10.1063/1.4908106.

```
Vrana, Péter, Department of Geometry, Budapest University of Technology and Economics, Egry József u. 1., 1111 Budapest, Christandl, Matthias, & Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen.
```*Asymptotic entanglement transformation between W and GHZ states*. United States. doi:10.1063/1.4908106.

```
Vrana, Péter, Department of Geometry, Budapest University of Technology and Economics, Egry József u. 1., 1111 Budapest, Christandl, Matthias, and Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen. Sun .
"Asymptotic entanglement transformation between W and GHZ states". United States.
doi:10.1063/1.4908106.
```

```
@article{osti_22403103,
```

title = {Asymptotic entanglement transformation between W and GHZ states},

author = {Vrana, Péter and Department of Geometry, Budapest University of Technology and Economics, Egry József u. 1., 1111 Budapest and Christandl, Matthias and Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen},

abstractNote = {We investigate entanglement transformations with stochastic local operations and classical communication in an asymptotic setting using the concepts of degeneration and border rank of tensors from algebraic complexity theory. Results well-known in that field imply that GHZ states can be transformed into W states at rate 1 for any number of parties. As a generalization, we find that the asymptotic conversion rate from GHZ states to Dicke states is bounded as the number of subsystems increases and the number of excitations is fixed. By generalizing constructions of Coppersmith and Winograd and by using monotones introduced by Strassen, we also compute the conversion rate from W to GHZ states.},

doi = {10.1063/1.4908106},

journal = {Journal of Mathematical Physics},

number = 2,

volume = 56,

place = {United States},

year = {Sun Feb 15 00:00:00 EST 2015},

month = {Sun Feb 15 00:00:00 EST 2015}

}

DOI: 10.1063/1.4908106

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