### Abstract

We discuss three important classes of three-qubit entangled states and their encoding into quantum gates, finite groups and Lie algebras. States of the GHZ and W-type correspond to pure tripartite and bipartite entanglement, respectively. We introduce another generic class B of three-qubit states, that have balanced entanglement over two and three parties. We show how to realize the largest cristallographie group W(E_{8}) in terms of three-qubit gates (with real entries) encoding states of type GHZ or W. Then, we describe a peculiar "condensation" of W(E_{8}) into the four-letter alternating group A_{4}, obtained from a chain of maximal subgroups. Group A_{4} is realized from two B-type generators and found to correspond to the Lie algebra sl(3, ℂ) ⊕ u(1). Possible applications of our findings to particle physics and the structure of genetic code are also mentioned.

Original language | English |
---|---|

Pages (from-to) | 39-51 |

Number of pages | 13 |

Journal | Reports on Mathematical Physics |

Volume | 67 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 2011 |

### Fingerprint

### Keywords

- Entanglement
- Lie algebras
- Quantum computation

### ASJC Scopus subject areas

- Mathematical Physics
- Statistical and Nonlinear Physics

### Cite this

*Reports on Mathematical Physics*,

*67*(1), 39-51. https://doi.org/10.1016/S0034-4877(11)00009-7

**Balanced tripartite entanglement, the alternating group A4 and the Lie algebra sl(3, ℂ)⊕ u(1).** / Planat, Michel; Lévay, P.; Saniga, Metod.

Research output: Contribution to journal › Article

*Reports on Mathematical Physics*, vol. 67, no. 1, pp. 39-51. https://doi.org/10.1016/S0034-4877(11)00009-7

}

TY - JOUR

T1 - Balanced tripartite entanglement, the alternating group A4 and the Lie algebra sl(3, ℂ)⊕ u(1)

AU - Planat, Michel

AU - Lévay, P.

AU - Saniga, Metod

PY - 2011/2

Y1 - 2011/2

N2 - We discuss three important classes of three-qubit entangled states and their encoding into quantum gates, finite groups and Lie algebras. States of the GHZ and W-type correspond to pure tripartite and bipartite entanglement, respectively. We introduce another generic class B of three-qubit states, that have balanced entanglement over two and three parties. We show how to realize the largest cristallographie group W(E8) in terms of three-qubit gates (with real entries) encoding states of type GHZ or W. Then, we describe a peculiar "condensation" of W(E8) into the four-letter alternating group A4, obtained from a chain of maximal subgroups. Group A4 is realized from two B-type generators and found to correspond to the Lie algebra sl(3, ℂ) ⊕ u(1). Possible applications of our findings to particle physics and the structure of genetic code are also mentioned.

AB - We discuss three important classes of three-qubit entangled states and their encoding into quantum gates, finite groups and Lie algebras. States of the GHZ and W-type correspond to pure tripartite and bipartite entanglement, respectively. We introduce another generic class B of three-qubit states, that have balanced entanglement over two and three parties. We show how to realize the largest cristallographie group W(E8) in terms of three-qubit gates (with real entries) encoding states of type GHZ or W. Then, we describe a peculiar "condensation" of W(E8) into the four-letter alternating group A4, obtained from a chain of maximal subgroups. Group A4 is realized from two B-type generators and found to correspond to the Lie algebra sl(3, ℂ) ⊕ u(1). Possible applications of our findings to particle physics and the structure of genetic code are also mentioned.

KW - Entanglement

KW - Lie algebras

KW - Quantum computation

UR - http://www.scopus.com/inward/record.url?scp=80052675643&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80052675643&partnerID=8YFLogxK

U2 - 10.1016/S0034-4877(11)00009-7

DO - 10.1016/S0034-4877(11)00009-7

M3 - Article

VL - 67

SP - 39

EP - 51

JO - Reports on Mathematical Physics

JF - Reports on Mathematical Physics

SN - 0034-4877

IS - 1

ER -