### 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 |
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Pages (from-to) | 39-51 |

Number of pages | 13 |

Journal | Reports on Mathematical Physics |

Volume | 67 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 1 2011 |

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### Keywords

- Entanglement
- Lie algebras
- Quantum computation

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Reports on Mathematical Physics*,

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