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

Michel Planat, Péter Lévay, Metod Saniga

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1 Citation (Scopus)


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.

Original languageEnglish
Pages (from-to)39-51
Number of pages13
JournalReports on Mathematical Physics
Issue number1
Publication statusPublished - Feb 1 2011



  • Entanglement
  • Lie algebras
  • Quantum computation

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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