Balanced Tripartite Entanglement, the Alternating Group A4 and the Lie Algebra sl(3,C) ⊕ u(1)

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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 cristallographic group W(E₈) in terms of three-qubit gates (with real entries) encoding states of type GHZ or W [M. Planat, Clifford group dipoles and the enactment of Weyl/Coxeter group W(E₈) by entangling gates, Preprint 0904.3691 (quant-ph)]. 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.