BALANCED TRIPARTITE ENTANGLEMENT, THE ALTERNATING GROUP A4 AND THE LIE ALGEBRA sl(3, C) ⊕ u(1)

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(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) circle plus u(1). Possible applications of our findings to particle physics and the structure of genetic code are also mentioned.