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 G H Z 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 G H Z 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 s l ( 3 ,   C ) ⊕ u ( 1 ) . Possible applications of our findings to particle physics and the structure of genetic code are also mentioned.