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 GHZGHZ and W  -type correspond to pure tripartite and bipartite entanglement, respectively. We introduce another generic class BB of three-qubit states, that have balanced entanglement over two and three parties. We show how to realize the largest cristallographic group W(E8)W(E8)in terms of three-qubit gates (with real entries) encoding states of type GHZGHZ or W.W. Then, we describe a peculiar “condensation” of W(E8)W(E8) into the four-letter alternating group A4,A4, obtained from a chain of maximal subgroups. Group A4A4 is realized from two B-type generators and found to correspond to the Lie algebra sl(3, C)⊕u(1).sl(3, C)⊕u(1). Possible applications of our findings to particle physics and the structure of genetic code are also mentioned.