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.
- Lie algebras
- Quantum computation
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics