2009/12/01 by Michel Planat, Péter Lévay, Peter Levay +1
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · Physics and Astronomy · #Algebra over a field #Bipartite graph #Class (philosophy) #Fractal and DNA sequence analysis #Group (periodic table) #Lie algebra #Lie group #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum entanglement #Simple Lie group #Spin representation #math-ph #math.GR #math.MP #math.RT #quant-ph
paper · pdf · doi:10.1016/s0034-4877(11)00009-7
published as Reports on Mathematical Physics 67, 1 (2010) 39-51 · 14 pages
arxiv created 2009/12/01 · openalex publication_date 2011/02/01 · arxiv updated 2015/05/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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, \it 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.