{
  "id": "0912.0172",
  "version": "v1",
  "published": "2009-12-01T14:57:29.000Z",
  "updated": "2009-12-01T14:57:29.000Z",
  "title": "Balanced Tripartite Entanglement, the Alternating Group A4 and the Lie Algebra $sl(3,C) \\oplus u(1)$",
  "authors": [
    "Michel Planat",
    "Peter Levay",
    "Metod Saniga"
  ],
  "comment": "14 pages",
  "journal": "Reports on Mathematical Physics 67, 1 (2010) 39-51",
  "categories": [
    "math-ph",
    "math.GR",
    "math.MP",
    "math.RT",
    "quant-ph"
  ],
  "abstract": "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 [M. Planat, {\\it Clifford group dipoles and the enactment of Weyl/Coxeter group $W(E_8)$ by entangling gates}, Preprint 0904.3691 (quant-ph)]. 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 $sl(3,\\mathbb{C})\\oplus u(1)$. Possible applications of our findings to particle physics and the structure of genetic code are also mentioned.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-12-01T14:57:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lie algebra",
      "balanced tripartite entanglement",
      "alternating group a4",
      "largest cristallographic group",
      "clifford group dipoles"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1016/S0034-4877(11)00009-7",
      "journal": "Reports on Mathematical Physics",
      "year": 2011,
      "month": "Feb",
      "volume": 67,
      "number": 1,
      "pages": 39
    },
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011RpMP...67...39P"
    }
  }
}