{
  "id": "1203.1364",
  "version": "v4",
  "published": "2012-03-07T01:47:24.000Z",
  "updated": "2013-01-30T02:04:47.000Z",
  "title": "Properties and construction of extreme bipartite states having positive partial transpose",
  "authors": [
    "Lin Chen",
    "Dragomir Z. Djokovic"
  ],
  "comment": "Updated version. 32 pages",
  "journal": "Communications in Mathematical Physics, 2013, vol. 323, issue 1, pp. 241-284",
  "doi": "10.1007/s00220-013-1770-6",
  "categories": [
    "math-ph",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "We consider a bipartite quantum system H_A x H_B with M=dim H_A and N=dim H_B. We study the set E of extreme points of the compact convex set of all states having positive partial transpose (PPT) and its subsets E_r={rho in E: rank rho=r}. Our main results pertain to the subsets E_r^{M,N} of E_r consisting of states whose reduced density operators have ranks M and N, respectively. The set E_1 is just the set of pure product states. It is known that E_r^{M,N} is empty for 1< r <= min(M,N) and for r=MN. We prove that also E_{MN-1}^{M,N} is empty. Leinaas, Myrheim and Sollid have conjectured that E_{M+N-2}^{M,N} is not empty for all M,N>2 and that E_r^{M,N} is empty for 1<r<M+N-2. We prove the first part of their conjecture. The second part is known to hold when min(M,N)=3 and we prove that it holds also when min(M,N)=4. This is a consequence of our result that E_{N+1}^{M,N} is empty if M,N>3. We introduce the notion of \"good\" states, show that all pure states are good and give a simple description of the good separable states. For a good state rho in E_{M+N-2}^{M,N}, we prove that the range of rho contains no product vectors and that the partial transpose of rho has rank M+N-2 as well. In the special case M=3, we construct good 3 x N extreme states of rank N+1 for all N>3.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-01-30T02:04:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "positive partial transpose",
      "extreme bipartite states",
      "properties",
      "construction",
      "compact convex set"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer",
      "journal": "Communications in Mathematical Physics",
      "year": 2013,
      "month": "Oct",
      "volume": 323,
      "number": 1,
      "pages": 241
    },
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013CMaPh.323..241C"
    }
  }
}