{
  "id": "1401.3181",
  "version": "v2",
  "published": "2014-01-14T13:29:36.000Z",
  "updated": "2015-06-01T01:38:28.000Z",
  "title": "Product vectors in the ranges of multi-partite states with positive partial transposes and permanents of matrices",
  "authors": [
    "Young-Hoon Kiem",
    "Seung-Hyeok Kye",
    "Joohan Na"
  ],
  "comment": "Accepted to Commun. Math. Phys., updated introduction",
  "journal": "Commun. Math. Phys. 338 (2015) 621-639",
  "doi": "10.1007/s00220-015-2385-x",
  "categories": [
    "quant-ph",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In this paper, we consider a system of homogeneous algebraic equations in complex variables and their conjugates, which arise naturally from the range criterion for separability of PPT states. We examine systematically these equations to get sufficient conditions for the existence of nontrivial solutions. This gives us possible upper bounds of ranks of PPT entangled edge states and their partial transposes. We will focus on the multi-partite cases which are much more delicate than the bi-partite cases. We use the notion of permanents of matrices as well as techniques from algebraic geometry through the discussion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-01-14T13:29:36.000Z",
      "abstract": "In order to determine if a given PPT state is separable or not, we have to search for product vectors in the ranges of the states and their partial transposes by the range criterion. This is equivalent to solving systems of algebraic equations in terms of complex variables and their conjugates. The purpose of this note is to examine systematically these equations to get sufficient conditions for the existence of required product vectors. We will focus on the multi-partite cases which are much more delicate than the bi-partite cases. We use the notion of permanents of matrices as well as techniques from algebraic geometry through the discussion.",
      "comment": "20 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-06-01T01:38:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81P15",
      "14F45",
      "55N45",
      "15A30",
      "46L05"
    ],
    "keywords": [
      "product vectors",
      "positive partial transposes",
      "multi-partite states",
      "permanents",
      "bi-partite cases"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer",
      "journal": "Communications in Mathematical Physics",
      "year": 2015,
      "month": "Sep",
      "volume": 338,
      "number": 2,
      "pages": 621
    },
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015CMaPh.338..621K"
    }
  }
}