{
  "id": "1401.2627",
  "version": "v2",
  "published": "2014-01-12T13:39:56.000Z",
  "updated": "2018-08-31T23:34:26.000Z",
  "title": "Characterization of multipartite entanglement in terms of local transformations",
  "authors": [
    "Youming Qiao",
    "Xiaoming Sun",
    "Nengkun Yu"
  ],
  "comment": "13 Pages, comments welcome",
  "categories": [
    "quant-ph"
  ],
  "abstract": "The degree of the generators of invariant polynomial rings of is a long standing open problem since the very initial study of the invariant theory in the 19th century. Motivated by its significant role in characterizing multipartite entanglement, we study the invariant polynomial rings of local unitary group---the tensor product of unitary group, and local general linear group---the tensor product of general linear group. For these two groups, we prove polynomial upper bounds on the degree of the generators of invariant polynomial rings. On the other hand, systematic methods are provided to to construct all homogenous polynomials that are invariant under these two groups for any fixed degree. Thus, our results can be regarded as a complete characterization of the invariant polynomial rings. As an interesting application, we show that multipartite entanglement is additive in the sense that two multipartite states are local unitary equivalent if and only if $r$-copies of them are LU equivalent for some $r$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-01-12T13:39:56.000Z",
      "title": "Characterization of multipartite entanglement",
      "abstract": "In this paper, we provide a characterization of multipartite entanglement in terms of equivalence classes of states under local unitary transformation (LU) by demonstrating a simple method to construct all homogenous polynomials that are invariant under local unitary group(LUIPs) for any fixed degree. We give an upper bound on the degree of the LUIP such that the ring of LUIPs can be generated by LUIPs with degree lower than the bound. Our characterization can directly generate an algorithm whose output is a generating set of LUIPs. By employing the concept of LUIPs, we prove that multipartite entanglement is additive in the sense that two multipartite states are LU equivalent if and only if $n$-copies of these two states are LU equivalent for some $n$. The method for studying LU equivalence is also used to classify the different types of multipartite mixed entanglement according to equivalence classes of states under stochastic local operations and classical communication (SLOCC), where the pure states case was previously studied by Gour and Wallach using another approach.",
      "comment": "6 Pages, comments welcome",
      "journal": null,
      "doi": null,
      "authors": [
        "Nengkun Yu",
        "Youming Qiao",
        "Xiaoming Sun"
      ]
    },
    {
      "version": "v2",
      "updated": "2018-08-31T23:34:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "multipartite entanglement",
      "characterization",
      "equivalence classes",
      "lu equivalent",
      "local unitary group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.2627Y"
    }
  }
}