{
  "id": "1812.01449",
  "version": "v1",
  "published": "2018-12-04T14:38:08.000Z",
  "updated": "2018-12-04T14:38:08.000Z",
  "title": "On decomposable correlation matrices",
  "authors": [
    "Benjamin Lovitz"
  ],
  "comment": "13 pages",
  "categories": [
    "quant-ph"
  ],
  "abstract": "Correlation matrices (positive semidefinite matrices with ones on the diagonal) are of fundamental interest in quantum information theory. In this work we introduce and study the set of $r$-decomposable correlation matrices: those that can be written as the Schur product of correlation matrices of rank at most $r$. We find that for all $r \\geq 2$, every $(r+1) \\times (r+1)$ correlation matrix is $r$-decomposable, and we construct ${(2r+1) \\times (2r+1)}$ correlation matrices that are not $r$-decomposable. One question this leaves open is whether every $4 \\times 4$ correlation matrix is $2$-decomposable, which we make partial progress toward resolving. We apply our results to an entanglement detection scenario.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-04T14:38:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "correlation matrix",
      "decomposable correlation matrices",
      "quantum information theory",
      "entanglement detection scenario",
      "positive semidefinite matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}