{
  "id": "2308.15389",
  "version": "v1",
  "published": "2023-08-29T15:32:58.000Z",
  "updated": "2023-08-29T15:32:58.000Z",
  "title": "Progress on the Kretschmann-Schlingemann-Werner Conjecture",
  "authors": [
    "Frederik vom Ende"
  ],
  "comment": "10+2 pages, to be submitted to J. Math. Phys",
  "categories": [
    "quant-ph",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Given any pair of completely positive, trace-preserving maps $\\Phi_1,\\Phi_2$ such that at least one of them has Kraus rank one, as well as any respective Stinespring isometries $V_1,V_2$, we prove that there exists a unitary $U$ on the environment such that $\\|V_1-({\\bf1}\\otimes U)V_2\\|_\\infty\\leq\\sqrt{2\\|\\Phi_1-\\Phi_2\\|_\\diamond}$. Moreover, we provide a simple example which shows that the factor $\\sqrt2$ on the right-hand side is optimal, and we conjecture that this inequality holds for every pair of channels.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-29T15:32:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "kretschmann-schlingemann-werner conjecture",
      "kraus rank",
      "simple example",
      "right-hand side",
      "inequality holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 2,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}