{
  "id": "1208.1076",
  "version": "v1",
  "published": "2012-08-06T03:24:17.000Z",
  "updated": "2012-08-06T03:24:17.000Z",
  "title": "Linear preservers and quantum information science",
  "authors": [
    "Ajda Fosner",
    "Zejun Huang",
    "Chi-Kwong Li",
    "Nung-Sing Sze"
  ],
  "comment": "13 pages",
  "categories": [
    "quant-ph",
    "math.FA"
  ],
  "abstract": "Let $m,n\\ge 2$ be positive integers, $M_m$ the set of $m\\times m$ complex matrices and $M_n$ the set of $n\\times n$ complex matrices. Regard $M_{mn}$ as the tensor space $M_m\\otimes M_n$. Suppose $|\\cdot|$ is the Ky Fan $k$-norm with $1 \\le k \\le mn$, or the Schatten $p$-norm with $1 \\le p \\le \\infty$ ($p\\ne 2$) on $M_{mn}$. It is shown that a linear map $\\phi: M_{mn} \\rightarrow M_{mn}$ satisfying $$|A\\otimes B| = |\\phi(A\\otimes B)|$$ for all $A \\in M_m$ and $B \\in M_n$ if and only if there are unitary $U, V \\in M_{mn}$ such that $\\phi$ has the form $A\\otimes B \\mapsto U(\\varphi_1(A) \\otimes \\varphi_2(B))V$, where $\\varphi_i(X)$ is either the identity map $X \\mapsto X$ or the transposition map $X \\mapsto X^t$. The results are extended to tensor space $M_{n_1} \\otimes ... \\otimes M_{n_m}$ of higher level. The connection of the problem to quantum information science is mentioned.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-08-06T03:24:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A69",
      "15A86",
      "15B57",
      "15A18"
    ],
    "keywords": [
      "quantum information science",
      "linear preservers",
      "tensor space",
      "complex matrices",
      "higher level"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.1076F"
    }
  }
}