{
  "id": "1609.07083",
  "version": "v1",
  "published": "2016-09-22T17:41:26.000Z",
  "updated": "2016-09-22T17:41:26.000Z",
  "title": "Sinkhorn-Knopp theorem for positive maps",
  "authors": [
    "Daniel Cariello"
  ],
  "comment": "This paper was submitted to Communications in Mathematical Physics on August 22 of 2016",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "A positive map $S:M_k\\rightarrow M_m$ is called doubly stochastic if $S(\\frac{Id}{\\sqrt{k}})=\\frac{Id}{\\sqrt{m}}$ and $S^*(\\frac{Id}{\\sqrt{m}})=\\frac{Id}{\\sqrt{k}}$. Here, we show that, for a positive map $T:M_k\\rightarrow M_m$, there are invertible matrices $X'\\in M_k$, $Y'\\in M_m$ such that $Y'T(X'XX'^*)Y'^*$ is doubly stochastic if and only if $T(Id)$ and $T^*(Id)$ are invertible matrices and there are orthogonal projections $V_i\\in M_k$, $W_i\\in M_m$, $1\\leq i\\leq s$, such that $\\mathbb{C}^k=\\bigoplus_{i=1}^s\\Im(V_i)$, $\\mathbb{C}^m=\\bigoplus_{i=1}^s\\Im(W_i)$ and, for every $i$, $T(V_iM_kV_i)\\subset W_iM_mW_i$, $\\frac{\\text{rank}(W_i)}{\\text{rank}(V_i)}=\\frac{m}{k}$ and $\\text{rank}(X)\\text{rank}(W_i)<\\text{rank}(T(X))\\text{rank}(V_i)$ for every positive semidefinite Hermitian matrix $X\\in V_iM_kV_i$ with $0<\\text{rank}(X)<\\text{rank}(V_i)$. In order to obtain this theorem, we generalize Sinkhorn and Knopp ideas of support and total support for positive maps and we adapt their iterative algorithm. This result provides a necessary and sufficent condition for the existence of the filter normal form, which is commonly used in Quantum Information Theory. We prove the existence of this normal form for states $A\\in M_k\\otimes M_m\\simeq M_{km}$ such that $\\dim(\\ker(A))<k-1$, if $k=m$, and $\\dim(\\ker(A))<\\min\\{k,m\\}$, if $k\\neq m$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-22T17:41:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "positive map",
      "sinkhorn-knopp theorem",
      "invertible matrices",
      "filter normal form",
      "positive semidefinite hermitian matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}