{
  "id": "1412.3743",
  "version": "v1",
  "published": "2014-12-11T18:05:00.000Z",
  "updated": "2014-12-11T18:05:00.000Z",
  "title": "Euclidean distance between Haar orthogonal and gaussian matrices",
  "authors": [
    "Carlos E. González-Guillén",
    "Carlos Palazuelos",
    "Ignacio Villanueva"
  ],
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "In this work we study a version of the general question of how well a Haar distributed orthogonal matrix can be approximated by a random gaussian matrix. Here, we consider a gaussian random matrix $Y_n$ of order $n$ and apply to it the Gram-Schmidt orthonormalization procedure by columns to obtain a Haar distributed orthogonal matrix $U_n$. If $F_i^m$ denotes the vector formed by the first $m$-coordinates of the $i$th row of $Y_n-\\sqrt{n}U_n$ and $\\alpha=\\frac{m}{n}$, our main result shows that the euclidean norm of $F_i^m$ converges exponentially fast to $\\sqrt{ \\left(2-\\frac{4}{3} \\frac{(1-(1 -\\alpha)^{3/2})}{\\alpha}\\right)m}$, up to negligible terms. To show the extent of this result, we use it to study the convergence of the supremum norm $\\epsilon_n(m)=\\sup_{1\\leq i \\leq n, 1\\leq j \\leq m} |y_{i,j}- \\sqrt{n}u_{i,j}|$ and we find a coupling that improves by a factor $\\sqrt{2}$ the recently proved best known upper bound of $\\epsilon_n(m)$. Applications of our results to Quantum Information Theory are also explained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-11T18:05:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "euclidean distance",
      "haar orthogonal",
      "gaussian matrices",
      "haar distributed orthogonal matrix",
      "random gaussian matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.3743G"
    }
  }
}