{
  "id": "1101.5041",
  "version": "v1",
  "published": "2011-01-26T12:23:46.000Z",
  "updated": "2011-01-26T12:23:46.000Z",
  "title": "The Symmetry Group of Gaussian States in $L^2 (\\mathbb{R}^n)$",
  "authors": [
    "K. R. Parthasarathy"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "This is a continuation of the expository article \\cite{krp} with some new remarks. Let $S_n$ denote the set of all Gaussian states in the complex Hilbert space $L^2 (\\mathbb{R}^n),$ $K_n$ the convex set of all momentum and position covariance matrices of order $2n$ in Gaussian states and let $\\mathcal{G}_n$ be the group of all unitary operators in $L^2 (\\mathbb{R}^n)$ conjugations by which leave $S_n$ invariant. Here we prove the following results. $K_n$ is a closed convex set for which a matrix $S$ is an extreme point if and only if $S=\\frac{1}{2} L^{T} L$ for some $L$ in the symplectic group $Sp (2n, \\mathbb{R}).$ Every element in $K_n$ is of the form $\\frac{1}{2} (L^{T} L + M^{T} M)$ for some $L,M$ in $Sp (2n, \\mathbb{R}).$ Every Gaussian state in $L^2 (\\mathbb{R}^n)$ can be purified to a Gaussian state in $L^2 (\\mathbb{R}^{2n}).$ Any element $U$ in the group $\\mathcal{G}_n$ is of the form $U = \\lambda W ({\\bm {\\alpha}}) \\Gamma (L)$ where $\\lambda$ is a complex scalar of modulus unity, ${\\bm {\\alpha}} \\in \\mathbb{C}^n,$ $L \\in Sp (2n, \\mathbb{R}),$ $W({\\bm {\\alpha}})$ is the Weyl operator corresponding to ${\\bm {\\alpha}} $ and $\\Gamma (L)$ is a unitary operator which implements the Bogolioubov automorphism of the Lie algebra generated by the canonical momentum and position observables induced by the symplectic linear transformation $L.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-01-26T12:23:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81S25",
      "60B15",
      "42A82",
      "81R30"
    ],
    "keywords": [
      "gaussian state",
      "symmetry group",
      "unitary operator",
      "complex hilbert space",
      "position covariance matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}