{
  "id": "1409.4941",
  "version": "v1",
  "published": "2014-09-17T10:49:37.000Z",
  "updated": "2014-09-17T10:49:37.000Z",
  "title": "Real numerical shadow and generalized B-splines",
  "authors": [
    "Charles F. Dunkl",
    "Piotr Gawron",
    "Łukasz Pawela",
    "Zbigniew Puchała",
    "Karol Życzkowski"
  ],
  "comment": "39 pages, 7 figures",
  "categories": [
    "math.FA",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "Restricted numerical shadow $P^X_A(z)$ of an operator $A$ of order $N$ is a probability distribution supported on the numerical range $W_X(A)$ restricted to a certain subset $X$ of the set of all pure states - normalized, one-dimensional vectors in ${\\mathbb C}^N$. Its value at point $z \\in {\\mathbb C}$ equals to the probability that the inner product $< u |A| u >$ is equal to $z$, where $u$ stands for a random complex vector from the set $X$ distributed according to the natural measure on this set, induced by the unitarily invariant Fubini-Study measure. For a Hermitian operator $A$ of order $N$ we derive an explicit formula for its shadow restricted to real states, $P^{\\mathbb R}_A(x)$, show relation of this density to the Dirichlet distribution and demonstrate that it forms a generalization of the $B$-spline. Furthermore, for operators acting on a space with tensor product structure, ${\\cal H}_A \\otimes {\\cal H}_B$, we analyze the shadow restricted to the set of maximally entangled states and derive distributions for operators of order N=4.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-17T10:49:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "real numerical shadow",
      "generalized b-splines",
      "tensor product structure",
      "unitarily invariant fubini-study measure",
      "random complex vector"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.4941D"
    }
  }
}