{
  "id": "2301.09074",
  "version": "v1",
  "published": "2023-01-22T08:08:51.000Z",
  "updated": "2023-01-22T08:08:51.000Z",
  "title": "Average Rényi Entropy of a Subsystem in Random Pure State",
  "authors": [
    "MuSeong Kim",
    "Mi-Ra Hwang",
    "Eylee Jung",
    "DaeKil Park"
  ],
  "comment": "14 pages, 3 figures",
  "categories": [
    "quant-ph",
    "hep-th"
  ],
  "abstract": "In this paper we examine the average R\\'{e}nyi entropy $S_{\\alpha}$ of a subsystem $A$ when the whole composite system $AB$ is a random pure state. We assume that the Hilbert space dimensions of $A$ and $AB$ are $m$ and $m n$ respectively. First, we compute the average R\\'{e}nyi entropy analytically for $m = \\alpha = 2$. We compare this analytical result with the approximate average R\\'{e}nyi entropy, which is shown to be very close. For general case we compute the average of the approximate R\\'{e}nyi entropy $\\widetilde{S}_{\\alpha} (m,n)$ analytically. When $1 \\ll n$, $\\widetilde{S}_{\\alpha} (m,n)$ reduces to $\\ln m - \\frac{\\alpha}{2 n} (m - m^{-1})$, which is in agreement with the asymptotic expression of the average von Neumann entropy. Based on the analytic result of $\\widetilde{S}_{\\alpha} (m,n)$ we plot the $\\ln m$-dependence of the quantum information derived from $\\widetilde{S}_{\\alpha} (m,n)$. It is remarkable to note that the nearly vanishing region of the information becomes shorten with increasing $\\alpha$, and eventually disappears in the limit of $\\alpha \\rightarrow \\infty$. The physical implication of the result is briefly discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-22T08:08:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random pure state",
      "average rényi entropy",
      "average von neumann entropy",
      "hilbert space dimensions",
      "general case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}