{
  "id": "1206.0082",
  "version": "v3",
  "published": "2012-06-01T05:11:29.000Z",
  "updated": "2012-08-31T06:29:25.000Z",
  "title": "Pretty good state transfer on double stars",
  "authors": [
    "Xiaoxia Fan",
    "Chris Godsil"
  ],
  "comment": "15 pages, 2 EPS figures",
  "categories": [
    "math.CO",
    "quant-ph"
  ],
  "abstract": "Let A be the adjacency matrix of a graph $X$ and suppose U(t)=exp(itA). We view A as acting on $\\cx^{V(X)}$ and take the standard basis of this space to be the vectors $e_u$ for $u$ in $V(X)$. Physicists say that we have perfect state transfer from vertex $u$ to $v$ at time $\\tau$ if there is a scalar $\\gamma$ such that $U(\\tau)e_u = \\gamma e_v$. (Since $U(t)$ is unitary, $\\norm\\gamma=1$.) For example, if $X$ is the $d$-cube and $u$ and $v$ are at distance $d$ then we have perfect state transfer from $u$ to $v$ at time $\\pi/2$. Despite the existence of this nice family, it has become clear that perfect state transfer is rare. Hence we consider a relaxation: we say that we have pretty good state transfer from $u$ to $v$ if there is a complex number $\\gamma$ and, for each positive real $\\epsilon$ there is a time $t$ such that $\\norm{U(t)e_u - \\gamma e_v} < \\epsilon$. Again we necessarily have $|\\gamma|=1$. Godsil, Kirkland, Severini and Smith showed that we have have pretty good state transfer between the end vertices of the path $P_n$ if and only $n+1$ is a power of two, a prime, or twice a prime. (There is perfect state transfer between the end vertices only for $P_2$ and $P_3$.) It is something of a surprise that the occurrence of pretty good state transfer is characterized by a number-theoretic condition. In this paper we study double-star graphs, which are trees with two vertices of degree $k+1$ and all other vertices with degree one. We prove that there is never perfect state transfer between the two vertices of degree $k+1$, and that there is pretty good state transfer between them if and only if $4k+1$ is a perfect square.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-08-31T06:29:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "perfect state transfer",
      "double stars",
      "end vertices",
      "study double-star graphs",
      "number-theoretic condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.0082F"
    }
  }
}