{
  "id": "1711.07585",
  "version": "v1",
  "published": "2017-11-21T00:25:38.000Z",
  "updated": "2017-11-21T00:25:38.000Z",
  "title": "Pure state `really' informationally complete with rank-1 POVM",
  "authors": [
    "Yu Wang",
    "Yun Shang"
  ],
  "categories": [
    "quant-ph",
    "cs.IT",
    "math.IT"
  ],
  "abstract": "What is the minimal number of elements in a rank-1 positive-operator-valued measure (POVM) which can uniquely determine any pure state in $d$-dimensional Hilbert space $\\mathcal{H}_d$? The known result is that the number is no less than $3d-2$. We show that this lower bound is not tight except for $d=2$ or 4. Then we give an upper bound of $4d-3$. For $d=2$, many rank-1 POVMs with four elements can determine any pure states in $\\mathcal{H}_2$. For $d=3$, we show eight is the minimal number by construction. For $d=4$, the minimal number is in the set of $\\{10,11,12,13\\}$. We show that if this number is greater than 10, an unsettled open problem can be solved that three orthonormal bases can not distinguish all pure states in $\\mathcal{H}_4$. For any dimension $d$, we construct $d+2k-2$ adaptive rank-1 positive operators for the reconstruction of any unknown pure state in $\\mathcal{H}_d$, where $1\\le k \\le d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-21T00:25:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "informationally complete",
      "minimal number",
      "unknown pure state",
      "dimensional hilbert space",
      "lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}