{
  "id": "1703.02657",
  "version": "v1",
  "published": "2017-03-08T01:12:30.000Z",
  "updated": "2017-03-08T01:12:30.000Z",
  "title": "Associating vectors in $\\CC^n$ with rank 2 projections in $\\RR^{2n}$: with applications",
  "authors": [
    "Peter G. Casazza",
    "Desai Cheng"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We will see that vectors in $\\CC^n$ have natural analogs as rank 2 projections in $\\RR^{2n}$ and that this association transfers many vector properties into properties of rank two projections on $\\RR^{2n}$. We believe that this association will answer many open problems in $\\CC^n$ where the corresponding problem in $\\RR^n$ has already been answered - and vice versa. As a application, we will see that phase retrieval (respectively, phase retrieval by projections) in $\\CC^n$ transfers to a variation of phase retrieval by rank 2 projections (respectively, phase retrieval by projections) on $\\RR^{2n}$. As a consequence, we will answer the open problem: Give the complex version of Edidin's Theorem \\cite{E} which classifies when projections do phase retrieval in $\\RR^n$. As another application we answer a longstanding open problem concerning fusion frames by showing that fusion frames in $\\CC^n$ associate with fusion frames in $\\RR^{2n}$ with twice the dimension. As another application, we will show that a family of mutually unbiased bases in $\\CC^n$ has a natural analog as a family of mutually unbiased rank 2 projections in $\\RR^{2n}$. The importance here is that there are very few real mutually unbiased bases but now there are unlimited numbers of real mutually unbiased rank 2 projections to be used in their place. As another application, we will give a variaton of Edidin's theorem which gives a surprising classification of norm retrieval. Finally, we will show that equiangular and biangular frames in $\\CC^n$ have an analog as equiangular and biangular rank 2 projections in $\\RR^{2n}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-08T01:12:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C15"
    ],
    "keywords": [
      "projections",
      "phase retrieval",
      "application",
      "associating vectors",
      "mutually unbiased rank"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}