{
  "id": "quant-ph/0605090",
  "version": "v1",
  "published": "2006-05-10T09:01:27.000Z",
  "updated": "2006-05-10T09:01:27.000Z",
  "title": "The Mutually Unbiased Bases Revisited",
  "authors": [
    "M. Combescure"
  ],
  "comment": "International Conference on Transport and Spectral Problems in Quantum Mechanics held in Honor of Jean-Michel Combes, Cergy Pontoise : France (2006)",
  "categories": [
    "quant-ph",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The study of Mutually Unbiased Bases continues to be developed vigorously, and presents several challenges in the Quantum Information Theory. Two orthonormal bases in $\\mathbb C^d, B {and} B'$ are said mutually unbiased if $\\forall b\\in B, b'\\in B'$ the scalar product $b\\cdot b'$ has modulus $d^{-1/2}$. In particular this property has been introduced in order to allow an optimization of the measurement-driven quantum evolution process of any state $\\psi \\in \\mathbb C^d$ when measured in the mutually unbiased bases $B\\_{j} {of} \\mathbb C^d$. At present it is an open problem to find the maximal umber of mutually Unbiased Bases when $d$ is not a power of a prime number. \\noindent In this article, we revisit the problem of finding Mutually Unbiased Bases (MUB's) in any dimension $d$. The method is very elementary, using the simple unitary matrices introduced by Schwinger in 1960, together with their diagonalizations. The Vandermonde matrix based on the $d$-th roots of unity plays a major role. This allows us to show the existence of a set of 3 MUB's in any dimension, to give conditions for existence of more than 3 MUB's for $d$ even or odd number, and to recover the known result of existence of $d+1$ MUB's for $d$ a prime number. Furthermore the construction of these MUB's is very explicit. As a by-product, we recover results about Gauss Sums, known in number theory, but which have apparently not been previously derived from MUB properties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-05-10T09:01:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "mutually unbiased bases",
      "measurement-driven quantum evolution process",
      "prime number",
      "simple unitary matrices",
      "quantum information theory"
    ],
    "tags": [
      "conference paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006quant.ph..5090C"
    }
  }
}