{
  "id": "2312.02583",
  "version": "v1",
  "published": "2023-12-05T08:55:57.000Z",
  "updated": "2023-12-05T08:55:57.000Z",
  "title": "A new class of distances on complex projective spaces",
  "authors": [
    "Rafał Bistroń",
    "Michał Eckstein",
    "Shmuel Friedland",
    "Tomasz Miller",
    "Karol Życzkowski"
  ],
  "comment": "31 pages",
  "categories": [
    "math-ph",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "The complex projective space $\\mathbb{P}(\\mathbb{C}^n)$ can be interpreted as the space of all quantum pure states of size $n$. A distance on this space, interesting from the perspective of quantum physics, can be induced from a classical distance defined on the $n$-point probability simplex by the `earth mover problem'. We show that this construction leads to a quantity satisfying the triangle inequality, which yields a true distance on complex projective space belonging to the family of quantum $2$-Wasserstein distances.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-05T08:55:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complex projective space",
      "quantum pure states",
      "earth mover problem",
      "point probability simplex",
      "triangle inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}