{
  "annotation_id": "b298c7b8-4e78-4092-b4fb-72eb8e78ba16",
  "date_created": "2026-03-02T18:02:16.356000Z",
  "date_modified": "2026-03-02T18:02:16.356000Z",
  "file_hash": "d2e0a24d8e40979743210932c879b0914d9216b96adcb2492ac8cffefb9dcabf",
  "private": false,
  "record": {
    "abstract": "This note deals with estimating the volume of the set of separable mixed\nquantum states when the dimension of the state space grows to infinity. This\nhas been studied recently for qubits; here we consider larger particles and\nconclude that, in all cases, the proportion of the states that are separable is\nsuper-exponentially small in the dimension of the set. We also show that the\npartial transpose criterion becomes imprecise when the dimension increases, and\nthat the lower bound $6^{-N/2}$ on the (Hilbert-Schmidt) inradius of the set of\nseparable states on N qubits obtained recently by Gurvits and Barnum is\nessentially optimal. We employ standard tools of classical convexity,\nhigh-dimensional probability and geometry of Banach spaces. One relatively\nnon-standard point is a formal introduction of the concept of projective tensor\nproducts of convex bodies, and an initial study of this concept.\n  PACS numbers: 03.65.Ud, 03.67.Mn, 03.65.Db, 02.40.Ft, 02.50.Cw\n  MSC-class: 46B28, 47B10, 47L05, 52A38, 81P68",
    "arxiv_id": "quant-ph/0503221",
    "authors": [
      "Guillaume Aubrun",
      "Stanislaw J. Szarek"
    ],
    "categories": [
      "quant-ph",
      "math.FA"
    ],
    "doi": "10.1103/PhysRevA.73.022109",
    "journal_ref": "Phys. Rev. A. 73, 022109 (2006)",
    "title": "Tensor products of convex sets and the volume of separable states on N   qudits",
    "url": "https://arxiv.org/abs/quant-ph/0503221"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "af24cbba-23ce-4d57-aceb-b2320adc381c",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}