{
  "annotation_id": "da3f7068-4031-4a4a-b3c8-80f5862a1ed1",
  "date_created": "2026-03-02T18:01:39.262000Z",
  "date_modified": "2026-03-02T18:01:39.262000Z",
  "file_hash": "fc5f3df356a8c5b0ab84f6d55173bb9c52206e6bcf10a11830ba010e48c44e97",
  "private": false,
  "record": {
    "abstract": "We show that every density matrix of an n-particle system prepared by a\nquantum network of constant depth is asymptotically commuting with the\nmean-field observables. We introduce certain pairs of hypersurfaces in the\nspace of density matrices and give lower bounds for the depth of a network\nwhich prepares states lying outside those pairs. The measurement of an\nobservable which is not asymptotically commuting with the mean-field\nobservables requires a network of depth in the order of log n, if one demands\nthe measurement to project the state into the eigenspace of the measured\nobservable.",
    "arxiv_id": "quant-ph/0003117",
    "authors": [
      "Dominik Janzing",
      "Thomas Beth"
    ],
    "categories": [
      "quant-ph"
    ],
    "title": "Remark on multi-particle observables and entangled states with constant   complexity",
    "url": "https://arxiv.org/abs/quant-ph/0003117"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "efae27c4-84a0-423a-9618-d399c32c368f",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}