{
  "annotation_id": "068e1d08-51f9-4a0e-bac1-c9460d3e2d4c",
  "date_created": "2026-03-02T18:02:30.285000Z",
  "date_modified": "2026-03-02T18:02:30.285000Z",
  "file_hash": "459cfa9f9e93c39b82940ca41bf754320b383119943684483f4d6dbcaac06e30",
  "private": false,
  "record": {
    "abstract": "We clarify the mathematical structure underlying unitary $t$-designs. These\nare sets of unitary matrices, evenly distributed in the sense that the average\nof any $t$-th order polynomial over the design equals the average over the\nentire unitary group. We present a simple necessary and sufficient criterion\nfor deciding if a set of matrices constitutes a design. Lower bounds for the\nnumber of elements of 2-designs are derived. We show how to turn mutually\nunbiased bases into approximate 2-designs whose cardinality is optimal in\nleading order. Designs of higher order are discussed and an example of a\nunitary 5-design is presented. We comment on the relation between unitary and\nspherical designs and outline methods for finding designs numerically or by\nsearching character tables of finite groups. Further, we sketch connections to\nproblems in linear optics and questions regarding typical entanglement.",
    "arxiv_id": "quant-ph/0611002",
    "authors": [
      "D. Gross",
      "K. Audenaert",
      "J. Eisert"
    ],
    "categories": [
      "quant-ph",
      "math-ph",
      "math.MP"
    ],
    "doi": "10.1063/1.2716992",
    "journal_ref": "J. Math. Phys. 48, 052104 (2007)",
    "title": "Evenly distributed unitaries: on the structure of unitary designs",
    "url": "https://arxiv.org/abs/quant-ph/0611002"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "a1d8409d-4fac-4ed5-9d97-0a451635b6a0",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}