{
  "annotation_id": "61fbd1aa-b3a6-4f46-b9ab-a702cbfe52a0",
  "date_created": "2026-03-02T18:02:22.877000Z",
  "date_modified": "2026-03-02T18:02:22.877000Z",
  "file_hash": "999854fb376b3fae4abb2390f05ac1dd8dd10e22b02d1c3fdd25a7d0372c742f",
  "private": false,
  "record": {
    "abstract": "Weyl\u0027s displacement operators for position and momentum commute if the\nproduct of the elementary displacements equals Planck\u0027s constant. Then, their\ncommon eigenstates constitute the Zak basis, each state specified by two phase\nparameters. Upon enforcing a periodic dependence on the phases, one gets a\none-to-one mapping of the Hilbert space on the line onto the Hilbert space on\nthe torus. The Fourier coefficients of the periodic Zak bases make up the\ndiscrete Zak bases. The two bases are mutually unbiased. We study these bases\nin detail, including a brief discussion of their relation to Aharonov\u0027s modular\noperators, and mention how they can be used to associate with the single degree\nof freedom of the line a pair of genuine qubits.",
    "arxiv_id": "quant-ph/0511234",
    "authors": [
      "Berthold-Georg Englert",
      "Kean Loon Lee",
      "Ady Mann",
      "Michael Revzen"
    ],
    "categories": [
      "quant-ph"
    ],
    "doi": "10.1088/0305-4470/39/7/011",
    "journal_ref": "Journal of Physics A: Mathematical and General 39 (2006) 1669-1682",
    "title": "Periodic and discrete Zak bases",
    "url": "https://arxiv.org/abs/quant-ph/0511234"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "bb3a83d9-83de-42eb-b407-756cf0a453e8",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}