{
  "annotation_id": "504c0b5c-dd85-4171-b5b8-9a3f591207d5",
  "date_created": "2026-03-02T18:02:30.289000Z",
  "date_modified": "2026-03-02T18:02:30.289000Z",
  "file_hash": "0b64e2fb23ab88685296c52f9b09b43c4a21d96b49d759ceb56614552c0c362a",
  "private": false,
  "record": {
    "abstract": "This thesis includes a survey of the results known for private and\napproximate private quantum channels. We develop the best known upper bound for\n$\\epsilon$-randomizing maps, $n+2\\log(1/\\epsilon)+c$ bits required to\n$\\epsilon$-randomize an arbitrary $n$-qubit state by improving a scheme of\nAmbainis and Smith \\cite{AS04} based on small bias spaces \\cite{NN90, AGHP92}.\nWe show by a probabilistic argument that in fact the great majority of random\nschemes using slightly more than this many bits of key are also\n$\\epsilon$-randomizing. We provide the first known non-trivial lower bound for\n$\\epsilon$-randomizing maps, and develop several conditions on them which we\nhope may be useful in proving stronger lower bounds in the future.",
    "arxiv_id": "quant-ph/0611037",
    "authors": [
      "Paul Dickinson"
    ],
    "categories": [
      "quant-ph"
    ],
    "title": "Approximate Private Quantum Channels",
    "url": "https://arxiv.org/abs/quant-ph/0611037"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "5101832b-782b-4f7f-89ec-41daea94872f",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}