{
  "annotation_id": "66ba0a41-4264-4d96-b4a1-ae8409564569",
  "date_created": "2026-03-02T18:01:42.541000Z",
  "date_modified": "2026-03-02T18:01:42.541000Z",
  "file_hash": "ab6be2ae370c48bd99771a48a89f02694fc01e18205966e28eacaed7dd1bffa4",
  "private": false,
  "record": {
    "abstract": "The relation between the requirement of efficient implementability and the\nproduct state representation of numbers is examined. Numbers are defined to be\nany model of the axioms of number theory or arithmetic. Efficient\nimplementability (EI) means that the basic arithmetic operations are physically\nimplementable and the space-time and thermodynamic resources needed to carry\nout the implementations are polynomial in the range of numbers considered.\nDifferent models of numbers are described to show the independence of both EI\nand the product state representation from the axioms. The relation between EI\nand the product state representation is examined. It is seen that the condition\nof a product state representation does not imply EI. Arguments used to refute\nthe converse implication, EI implies a product state representation, seem\nreasonable; but they are not conclusive. Thus this implication remains an open\nquestion.",
    "arxiv_id": "quant-ph/0104061",
    "authors": [
      "Paul Benioff"
    ],
    "categories": [
      "quant-ph"
    ],
    "doi": "10.1103/PhysRevA.64.052310",
    "journal_ref": "Phys. Rev. 64A, 052310 (2001)",
    "title": "Efficient Implementation and the Product State Representation of Numbers",
    "url": "https://arxiv.org/abs/quant-ph/0104061"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "fce8c356-acf9-4317-a354-400b7280b177",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}