{
  "id": "2010.07902",
  "version": "v1",
  "published": "2020-10-15T17:28:36.000Z",
  "updated": "2020-10-15T17:28:36.000Z",
  "title": "Entropic proofs of Singleton bounds for quantum error-correcting codes",
  "authors": [
    "Markus Grassl",
    "Felix Huber",
    "Andreas Winter"
  ],
  "comment": "13 pages, 5 figures. Happy birthday Lorenzo!",
  "categories": [
    "quant-ph",
    "cs.IT",
    "math.IT"
  ],
  "abstract": "We show that a relatively simple reasoning using von Neumann entropy inequalities yields a robust proof of the quantum Singleton bound for quantum error-correcting codes (QECC). For entanglement-assisted quantum error-correcting codes (EAQECC) and catalytic codes (CQECC), the generalised quantum Singleton bound was believed to hold for many years until recently one of us found a counterexample [MG, arXiv:2007.01249]. Here, we rectify this state of affairs by proving the correct generalised quantum Singleton bound for CQECC, extending the above-mentioned proof method for QECC; we also prove information-theoretically tight bounds on the entanglement-communication tradeoff for EAQECC. All of the bounds relate block length $n$ and code length k for given minimum distance d and we show that they are robust, in the sense that they hold with small perturbations for codes which only correct most of the erasure errors of less than d letters. In contrast to the classical case, the bounds take on qualitatively different forms depending on whether the minimum distance is smaller or larger than half the block length.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-15T17:28:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quantum error-correcting codes",
      "generalised quantum singleton bound",
      "entropic proofs",
      "von neumann entropy inequalities yields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}