{
  "id": "2407.11118",
  "version": "v1",
  "published": "2024-07-15T18:00:01.000Z",
  "updated": "2024-07-15T18:00:01.000Z",
  "title": "Tight lower bound on the error exponent of classical-quantum channels",
  "authors": [
    "Joseph M. Renes"
  ],
  "comment": "14 pages, no figures. This work extends arXiv:2207.08899 to the case of general CQ channels",
  "categories": [
    "quant-ph",
    "cs.IT",
    "math.IT"
  ],
  "abstract": "A fundamental quantity of interest in Shannon theory, classical or quantum, is the error exponent of a given channel $W$ and rate $R$: the constant $E(W,R)$ which governs the exponential decay of decoding error when using ever larger optimal codes of fixed rate $R$ to communicate over ever more (memoryless) instances of a given channel $W$. Nearly matching lower and upper bounds are well-known for classical channels. Here I show a lower bound on the error exponent of communication over arbitrary classical-quantum (CQ) channels which matches Dalai's sphere-packing upper bound [IEEE TIT 59, 8027 (2013)] for rates above a critical value, exactly analogous to the case of classical channels. Unlike the classical case, however, the argument does not proceed via a refined analysis of a suitable decoder, but instead by leveraging a bound by Hayashi on the error exponent of the cryptographic task of privacy amplification [CMP 333, 335 (2015)]. This bound is then related to the coding problem via tight entropic uncertainty relations and Gallager's method of constructing capacity-achieving parity-check codes for arbitrary channels. Along the way, I find a lower bound on the error exponent of the task of compression of classical information relative to quantum side information that matches the sphere-packing upper bound of Cheng et al. [IEEE TIT 67, 902 (2021)]. In turn, the polynomial prefactors to the sphere-packing bound found by Cheng et al. may be translated to the privacy amplification problem, sharpening a recent result by Li, Yao, and Hayashi [IEEE TIT 69, 1680 (2023)], at least for linear randomness extractors.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-15T18:00:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "error exponent",
      "tight lower bound",
      "classical-quantum channels",
      "ieee tit",
      "matches dalais sphere-packing upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}