{
  "id": "classical-capacity-of-generalized-amplitude-damping",
  "title": "Classical capacity of generalized amplitude damping",
  "description": "Every qubit generalized amplitude-damping channel, for all damping strengths and stationary populations, has Holevo information of n uses equal to n times that of one use, an explicit maximum over one variable attained by two pure states with opposite phases; the channel is also additive with every finite-dimensional partner channel for the Holevo information, the minimum output entropy and the regularized Holevo information. The proof: entropy differences as integrals of log-determinants; a jointly convex Schur-complement log-determinant and a concave scalar deficit; an entropy inequality for block matrices with a zero block, extended to the thermal block structure; a site-by-site bound for the output entropy of n uses in terms of the input populations. One named hypothesis, the classical coding theorem, turns the results on Holevo information into results on the classical capacity. The course is an exposition of OpenAI's proof (2026).",
  "units": [
    {
      "id": "quantum-channels-and-holevo-information",
      "title": "Quantum channels and Holevo information",
      "reader": "quantum-channels-and-holevo-information.html",
      "source": "src/quantum-channels-and-holevo-information.md",
      "source_sha256": "1c745458493aa166ee845a6a322e6ffedb5fd6b72208807b8c87d479fc84dfd3",
      "status": "author_self_checked"
    },
    {
      "id": "entropy-differences-and-log-determinants",
      "title": "Entropy differences and log-determinants",
      "reader": "entropy-differences-and-log-determinants.html",
      "source": "src/entropy-differences-and-log-determinants.md",
      "source_sha256": "14465c79a018f561b8cd0297bb861a60e5351f2e482965e5698c51e0d561a8fc",
      "status": "author_self_checked"
    },
    {
      "id": "block-entropy-inequalities",
      "title": "Block entropy inequalities",
      "reader": "block-entropy-inequalities.html",
      "source": "src/block-entropy-inequalities.md",
      "source_sha256": "cf31407982cdf38ec24e6e2914283a92f82995d8586193e11088df424b200aa3",
      "status": "author_self_checked"
    },
    {
      "id": "output-entropy-of-damping-blocks",
      "title": "Output entropy of damping blocks",
      "reader": "output-entropy-of-damping-blocks.html",
      "source": "src/output-entropy-of-damping-blocks.md",
      "source_sha256": "83e7dbf985038af9955b1f16f769d3b22b29486a31701049c3ee7b4de1774f9d",
      "status": "author_self_checked"
    },
    {
      "id": "additivity-with-an-arbitrary-partner",
      "title": "Additivity with an arbitrary partner",
      "reader": "additivity-with-an-arbitrary-partner.html",
      "source": "src/additivity-with-an-arbitrary-partner.md",
      "source_sha256": "8587113b34d21e3b7677a6f4a9a49169881f732be5b052daa8736930a4ce6f89",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "course Entropy of automorphisms, lesson Operator convex functions and the continuity of entropy: Shannon's inequalities (Proposition 4.1) and the Fannes-Audenaert inequality (Theorem 7.1)",
    "linear algebra: the spectral theorem, singular value decompositions, determinants and Schur complements, Hilbert-Schmidt norms",
    "named hypothesis (CT), the classical coding theorem of Holevo and of Schumacher and Westmoreland, stated in the first lesson; not proved in the programme"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}