{
  "abstract": "For a channel $Φ$ on $M_d$ written in the generalised Bloch parameterisation $r \\mapsto Ar+c$, we prove that entanglement breaking implies $\\|A\\|_*^2 + \\frac{d(d-1)}{2}\\,|c|^2 \\;\\le\\; (d-1)^2 ,$ where $\\|\\cdot\\|_*$ denotes the nuclear norm. Read on the Choi state of the channel, the inequality is an instance of the covariance-matrix criterion of Gittsovich et al.; what is added here is an elementary proof in Holevo form and a complete description of the equality cases. Apart from the constant channel onto a pure state, which shows that the coefficient $d(d-1)/2$ cannot be increased, equality holds exactly for the measure-and-prepare channels with rank-one effects $α_k|e_k\\rangle\\langle e_k|$ and pure outputs $|ψ_k\\rangle\\langleψ_k|$ whose overlaps are an affine image of those of the measurement directions, $|\\langleψ_j|ψ_k\\rangle|^2 = P + (1-P)\\,(d|\\langle e_j|e_k\\rangle|^2 - 1)/(d-1)$ for a common $P = \\mathrm{Tr}\\,Φ(\\mathbf{1}/d)^2$. These include the completely dephasing channel, the re-preparation channel of every rank-one POVM, a one-parameter family interpolating between dephasing and the constant channel, and non-unital non-projective examples already at $d = 2$. The qubit case reads $\\|A\\|_*^2+|c|^2\\le1$ and extends to full rank a rank-two condition of Ruskai. The proof sharpens Ruskai's bound $\\|A\\|_*\\le1$ by retaining the POVM completeness relation $\\sum_k f_k=0$, which her argument discards; this recentres $A$ from a second moment into a cross-covariance, and is the entire content of the improvement.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27906",
  "author_total": 1,
  "authors": [
    {
      "name": "Eran Kopel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dzd2rv0eqtzp5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27906",
  "fulltext_url": null,
  "id": "dzd2rv0eqtzp5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27906",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Information and Cryptography"
    ]
  ],
  "openalex": "W7214198911",
  "pdf_url": "https://arxiv.org/pdf/2609.27906",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dzd2rv0eqtzp5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:52Z",
      "native_id": "2609.27906",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27906"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "A sharp norm inequality for entanglement-breaking channels",
  "title_text": "A sharp norm inequality for entanglement-breaking channels",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/dzd2rv0eqtzp5",
  "venue": "arXiv",
  "versions": []
}