{
  "abstract": "We study maximum purity and minimum von Neumann entropy of absolutely positive partial transpose (APPT) states, together with the relative volume of their spectral sets. A consequence of Hildebrand's criterion gives an explicit outer spectral polytope, whose vertices we classify. Optimizing purity over this polytope, together with the Song--Chen result for the $2\\otimes3$ system, yields, for every $mn\\ge6$, an explicit upper bound on the purity of APPT states that is asymptotic to $4/(3mn)$. This improves the previous $2/(mn)$ upper bound for absolutely separable states. The new bound applies to every APPT state, a class containing all absolutely separable states, and is sharp for every $2\\otimes n$ system with $n\\ge3$. For every $3\\otimes n$ system with $n\\ge3$, however, the unique outer-polytope maximizer is not APPT, so the bound is strict and disproves the Dũng--Khôi qutrit--qudit conjecture. The polytope also gives an explicit entropy lower bound in arbitrary bipartite dimensions and, together with the Song--Chen extreme-point classification, the exact minimum entropy for every $2\\otimes n$ system. Finally, exact formulas for the relative volumes of an inner polytope and the outer spectral polytope give explicit two-sided bounds on the qubit--qudit relative spectral volume $a_n$ whose ratio is less than $4$ and tends to $3$. Consequently, $a_n=Θ\\!\\left(\\sqrt n(4/27)^n\\right)$, and the relative volume of the qubit--qudit APPT spectral set (equivalently, the absolutely separable spectral set) has exact exponential decay rate $\\ln(27/4)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18568",
  "author_total": 1,
  "authors": [
    {
      "name": "Anh T. Tran"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5b64zfa1fk053.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18568",
  "fulltext_url": null,
  "id": "5b64zfa1fk053",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18568",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Information and Cryptography"
    ]
  ],
  "openalex": "W7213501855",
  "pdf_url": "https://arxiv.org/pdf/2609.18568",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5b64zfa1fk053.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:01Z",
      "native_id": "2609.18568",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18568"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Spectral Optimization for Absolutely PPT States: Purity, Entropy, and Volume Decay",
  "title_text": "Spectral Optimization for Absolutely PPT States: Purity, Entropy, and Volume Decay",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/5b64zfa1fk053",
  "venue": "arXiv",
  "versions": []
}