{
  "abstract": "For a bipartite state $ρ$, information about the spectrum of its partial transpose $ρ^{Γ_B}$ can be inferred from measurements on multiple copies of $ρ$, without full state tomography. This raises a natural question: which eigenvalue lists can arise as $\\operatorname{spec}(ρ^{Γ_B})$ for a density operator $ρ$? We completely solve this inverse eigenvalue problem for two qubits. Every nonnegative trace-one spectrum is realized as $\\operatorname{spec}(ρ^{Γ_B})$ by some PPT state $ρ$, whereas an ordered candidate eigenvalue list $(x,y,z,-q)$, with $x\\ge y\\ge z\\ge0$, $q>0$, and $x+y+z-q=1$, is realized by an NPT state iff $q\\le y$ and $qy\\le xz$. Sufficiency in the latter case is established by an explicit $X$ state whose quantum steering ellipsoid has center $c=(y-q)/(1-z)$ and normalized volume $V/V_{\\max}(c)=qy/(xz)$, providing a geometric interpretation of the inequalities $q\\le y$ and $qy\\le xz$ as the allowed ellipsoid-center region and the fixed-center volume bound. Beyond this geometric picture, the two-qubit inverse theorem also yields exact negativity bounds from the two lowest nontrivial PT moments. Given fixed values of $p_2=Tr[(ρ^{Γ_B})^2]$ and $p_3=Tr[(ρ^{Γ_B})^3]$, we determine the exact minimum and maximum negativity over all two-qubit states subject to these moment constraints. When no PPT state is consistent with the pair $(p_2,p_3)$, the minimum is attained either at $x=y$ or $qy=xz$, while the maximum is attained either at $y=z$ or $q=y$. Finally, we show how the two-qubit inequalities persist as necessary constraints for the inverse eigenvalue problem in qubit--qudit systems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29068",
  "author_total": 3,
  "authors": [
    {
      "name": "Ruoting Dou"
    },
    {
      "name": "Shengjun Wu"
    },
    {
      "name": "Zeng-Bing Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dkyp9x653s3vj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29068",
  "fulltext_url": null,
  "id": "dkyp9x653s3vj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29068",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Information and Cryptography"
    ]
  ],
  "openalex": "W7204956803",
  "pdf_url": "https://arxiv.org/pdf/2608.29068",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dkyp9x653s3vj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:41Z",
      "native_id": "2608.29068",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29068"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "The Inverse Eigenvalue Problem for Partial Transposes of Two-Qubit States",
  "title_text": "The Inverse Eigenvalue Problem for Partial Transposes of Two-Qubit States",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:41Z",
  "url": "https://research.draconitic.com/paper/dkyp9x653s3vj",
  "venue": "arXiv",
  "versions": []
}