{
  "abstract": "Let $ρ$ and $σ$ be density operators on a separable Hilbert space. For $0<α<1$, we study the directed trace-norm overlap $Φ_α(ρ,σ)=\\|ρ^ασ^{1-α}\\|_1$ and its symmetrized form $\\mathcal F_α(ρ,σ)=\\frac12\\bigl(Φ_α(ρ,σ)+Φ_{1-α}(ρ,σ)\\bigr)$. At $α=\\tfrac12$, both quantities coincide with the root Uhlmann--Jozsa fidelity. Our aim is not to introduce a new notion of fidelity, but to understand how these overlaps vary with the parameter and when equality occurs in the resulting inequalities. We first prove that $α\\mapstoΦ_α(ρ,σ)$ is log-convex. This yields a sharp lower bound for $\\mathcal F_α$ in terms of the root fidelity, together with a stronger intermediate geometric-mean bound. We also show that $Φ_α$ is the trace functional $Q_{α,1/2}$ associated with the $α$-$z$ Renyi divergence, which connects the directed overlap with the known $α$-$z$ Renyi theory. We then determine the exact data-processing behavior of the symmetrized family. Universal monotonicity under quantum channels holds only at $α=\\tfrac12$; for every other value of $α$, it already fails under diagonal pinching of faithful real qubit states. In finite dimensions, we give a complete spectral description of the equality cases and provide explicit noncommuting examples. We also discuss some natural questions about overlap-preserving maps. Finally, we prove that the difference between the Petz overlap and the directed trace-norm overlap vanishes exactly when $ρ$ and $σ$ commute, without requiring either faithfulness or any additional support assumption.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14203",
  "author_total": 2,
  "authors": [
    {
      "name": "Zahra Maleki Khouzani"
    },
    {
      "name": "Seyed Mahmoud Manjegani"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/86jx6c3vfbtxw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14203",
  "fulltext_url": null,
  "id": "86jx6c3vfbtxw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14203",
  "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",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7213360840",
  "pdf_url": "https://arxiv.org/pdf/2609.14203",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/86jx6c3vfbtxw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:25Z",
      "native_id": "2609.14203",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14203"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Trace-Norm Overlaps of Quantum States: Interpolation, Equality, and Data-Processing Rigidity",
  "title_text": "Trace-Norm Overlaps of Quantum States: Interpolation, Equality, and Data-Processing Rigidity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/86jx6c3vfbtxw",
  "venue": "arXiv",
  "versions": []
}