{
  "abstract": "We all know that a density matrix $ρ$ is pure if and only if $\\mathrm{Tr} \\,ρ^2 = 1$. But is this the only way to prove the purity of $ρ$ using the trace of its powers? Here I systematically study the necessary and sufficient conditions of the more general question of guaranteeing that all eigenvalues of a Hermitian matrix belong to a specified set through its trace of powers. More importantly, these characterization results are automatically witnesses certifying a quantum state as entangled, a Hermitian operator has least one negative eigenvalue and a linear operator as non-Hermitian. I demonstrate the effectiveness of these witnesses, analyze their performance, and study their strength, weakness together with resource requirement through numerical simulation as well as analytical work. I also discuss briefly the effects of numerical stability, uncertainty in measurement and rounding errors on these problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09557",
  "author_total": 1,
  "authors": [
    {
      "name": "H. F. Chau"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8j3zdsx9m01vq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09557",
  "fulltext_url": null,
  "id": "8j3zdsx9m01vq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09557",
  "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",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Information and Cryptography"
    ]
  ],
  "openalex": "W7212112316",
  "pdf_url": "https://arxiv.org/pdf/2609.09557",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8j3zdsx9m01vq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:12Z",
      "native_id": "2609.09557",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09557"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "From Certifying Rank $k$ Projectors To Non-Positivity, Entanglement, And Non-Hermitian Witnesses Through Traces Of Matrix Powers",
  "title_text": "From Certifying Rank k Projectors To Non-Positivity, Entanglement, And Non-Hermitian Witnesses Through Traces Of Matrix Powers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:12Z",
  "url": "https://research.draconitic.com/paper/8j3zdsx9m01vq",
  "venue": "arXiv",
  "versions": []
}