{
  "abstract": "A state over time (stote) pairs a state $ρ$ with a channel $\\mathcal{E}$ as the Hermitian, generally non-positive operator $(ρ\\otimes\\mathbf{1})\\star J_\\mathcal{E}$. Characterizing the convex cone $\\widehat{\\mathcal{Q}}$ these operators generate was posed as an open problem in~\\cite{HoogstederCalsamigliaWinter2025}. We exhibit a linear copy of the bipartite separable cone as an exposed face of $\\widehat{\\mathcal{Q}}$: for output systems of the form $B = A'R$ with $A'$ a copy of the input, there is an explicit witness $W_0\\in\\widehat{\\mathcal{Q}}^*$ and an explicit injective linear map $L$ with $σ\\in \\mathrm{SEP}_{+}(A{:}R)\\iff L(σ)\\in\\widehat{\\mathcal{Q}}(A{:}A'R)$. Composing with the NP-hardness of separability shows that exact membership, weak membership, and weak optimization are NP-hard for $\\widehat{\\mathcal{Q}}$ and its dual. A quantitative stability bound makes the weak-membership reduction valid at inverse-polynomial accuracy. Thus polynomial-time membership tests do not exist unless $\\mathrm{P}=\\mathrm{NP}$. Independently of this complexity assumption, Fawzi's theorem implies that $\\widehat{\\mathcal{Q}}(A{:}A'R)$ admits no finite semidefinite representation when $d=\\dim A\\ge2$, $r=\\dim R\\ge2$, and $d+r>5$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34972",
  "author_total": 1,
  "authors": [
    {
      "name": "Matt Hoogsteder-Riera"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/702dcjd0326h8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34972",
  "fulltext_url": null,
  "id": "702dcjd0326h8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34972",
  "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",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Information and Cryptography"
    ]
  ],
  "openalex": "W7223966911",
  "pdf_url": "https://arxiv.org/pdf/2610.34972",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/702dcjd0326h8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:35Z",
      "native_id": "2610.34972",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34972"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Membership in the cone of quantum states over time is NP-hard",
  "title_text": "Membership in the cone of quantum states over time is NP-hard",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/702dcjd0326h8",
  "venue": "arXiv",
  "versions": []
}