{
  "abstract": "We prove quantitative lower bounds on the semidefinite extension complexity of the set of separable quantum states on $\\mathbb{C}^d\\otimes\\mathbb{C}^d$. We consider semidefinite programs (SDPs) that approximate the maximum acceptance probability of a measurement over separable states, the optimization problem underlying QMA(2). In the extended-formulation model of Harrow, Natarajan, and Wu (HNW), all measurements share a common feasible region and an objective-independent embedding of product states that exactly reproduces their acceptance probabilities. For every $0<θ<2/7$, there are constants $c_θ,a_θ>0$ such that, for sufficiently large $d$, any such SDP with uniform additive error $0<a\\le a_θ$ has size at least $d^{c_θ\\min\\{a^{-1/3},d^θ\\}}$. The bound applies at sufficiently small constant error, is superpolynomial in $d$ whenever $a=o(1)$, and becomes $d^{Ω(d^θ)}$ when $a\\le d^{-3θ}$, improving HNW's quasipolynomial bound at inverse- square error. The same bound holds for any SDP-representable convex set of states that contains all separable states and lies within trace distance $a$ of them, giving a quantitative counterpart to Fawzi's theorem that the separable set has no exact semidefinite representation. Our proof combines the quantitative pseudo-density theorem of Lee, Raghavendra, and Steurer with explicit block-positive operators and Chebyshev amplification. Our main results are supported by Lean proofs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09033",
  "author_total": 3,
  "authors": [
    {
      "name": "Sevag Gharibian"
    },
    {
      "name": "Carsten Hecht"
    },
    {
      "name": "Dorian Rudolph"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/egm7xagezzeb2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09033",
  "fulltext_url": null,
  "id": "egm7xagezzeb2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09033",
  "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",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7212083613",
  "pdf_url": "https://arxiv.org/pdf/2609.09033",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/egm7xagezzeb2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:52Z",
      "native_id": "2609.09033",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09033"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Semidefinite extension complexity of the separable set, with applications to approximate disentanglers",
  "title_text": "Semidefinite extension complexity of the separable set, with applications to approximate disentanglers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:52Z",
  "url": "https://research.draconitic.com/paper/egm7xagezzeb2",
  "venue": "arXiv",
  "versions": []
}