{
  "abstract": "Absolutely maximally entangled states of four six-level systems, AME$(4,6)$, are the quantum solutions of Euler's problem of the thirty-six officers. Until recently all reported solutions were complex, and it was conjectured that no real orthogonal 2-unitary matrix of order $36$ exists. We give real AME$(4,6)$ states with $60$ non-zero coefficients in $\\{\\pm1,\\sqrt2\\}/\\sqrt{72}$. Of their thirty-six officers, $12$ are classical, as some must be whenever fewer than $72$ coefficients are non-zero. They lie in the $-1$ eigenspace of an order-$4$ local symmetry, and on their support the AME conditions become linear. We derive two closed-form families of AME$(4,6)$ states that meet at one of these states: a $10$-parameter family that is supported on $72$ basis states, and an $11$-parameter family on $60$ basis states. At each of its two ends the first family loses $12$ coefficients, and the AME states on the remaining $60$ basis states form an $11$-parameter family: the second family at one end, and an analogous family at the other. These two $60$-term families contain $4096$ real AME states up to diagonal local unitaries, $1024$ of which are limits of the first family. They fall into exactly $20$ classes under local unitaries and permutations of the parties, and we certify their separation with interval arithmetic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16673",
  "author_total": 1,
  "authors": [
    {
      "name": "Palash Pandya"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d55ta7tbvb84j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16673",
  "fulltext_url": null,
  "id": "d55ta7tbvb84j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16673",
  "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 Information and Cryptography"
    ]
  ],
  "openalex": "W7221804196",
  "pdf_url": "https://arxiv.org/pdf/2610.16673",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d55ta7tbvb84j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:06Z",
      "native_id": "2610.16673",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16673"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Euler's thirty-six officers need not all be entangled: real AME(4,6), families and classification",
  "title_text": "Euler's thirty-six officers need not all be entangled: real AME(4,6), families and classification",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:06Z",
  "url": "https://research.draconitic.com/paper/d55ta7tbvb84j",
  "venue": "arXiv",
  "versions": []
}