{
  "abstract": "For a quantum code defined by an isometry $T:H_{\\rm code} \\to H_{\\rm phys}$, an operator map is a *-homomorphism $φ: B(H_{\\rm code}) \\to B(H_{\\rm phys})$ from the space of operators on the logical Hilbert space to the space of operators on the physical Hilbert space satisfying $φ({\\cal O}) T = T {\\cal O}$ for every operator ${\\cal O}$ on $H_{\\rm code}$. This can additionally be taken to be unital $φ(I_{code}) = I_{phys}$ if and only if $\\dim H_{\\rm code}$ divides $\\dim H_{\\rm phys}$. In this case, $φ(B(H_{\\rm code})$ is a von Neumann factor subalgebra of $B(H_{\\rm phys})$ and we call the pair $(T,φ)$ a {\\it factor code}. A code network is a family of Hilbert spaces $\\{{\\cal H}_i\\}$ indexed by a partially ordered set with isometries $T_{ji}:{\\cal H}_i \\to {\\cal H}_j$ for $i < j$ satisfying $T_{kj} T_{ji} = T_{ki}$ for $i < j < k$. In this paper, we ask when this can be promoted to a {\\it factor code network} by defining unital operator maps $φ_{ji}$ so that $(T_{ji}, φ_{ji})$ is a factor code and $φ_{kj} φ_{ji} = φ_{ki}$. We show that the divisibility conditions on Hilbert space dimensions are sufficient for any code network with a tree structure but not in general. For a diamond-shaped network, we show that promotion is possible if and only if the principle angles between $T_{42} H_2$ and $T_{43} H_3$ all have multiplicity divisible by $H_1$. For incomparable $i_1,i_2 < j$, we characterize when $φ_{j i_1} (B(H_{i_1}))$ and $φ_{j i_2} (B(H_{i_2}))$ commute, when they generate all of $B(H_j)$, and when they intersect only on multiples of the identity. For $i < j_1,j_2 < k$, we characterize when $φ_{k j_1} (B(H_{j_1})) \\cap φ_{k j_2} (B(H_{j_2})) = φ_{k i} (B(H_{i}))$ and when these algebras together with $B(H_k)$ form a nondegenerate commuting square.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00469",
  "author_total": 4,
  "authors": [
    {
      "name": "Abhisek Sahu"
    },
    {
      "name": "Jeremy van der Heijden"
    },
    {
      "name": "Mark Van Raamsdonk"
    },
    {
      "name": "Rana Zibakhsh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/csnsgnqgc12sd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00469",
  "fulltext_url": null,
  "id": "csnsgnqgc12sd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00469",
  "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",
      "hep-th"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7216012199",
  "pdf_url": "https://arxiv.org/pdf/2610.00469",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/csnsgnqgc12sd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:03Z",
      "native_id": "2610.00469",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00469"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Factor Code Networks",
  "title_text": "Factor Code Networks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:03Z",
  "url": "https://research.draconitic.com/paper/csnsgnqgc12sd",
  "venue": "arXiv",
  "versions": []
}