{
  "abstract": "The entangling power of a unitary operator acting on a bipartite Hilbert space measures the entanglement it generates from product states, averaged over the inputs. A finite-dimensional unitary has a spectral decomposition $U=\\sum_{a=1}^{n}e^{iθ_a}P_a$, where $e^{iθ_a}$ are the eigenvalues, $P_a$ the corresponding eigen-projectors, and $n$ is the number of distinct eigenvalues. After removing an overall phase, the entangling power is a function on the $(n-1)$-torus of relative eigenphases at fixed spectral projectors. We prove that this function is stationary at all $2^{n-1}$ points on the torus where every relative phase is $0$ or $π$, which we define as \\textit{corners}. Up to an overall phase, $U$ at each corner is a generalized reflection $R=\\mathbb{I}-2Q$ satisfying $R^2=\\mathbb{I}$, where $Q$ is the sum of spectral projectors whose relative phase is $π$. At the corner the entangling power is expressed in terms of seven local-unitary invariants of $Q$. A unitary gate $U$ can be realized as a corner of some projector family if and only if $U^2\\propto\\mathbb{I}$, a condition satisfied by many Clifford and non-Clifford gates. We illustrate the theorem with two-qubit gates, $SU(N)$ channel decompositions, and two-site spin chains, obtaining examples of minima, maxima, and saddle points. In addition, a corner that is a saddle point on the full phase torus can appear as a local maximum or minimum along different time-evolution trajectories.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09276",
  "author_total": 2,
  "authors": [
    {
      "name": "Ian Low"
    },
    {
      "name": "Navin McGinnis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fpvxjc99ff800.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09276",
  "fulltext_url": null,
  "id": "fpvxjc99ff800",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09276",
  "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",
      "hep-ph"
    ],
    [
      "arxiv",
      "hep-th"
    ],
    [
      "arxiv",
      "nucl-th"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Information and Cryptography"
    ]
  ],
  "openalex": "W7212149594",
  "pdf_url": "https://arxiv.org/pdf/2609.09276",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fpvxjc99ff800.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:12Z",
      "native_id": "2609.09276",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09276"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:hep-ph",
      "subfield": "high-energy"
    },
    {
      "field": "physics",
      "origin": "arxiv:nucl-th",
      "subfield": "nuclear"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Universal Entanglement Dynamics of Unitary Operators",
  "title_text": "Universal Entanglement Dynamics of Unitary Operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:12Z",
  "url": "https://research.draconitic.com/paper/fpvxjc99ff800",
  "venue": "arXiv",
  "versions": []
}