{
  "abstract": "Determining the minimal number of intensity measurements required for phase retrieval in $\\mathbb{C}^4$ has been a long-standing open problem. Prior to this work, the best-known results implied that this minimum was either $10$ or $11$. In this paper, we leverage characteristic classes and cohomology groups from differential topology to prove that no family of $10$ vectors in $\\mathbb{C}^4$ possesses the phase retrieval property. Combining our lower bound with Vinzant's explicit eleven-vector construction establishes that the exact minimum is $11$. Our result yields a significant consequence for pure state quantum tomography, namely, a rank-one POVM on $\\mathbb{C}^4$ requires exactly $11$ elements to be informationally complete for pure states. This further implies that three orthonormal bases are insufficient to uniquely distinguish all pure states in $\\mathbb{C}^4$. Because four orthonormal bases are already known to be sufficient, we conclude that exactly four bases are required, thereby completely resolving the problem left in [C. Carmeli, T. Heinosaari, J. Schultz, A. Toigo, Eur. Phys. J. D].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.27719",
  "author_total": 2,
  "authors": [
    {
      "name": "Meng Huang"
    },
    {
      "name": "Jihun Kim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/113tj96gcxxbn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.27719",
  "fulltext_url": null,
  "id": "113tj96gcxxbn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.27719",
  "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",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Information and Cryptography"
    ]
  ],
  "openalex": "W7172012217",
  "pdf_url": "https://arxiv.org/pdf/2607.27719",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/113tj96gcxxbn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:41Z",
      "native_id": "2607.27719",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.27719"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Phase Retrieval in $\\mathbb C^4$ Requires Exactly Eleven Measurements",
  "title_text": "Phase Retrieval in ℂ^4 Requires Exactly Eleven Measurements",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/113tj96gcxxbn",
  "venue": "arXiv",
  "versions": []
}