{
  "abstract": "Let $d\\geq 2$ and let $\\bf{f}_1,\\ldots,\\bf{f}_m\\in\\mathbb C^d$. We prove that if $m\\leq 2d-1$, then the intensity measurement map \\[ \\bf{x}\\longmapsto \\bigl( |\\langle \\bf{x},\\bf{f}_1\\rangle|^2, \\ldots, |\\langle \\bf{x},\\bf{f}_m\\rangle|^2 \\bigr) \\] fails to recover almost every signal in $\\mathbb C^d$ uniquely up to a global phase factor. Combined with the known generic sufficiency of $2d$ measurements, our result establishes that the minimum number of measurements required for almost-everywhere phase retrieval in $\\mathbb C^d$ is exactly $2d$. This resolves an open problem in phase retrieval by determining the exact measurement threshold for almost-everywhere phase retrieval in ${\\mathbb C}^d$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.15003",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhiqiang Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7g42qw686cdpv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.15003",
  "fulltext_url": null,
  "id": "7g42qw686cdpv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.15003",
  "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",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.IT"
    ],
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      "arxiv",
      "math.NA"
    ],
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      "openalex",
      "2206|Sparse and Compressive Sensing Techniques"
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  "openalex": "W7203707638",
  "pdf_url": "https://arxiv.org/pdf/2608.15003",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7g42qw686cdpv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.15003",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.15003"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "The Minimum Number of Measurements for Almost-Everywhere Complex Phase Retrieval",
  "title_text": "The Minimum Number of Measurements for Almost-Everywhere Complex Phase Retrieval",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/7g42qw686cdpv",
  "venue": "arXiv",
  "versions": []
}