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The Minimum Number of Measurements for Almost-Everywhere Complex Phase Retrieval

  • Zhiqiang Xu
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Abstract

Let d≥ 2 and let bff1,…,bffm∈ℂd. We prove that if m≤ 2d−1, then the intensity measurement map bfxlongmapsto ( |⟨ bfx,bff1⟩|2, …, |⟨ bfx,bffm⟩|2 ) fails to recover almost every signal in ℂd 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 ℂd is exactly 2d. This resolves an open problem in phase retrieval by determining the exact measurement threshold for almost-everywhere phase retrieval in ℂd.

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Cite

BibTeXRIS
BibTeX
@misc{xu2026minimum,
  title = {{The Minimum Number of Measurements for Almost-Everywhere Complex Phase Retrieval}},
  author = {Zhiqiang Xu},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.15003},
  eprint = {2608.15003},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.15003},
}
RIS
TY  - UNPB
TI  - The Minimum Number of Measurements for Almost-Everywhere Complex Phase Retrieval
AU  - Zhiqiang Xu
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.15003
UR  - https://arxiv.org/abs/2608.15003
AB  - Let d≥ 2 and let bff_1,…,bff_m∈ℂ^d. We prove that if m≤ 2d−1, then the intensity measurement map bfxlongmapsto ( |⟨ bfx,bff_1⟩|^2, …, |⟨ bfx,bff_m⟩|^2 ) fails to recover almost every signal in 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 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 ℂ^d.
ER  -