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PreprintarXiv licencearXiv

Phase Retrieval in ℂ4 Requires Exactly Eleven Measurements

  • Meng Huang
  • Jihun Kim
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Abstract

Determining the minimal number of intensity measurements required for phase retrieval in ℂ4 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 ℂ4 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 ℂ4 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 ℂ4. 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].

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Cite

BibTeXRIS
BibTeX
@misc{huang2026phase,
  title = {{Phase Retrieval in $\mathbb C^4$ Requires Exactly Eleven Measurements}},
  author = {Meng Huang and Jihun Kim},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2607.27719},
  eprint = {2607.27719},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.27719},
}
RIS
TY  - UNPB
TI  - Phase Retrieval in ℂ^4 Requires Exactly Eleven Measurements
AU  - Meng Huang
AU  - Jihun Kim
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2607.27719
UR  - https://arxiv.org/abs/2607.27719
AB  - Determining the minimal number of intensity measurements required for phase retrieval in 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 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 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 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].
ER  -