The minimum number of quadratic measurements for generalized phase retrieval
Let $\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}$ and $d \ge 2$. Self-adjoint matrices $A_1, \dots, A_m \in \mathbb{F}^{d \times d}$, of any rank and not necessarily positive semidefinite, are phase retrievable if for all $x, y \in \mathbb{F}^d$, $$x^* A_j x = y^* A_j y \ \ (j = 1, \dots, m) \quad \Longrightarrow \quad x = cy \ \text{ for some } c \in \mathbb{F} \text{ with } |c| = 1,$$ and $m_{\mathbb{F}}(d)$ is the least $m$ for which such matrices exist. Wang and Xu showed that $m_{\mathbb{R}}(d) \le 2d - 1$ for odd $d$ and $m_{\mathbb{R}}(d) \le 2d - 2$ for even $d$, with equality when $d = 2^k + 1$ and when $d = 2^k + 2$ ($k \ge 1$). They also showed $m_{\mathbb{C}}(2) = 3$ and found $m_{\mathbb{C}}(d)$ for several families of $d$, for instance $m_{\mathbb{C}}(d) = 4d - 4$ for $d = 2^k + 1$ with $k \ge 2$. Apart from $m_{\mathbb{C}}(2) \ge 3$, their lower bounds come from results on embeddings of projective spaces in Euclidean spaces.
Problem. Determine $m_{\mathbb{R}}(d)$ and $m_{\mathbb{C}}(d)$ exactly for every $d \ge 2$.
Source: Wang and Xu (2019), Section 5, which asks for the minimal numbers m_ℝ(d) and m_ℂ(d); posed as open problems by Xu (2026), Problems 4.1 and 4.3.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- AT, NA
- Keywords
- OpenProblemsInNLA, phase retrieval, projective spaces, quadratic measurements
- Literature checked
- Oct. 8, 2026 by clemens-admin
References
@article{wang2019,
author = {Wang, Yang and Xu, Zhiqiang},
title = {Generalized phase retrieval: Measurement number, matrix recovery and beyond},
journal = {Appl. Comput. Harmon. Anal.},
volume = {47},
number = {2},
year = {2019},
pages = {423--446},
doi = {10.1016/j.acha.2017.09.003},
eprint = {1605.08034},
archivePrefix = {arXiv},
}
@article{xu2026,
author = {Xu, Zhiqiang},
title = {Signal Recovery on Algebraic Varieties Using Linear Samples},
journal = {Acta Math. Sin. (Engl. Ser.)},
volume = {42},
number = {3},
year = {2026},
pages = {740--754},
doi = {10.1007/s10114-026-5299-y},
eprint = {2506.17572},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-fr11,
author = {Townsend, Alex},
title = {{FR-11}: The minimum number of quadratic measurements for generalized phase retrieval},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/frames-and-matrix-designs/FR-11/README.md},
note = {Read at commit eb7c9d6, 2026-10-07},
}
Resolution claims
The note gives twelve explicitly specified integral symmetric $7\times7$ matrices whose quadratic measurements distinguish every vector in $\mathbb{R}^7$ up to a global sign, including the zero vector. Every matrix entry has absolute value at most $144$. The construction restricts multiplication of complex cubic polynomials to a real hyperplane and then takes a real-linear quotient with a two-dimensional kernel. Nonsingularity is …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-08 | Open → Partially resolved | The registry's curators | An explicit twelve-measurement construction in real dimension seven v1 recorded from OpenProblemsInNLA @ eb7c9d6, manuscripts/twelve_measurements.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00200, "The minimum number of quadratic measurements for generalized phase retrieval", clopen.solutions.