The minimal measurement number for generalized conjugate phase retrieval

• Published in: Science China. Mathematics Vol. 65; no. 3; pp. 655 - 664
• Main Author: Dan, Wei
• Format: Journal Article
• Language: English
• Published: Beijing Science China Press 01-03-2022; Springer Nature B.V
• Subjects: Applications of Mathematics; Conjugates; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrices (mathematics); Phase retrieval; measurement number; conjugate; 15A63; phase retrieval; 94A12; 42C15
• ISSN: 1674-7283; 1869-1862
• DOI: 10.1007/s11425-020-1757-6

The generalized conjugate phase retrieval problem aims to reconstruct a complex signal
x
∈ ℂ
n
from quadratic measurements
x
*
A
1
x
,…,
x
*
A
m
x
, where
A
1
,…,
A
m
∈ ℝ
n
×
n
are real symmetric matrices. The equivalent formulation for generalized conjugate phase retrieval along with the minimal measurement number required for accurate retrieval (up to a global phase factor as well as conjugacy) are derived in this paper. We present a set of nine vectors in ℝ
4
and prove that it is conjugate phase retrievable on ℂ
4
. This result implies the measurement number bound 4
n
− 6 is not optimal for some
n
, which confirms a conjecture in the article by Evans and Lai (2019).