The Minimal Measurement Number for Almost-Everywhere Complex Phase Retrieval

2026-08-14

AI Summary

This study addresses the minimum number of measurements required for almost everywhere unique reconstruction in complex phase retrieval by integrating algebraic geometry and measure theory. The analysis demonstrates that 2d−1 measurements are insufficient to guarantee uniqueness and establishes, for the first time, that 2d constitutes the exact lower bound for almost everywhere unique recovery in complex spaces. By resolving a longstanding controversy regarding measurement thresholds, this work definitively characterizes the fundamental identifiability limits of complex phase retrieval. Consequently, these findings provide a rigorous mathematical foundation and theoretical support for the design of related algorithms and system architectures, thereby clarifying the precise conditions necessary for successful signal recovery in this domain.

Abstract

Let $d\geq2$ and let $\bm{f}_1,\ldots,\bm{f}_m\in\mathbb C^d$. We prove that if \(m\leq 2d-1\), then the intensity measurement map 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 minimal number of measurements required for almost-everywhere phase retrieval in $\mathbb C^d$ is exactly $2d$.

Problem

• Complex Phase Retrieval
• Minimal Measurement Number
• Almost-Everywhere Recovery

Innovation

• Complex Phase Retrieval
• Minimal Measurement Number
• Almost-Everywhere Recovery
• Intensity Measurements