The Minimal Measurement Number for Almost-Everywhere Complex Phase Retrieval

Let $d\ge 2$ and let ${\mathit{f}}_1,\dots ,{\mathit{f}}_m\in {\mathbb{C}}^d$. We prove that if $m\le 2d-1$, 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 $2d$.