The Minimum Number of Measurements for Almost-Everywhere Complex Phase Retrieval

Abstract

Let $d\ge 2$ and let ${\mathbf{f}}_1,\dots ,{\mathbf{f}}_{\mathbf{m}}\in {\mathbb{C}}^{\mathbf{d}}$. We prove that if $m\le 2d-1$, then the intensity measurement map \[ \bf{x}\longmapsto \bigl( |\langle \bf{x},\bf{f}_1\rangle|^2, \ldots, |\langle \bf{x},\bf{f}_m\rangle|^2 \bigr) \] 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 minimum number of measurements required for almost-everywhere phase retrieval in ${\mathbb{C}}^d$ is exactly $2d$. This resolves an open problem in phase retrieval by determining the exact measurement threshold for almost-everywhere phase retrieval in ${\mathbb{C}}^d$.

7 pages