Classifying weak phase retrieval

We will give several surprising equivalences and consequences of weak phase retrieval. These results give a complete understanding of the difference between weak phase retrieval and phase retrieval. We also answer two longstanding open problems on weak phase retrieval: (1) We show that the families of weak phase retrievable frames $\{x_{i}\}_{i=1}^{m}$ in $\mathbb{R}^n$ are not dense in the family of $m$-element sets of vectors in $\mathbb{R}^n$ for all $m\ge 2n-2$; (2) We show that any frame $\{x_i\}_{i=1}^{2n-2}$ containing one or more canonical basis vectors in $\mathbb{R}^n$ cannot do weak phase retrieval. We provide numerous examples to show that the obtained results are best possible.