  Phase retrieval and phaseless reconstruction for Hilbert space frames is a
very active area of research. Recently, it was shown that these concepts are
equivalent. In this paper, we make a detailed study of a weakening of these
concepts to weak phase retrieval and weak phaseless reconstruction. We will
give several necessary and/or sufficient conditions for frames to have these
weak properties. We will prove three surprising results: (1) Weak phaseless
reconstruction is equivalent to phaseless reconstruction. I.e. It never was
"weak"; (2) Weak phase retrieval is not equivalent to weak phaseless
reconstruction; (3) Weak phase retrieval requires at least $2m-2$ vectors in an
m-dimensional Hilbert space. We also gives several examples illustrating the
relationship between these concepts.
