Phase retrieval by projections (Q2822576)

This paper gives answers or makes incremental steps toward the following questions: {\parindent=6mm NEWLINE\begin{itemize}\item[(1)] Can families of subspaces for which such measurements are injective be completely classified?NEWLINE\item[(2)] What is the minimal number of subspaces required to have injectivity?NEWLINE\item[(3)] How closely does this problem compare to the usual phase retrieval problem with families of measurement vectors? NEWLINENEWLINE\end{itemize}}NEWLINENEWLINE``Several characterizations of subspaces which yield injective measurements, and through a concrete construction'' are provided and ``the surprising result that phase retrieval can be achieved with \(2M-1\) projections of arbitrary rank in \(M\)-dimensional Hilbert spaces'' is proved. Several open problems are finally presented and discussed regarding issues unique to the phase retrieval problem with subspaces.