Stability of Phase Retrievable Frames

In this paper we study the property of phase retrievability by redundant sysems of vectors under perturbations of the frame set. Specifically we show that if a set $\fc$ of $m$ vectors in the complex Hilbert space of dimension n allows for vector reconstruction from magnitudes of its coefficients, then there is a perturbation bound $\rho$ so that any frame set within $\rho$ from $\fc$ has the same property. In particular this proves the recent construction in \cite{BH13} is stable under perturbations. By the same token we reduce the critical cardinality conjectured in \cite{BCMN13a} to proving a stability result for non phase-retrievable frames.