  We consider the problem of recovering a complex vector $\mathbf{x}\in
\mathbb{C}^n$ from $m$ quadratic measurements $\{\langle A_i\mathbf{x},
\mathbf{x}\rangle\}_{i=1}^m$. This problem, known as quadratic feasibility,
encompasses the well known phase retrieval problem and has applications in a
wide range of important areas including power system state estimation and x-ray
crystallography. In general, not only is the the quadratic feasibility problem
NP-hard to solve, but it may in fact be unidentifiable. In this paper, we
establish conditions under which this problem becomes {identifiable}, and
further prove isometry properties in the case when the matrices
$\{A_i\}_{i=1}^m$ are Hermitian matrices sampled from a complex Gaussian
distribution. Moreover, we explore a nonconvex {optimization} formulation of
this problem, and establish salient features of the associated optimization
landscape that enables gradient algorithms with an arbitrary initialization to
converge to a \emph{globally optimal} point with a high probability. Our
results also reveal sample complexity requirements for successfully identifying
a feasible solution in these contexts.
