  Phase retrieval refers to the problem of reconstructing an unknown vector
$x_0 \in \mathbb{C}^n$ or $x_0 \in \mathbb{R}^n $ from $m$ measurements of the
form $y_i = \big\vert \langle \xi^{\left(i\right)}, x_0 \rangle \big\vert^2 $,
where $ \left\{ \xi^{\left(i\right)} \right\}^m_{i=1} \subset \mathbb{C}^m $
are known measurement vectors. While Gaussian measurements allow for recovery
of arbitrary signals provided the number of measurements scales at least
linearly in the number of dimensions, it has been shown that ambiguities may
arise for certain other classes of measurements $ \left\{ \xi^{\left(i\right)}
\right\}^{m}_{i=1}$ such as Bernoulli measurements or Fourier measurements. In
this paper, we will prove that even when a subgaussian vector $
\xi^{\left(i\right)} \in \mathbb{C}^m $ does not fulfill a small-ball
probability assumption, the PhaseLift method is still able to reconstruct a
large class of signals $x_0 \in \mathbb{R}^n$ from the measurements. This
extends recent work by Krahmer and Liu from the real-valued to the
complex-valued case. However, our proof strategy is quite different and we
expect some of the new proof ideas to be useful in several other measurement
scenarios as well. We then extend our results $x_0 \in \mathbb{C}^n $ up to an
additional assumption which, as we show, is necessary.
