  In this short note, we consider the worst case noise robustness of any phase
retrieval algorithm which aims to reconstruct all nonvanishing vectors
$\mathbf{x} \in \mathbb{C}^d$ (up to a single global phase multiple) from the
magnitudes of an arbitrary collection of local correlation measurements.
Examples of such measurements include both spectrogram measurements of
$\mathbf{x}$ using locally supported windows and masked Fourier transform
intensity measurements of $\mathbf{x}$ using bandlimited masks. As a result,
the robustness results considered herein apply to a wide range of both
ptychographic and Fourier ptychographic imaging scenarios. In particular, the
main results imply that the accurate recovery of high-resolution images of
extremely large samples using highly localized probes is likely to require an
extremely large number of measurements in order to be robust to worst case
measurement noise, independent of the recovery algorithm employed. Furthermore,
recent pushes to achieve high-speed and high-resolution ptychographic imaging
of integrated circuits for process verification and failure analysis will
likely need to carefully balance probe design (e.g., their effective
time-frequency support) against the total number of measurements acquired in
order for their imaging techniques to be stable to measurement noise, no matter
what reconstruction algorithms are applied.
