  Motivated by the importance of optical microscopes to science and
engineering, scientists have pondered for centuries how to improve their
resolution and the existence of fundamental resolution limits. In recent years,
a new class of microscopes that overcome a long-held belief about the
resolution have revolutionized biological imaging. Termed "superresolution"
microscopy, these techniques work by accurately locating optical point sources
from far field. To investigate the fundamental localization limits, here I
derive quantum lower bounds on the error of locating point sources in free
space, taking full account of the quantum, nonparaxial, and vectoral nature of
photons. These bounds are valid for any measurement technique, as long as it
obeys quantum mechanics, and serve as general no-go theorems for the resolution
of microscopes. To arrive at analytic results, I focus mainly on the cases of
one and two classical monochromatic sources with an initial vacuum optical
state. For one source, a lower bound on the root-mean-square position
estimation error is on the order of $\lambda_0/\sqrt{N}$, where $\lambda_0$ is
the free-space wavelength and $N$ is the average number of radiated photons.
For two sources, owing to the statistical effect of nuisance parameters, the
error bound diverges when their radiated fields overlap significantly. The use
of squeezed light to enhance further the accuracy of locating one classical
point source and the localization limits for partially coherent sources and
single-photon sources are also discussed. The presented theory establishes a
rigorous quantum statiscal inference framework for the study of superresolution
microscopy and points to the possibility of using quantum techniques for true
resolution enhancement.
