  An ideal direct imaging coronagraph, which selectively rejects the
fundamental mode of a telescope, has been shown to achieve the quantum
information limits for exoplanet detection and localization. In this study, we
experimentally implement this quantum-optimal coronagraph using spatial mode
(de)multiplexing. Our benchtop system includes a forward and inverse pass
through a free-space programmable spatial mode sorter, designed to isolate
photons in a point spread function (PSF)-adapted basis. During the forward
pass, the fundamental mode is rejected, effectively eliminating light from an
on-axis point-like star. On the inverse pass, the remaining modes are
coherently recombined, enabling direct imaging of a faint companion. We develop
a probabilistic measurement model that accounts for combined effects of
fundamental shot noise and experimental noise specific to our benchtop setup,
such as modal cross-talk, dark noise, and ambient background illumination. We
leverage this measurement model to formulate a maximum-likelihood estimator of
the exoplanet position given an image captured with the coronagraph. Using this
approach, we successfully localize an artificial exoplanet at sub-diffraction
distances $(<\sigma)$ from its host star under a 1000:1 star-planet contrast
ratio. Our system accurately localizes the exoplanet up to an absolute error
$<0.03\sigma$ over the separation range $[0,\,0.6]\sigma$. Finally, we
numerically evaluate the precision of our experimental coronagraph against
state-of-the-art coronagraphs subject to comparable noise models.
