  In recent years, weak lensing of the cosmic microwave background (CMB) has
emerged as a powerful tool to probe fundamental physics, such as neutrino
masses, primordial non-Gaussianity, dark energy, and modified gravity. The
prime target of CMB lensing surveys is the lensing potential, which is
reconstructed from observed CMB temperature $T$ and polarization $E$ and $B$
fields. Until very recently, this reconstruction has been performed with
quadratic estimators (QEs), which, although known to be suboptimal for
high-sensitivity experiments, are numerically efficient, and useful to make
forecasts and cross-check the results of more sophisticated likelihood-based
methods. It is expected that ongoing and near-future CMB experiments such as
AdvACT, SPT-3G and the Simons Observatory (SO), will also rely on QEs. Here, we
review different QEs, and clarify their differences. In particular, we show
that the Hu-Okamoto (HO02) estimator is not the absolute optimal lensing
estimator that can be constructed out of quadratic combinations of $T, E$ and
$B$ fields. Instead, we derive the global-minimum-variance (GMV) lensing
quadratic estimator. Although this estimator can be found elsewhere in the
literature, it was erroneously described as equivalent to the HO02 estimator,
and has never been used in real data analyses. Here, we show explicitly that
the HO02 estimator is suboptimal to the GMV estimator, with a reconstruction
noise larger by up to $\sim 9\%$ for a SO-like experiment. We further show that
the QE used in the Planck, and recent SPT lensing analysis are suboptimal to
both the HO02 and GMV estimator, and would have a reconstruction noise up to
$\sim 11\%$ larger than that of the GMV estimator for a SO-like experiment. In
addition to clarifying differences between different QEs, this work should thus
provide motivation to implement the GMV estimator in future lensing analyses
relying on QEs.
