CMB lensing reconstruction using cut sky polarization maps and pure-$B$
  modes
  Detailed measurements of the CMB lensing signal are an important scientific
goal of ongoing ground-based CMB polarization experiments, which are mapping
the CMB at high resolution over small patches of the sky. In this work we
simulate CMB polarization lensing reconstruction for the $EE$ and $EB$
quadratic estimators with current-generation noise levels and resolution, and
show that without boundary effects the known and expected zeroth and first
order $N^{(0)}$ and $N^{(1)}$ biases provide an adequate model for non-signal
contributions to the lensing power spectrum estimators. Small sky areas present
a number of additional challenges for polarization lensing reconstruction,
including leakage of $E$ modes into $B$ modes. We show how simple windowed
estimators using filtered pure-$B$ modes can greatly reduce the mask-induced
mean-field lensing signal and reduce variance in the estimators. This provides
a simple method (used with recent observations) that gives an alternative to
more optimal but expensive inverse-variance filtering.
