  Compensated isocurvature perturbations are opposite spatial fluctuations in
the baryon and dark matter (DM) densities. They arise in the curvaton model and
some models of baryogenesis. While the gravitational effects of baryon
fluctuations are compensated by those of DM, leaving no observable impacts on
the cosmic microwave background (CMB) at first order, they modulate the sound
horizon at recombination, thereby correlating CMB anisotropies at different
multipoles. As a result, CIPs can be reconstructed using quadratic estimators
similarly to CMB detection of gravitational lensing. Because of these
similarities, however, the CIP estimators are biased with lensing contributions
that must be subtracted. These lensing contributions for CMB polarization
measurement of CIPs are found to roughly triple the noise power of the total
CIP estimator on large scales. In addition, the cross power with temperature
and $E$-mode polarization are contaminated by lensing-ISW (integrated
Sachs-Wolfe) correlations and reionization-lensing correlations respectively.
For a cosmic-variance-limited (CVL) temperature and polarization experiment
measuring out to multipoles $l_{\max} = 2500$, the lensing noise raises the
detection threshold by a factor of 1.5, leaving a $2.7\sigma$ detection
possible for the maximal CIP signal in the curvaton model.
