De-lensing cosmic birefringence with joint reconstruction

Reconstruction of anisotropic cosmic birefringence from the CMB is limited by lensing $B$-mode sample variance and partially degenerate with the lensing curl. We study the joint reconstruction of lensing (gradient and curl) and birefringence at the maximum-a-posteriori point, whose spectra are near-optimal statistics. We predict the joint response and noise analytically, validate them on simulations, and show unbiased inference of its amplitude. Errors will shrink $5$-fold relative to quadratic estimators at SPT-3G+ depths, promising much tighter constraints on parity-violating new physics, owing to the removal of the lensing $B$-polarization sample variance.