  Estimating the cross-correlation power spectra of cosmic microwave background
(CMB), in particular, the T B and EB spectra, is important for testing parity
symmetry in cosmology and diagnosing insidious instruments systematics. The
Quadratic Maximum Likelihood (QML) estimator provides the optimal estimates of
power spectra, but it is computationally very expensive. The hybrid pseudo-Cl
estimator is computationally fast but performs poorly on large scales. As a
natural extension of previous work (Chen et al. 2021), in this article, we
present a new unbiased estimator based on the Smith-Zaldarriaga (SZ) approach
of E-B separation and scalar QML approach to reconstruct the cross-correlation
power spectrum, called QML-SZ estimator. Our new estimator relies on the
ability to construct scalar maps, which allows us to use a scalar QML estimator
to obtain the cross-correlation power spectrum. By reducing the pixel number
and algorithm complexity, the computational cost is nearly one order of
magnitude smaller and the running time is nearly two orders of magnitude faster
in the test situations.
