Appendix B: The general error matrix of the E and B multipolar coefficients

 Assuming that the data treatment has removed all correlation between different directions in the sky, the matrix elements of the error matrix of the E and B multipolar coefficients are:

$$\begin{eqnarray} & <\delta a_{^E_B, lm}\,{\delta a_{^E_B, l''m''}}^*\gt = \frac{... ... {{}_{-2}Y_{lm}}^*\,{}_{2}Y_{l''m''}]({\hat{\bf n}}_k), \nonumber \end{eqnarray}$$

where $\bf{N}({\hat{\bf n}}_k)$ is the noise matrix of the Stokes parameters Q and U in the direction ${\hat{\bf n}}_k$ of pixel  k.

Acknowledgements

We thank Alex Kim for useful suggestions and for reading the manuscript, and the referee, Uros Seljak, who suggested to consider the implications on the E and B scalar parameters.