the real SPIRE map, and $R'_2=R_2-C$ the simulated SPIRE map. We have checked that the cross-correlation result (calculated using Eq. (5)) as well as its error does not depend on the value of $C$ at all. Although this is a reasonable choice of estimator, we have not investigated whether it is optimal.

The simulated SPIRE are constructed by randomising the pixels of the real map. It allows us to take into account of the effect of not only instrumental noise, but also confusion noise and cirrus. The random quasar samples are generated according to the angular completeness maps in the overlapping area between the SDSS DR7 and SDSS-III DR9 quasar sample and the *Herschel* surveys. As mentioned before, we assume the angular completeness fraction is uniform for the DR7 quasars. So, we simply generate uniformly distribution points as the random DR7 quasar sample. Admittedly, the real angular completeness level for the DR7 quasars might vary across our fields. However, as what we are concerned with in this paper is the cross-correlation signal between the quasars and the CIB measured by [*H*erschel]{}-SPIRE, the variation in the DR7 QSO completeness is not expected to bias our results as long as the completeness 