  We investigate the power spectrum of the distortion of Cosmic Microwave
Background (CMB) due to the decay of the primordial magnetic fields. It is
known that there are two-types of the CMB distortions, so-called \mu- and
y-types and we find that the signal of the y-type distortion becomes larger
than that of the \mu-type one. We also discuss cross power spectra between the
CMB distortions and the CMB temperature anisotropy, which are naturally
generated due to the existence of the primordial magnetic fields. We find that
such cross power spectra have small amplitudes compared with the auto-power
spectra of the CMB distortions because of the Silk damping effect of the
temperature anisotropy. We also investigate the possibility of detecting such
signal in the future CMB experiments, including not only absolutely calibrated
experiments such as PIXIE but also relatively calibrated experiments such as
LiteBIRD and CMBpol.
