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Part of the CMB polarization signal in the direction of galaxy clusters isproduced by Thomson scattering of the CMB temperature quadrupole. In principlethis allows measurement of the CMB power spectrum harmonic $C_2(z)$ with higheraccuracy (at $z>0$) than the cosmic variance limit imposed by sample varianceon one CMB sky. However the observed signals are statistically correlated ifthe comoving separation between the clusters is small enough. Thus one cannotreduce the sample variance by more than roughly the number of separate regionsavailable which produce uncorrelated signals, as first pointed out byKamionkowski and Loeb. In this paper we analyze in detail the procedureoutlined by Kamionkowski and Loeb, computing the correlation of thepolarization signals by considering the variation of the spherical harmonicexpansion coefficients of the temperature anisotropy on our past light cone. Wefind that in fact the cosmic variance is not reducible below the single sky CMBvalue using the cluster method. Thus this method is not likely to be of use forreconstruction of the primordial power spectrum. However the method does yielda measurement of $C_2$ as a function of redshift with increasing accuracy athigher redshift, and thus potentially a probe of the mechanism which may havesuppressed the quadrupole. (abridged)Comment: 21 pages, 7 figures; replaced with version accepted by Phys.Rev.D,  title change

Analysis of the Kamionkowski-Loeb method of reducing cosmic variance with CMB polarization