  We propose several pseudoentropy measures that agree well with the Wehrl
entropy, but are significantly faster to compute. All of them are rotationally
invariant measures of entanglement very sensitive to non-Gaussianity,
anisotropy, and statistical dependence of spherical harmonic coefficients. We
provide a simple proof that the projection pseudoentropy converges to the Wehrl
entropy with increasing dimensionality of the ancilla projection space.
Furthermore, for $l=2$, we show that both the Wehrl entropy and the angular
pseudoentropy can be expressed as functions of the squared chordal distance of
multipole vectors. We also show that the angular pseudoentropy can distinguish
between Gaussian and non-Gaussian temperature fluctuations at large multipoles
and henceforth provides a non-brute-force method for identifying
non-Gaussianities. This allows us to study possible hints of statistical
anisotropy and non-Gaussianity in the CMB up to multipole $l=1000$ using Planck
2015/2018, and WMAP 7-yr data. We find that $l=5$ and $28$ have a large entropy
at $2$--$3\sigma$ significance and a slight hint towards a connection of this
with the cosmic dipole. On a wider range of large angular scales we do not find
indications of isotropy/Gaussianity violation. We also find a small-scale
range, $l\in[895,905]$, that is incompatible with the assumptions at about
$3\sigma$ level, although how much this significance can be reduced by taking
into account the selection effect is left as an open question. We find overall
similar results in our analysis of the 2015 and the 2018 data. Finally, we also
demonstrate how a range of angular momenta can be studied with the range
angular pseudoentropy. Our main purpose is to introduce the methods, analyze
their mathematical background, and demonstrate their usage for providing
researchers in this field with an additional tool.
