In recent years , high-resolution cosmic microwave background ( CMB ) measurements have opened up the possibility to explore statistical features of the temperature fluctuations down to very small angular scales .
One method that has been used is the Wehrl entropy , which is , however , extremely costly in terms of computational time .
Here , we propose several different pseudoentropy measures ( projection , angular , and quadratic ) that agree well with the Wehrl entropy , but are significantly faster to compute .
All of the presented alternatives are rotationally invariant measures of entanglement after identifying each multipole l of temperature fluctuations with a spin- l quantum state and are very sensitive to non-Gaussianity , anisotropy , and statistical dependence of spherical harmonic coefficients in the data .
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 one-dimensional functions of the squared chordal distance of multipole vectors , giving a tight connection between the two measures .
We also show that the angular pseudoentropy can clearly 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 , Planck 2018 , and WMAP 7-yr full sky data .
We find that l = 5 and l = 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 violation of isotropy or Gaussianity .
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 , i.e. , , how likely it is to find ranges of a certain size with the observed features , and inhomogeneous noise is left as an open question .
Furthermore , 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 , which measures averages and correlations of different multipoles .
Our main purpose in this work is to introduce the methods , analyze their mathematical background , and demonstrate their usage for providing researchers in this field with an additional tool .
We believe that the formalism developed here can underpin future studies of the Gaussianity and isotropy of the CMB and help to identify deviations , especially at small angular scales .