  We compute the entanglement entropy, mutual information and quantum discord
of the Cosmic Microwave Background (CMB) fluctuations in real space. To that
end, we first show that measurements of these fluctuations at two distinct
spatial locations can be described by a bipartite, continuous Gaussian system.
This leads to explicit formulas for the mutual information and the quantum
discord in terms of the Fourier-space power spectra of the curvature
perturbation. We then find that quantum entanglement, that builds up in Fourier
space between opposite wave momenta as an effect of quantum squeezing, is
transferred to real space. In particular, both the mutual information and
quantum discord, which decay as the fourth power of the distance between the
two measurements in flat space time, asymptotes a constant in cosmological
backgrounds. At the scales probed in the CMB however, they are highly
suppressed, while they can reach order-one values at much smaller scales, where
primordial black holes could have formed.
