Real-space entanglement in the Cosmic Microwave Background

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.