Entanglement entropy in de Sitter spacetime

We study the dynamics of the entanglement entropy in higher-dimensional de Sitter spacetime. We find that the basic properties of the entanglement entropy of pure states are not complied in both one- and two-sided de Sitter spaces. We specify several types of Cauchy surfaces in de Sitter and discover that the entropies of pure states defined on them are either non-zero or diverge in the infrared depending on what patch we are in. We discover that the well-known complementarity property, which says that the entropies of a region and its complement that combine together into a Cauchy surface should be equal for the case of a pure state, is not obeyed.