Ryu and Takayanagi have conjectured a simple formula for the entanglement entropy of an arbitrary spatial region in an arbitrary holographic field theory. This conjecture, which remains unproven, makes a number of interesting predictions. A rather dramatic one is that the mutual information between separated regions drops to zero at a finite separation. We will present independent evidence for this prediction in two-dimensional conformal field theories, based on calculations of entanglement Renyi entropies using both holographic and CFT techniques. The necessary background material from quantum information theory will be explained along the way.