  Assuming the Bousso bound, we prove a singularity theorem: if the light rays
entering a hyperentropic region contract, then at least one light ray must be
incomplete. "Hyperentropic" means that the entropy of the region exceeds the
Bekenstein-Hawking entropy of its spatial boundary. Our theorem provides a
direct link between singularities and quantum information. The hyperentropic
condition replaces the noncompactness assumption in Penrose's theorem, so our
theorem is applicable even in a closed universe. In an asymptotically de Sitter
spacetime, for example, a big bang singularity can be diagnosed from the
presence of dilute radiation at arbitrarily late times. In asymptotically flat
space, Penrose's theorem can be recovered by adding soft radiation.
