Entropy Bounds and Entanglement

The generalized covariant entropy bound, or Bousso bound, is a holographic bound on the entropy of a region of space in a gravitational theory. It bounds the entropy passing through certain null surfaces. The bound remains nontrivial in the weak-gravity limit, and provides non-trivial constraints on the entropy of ordinary quantum states even in a regime where gravity is negligible. In the first half of this thesis, we present a proof of the Bousso bound in the weak-gravity regime within the framework of quantum field theory. The bound uses techniques from quantum information theory which relate the energy and entropy of quantum states. We present two proofs of the bound in free and interacting field theory. In the second half, we present a generalization of the Bousso bound called the quantum focussing conjecture. Our conjecture is a bound on the rate of entropy generation in a quantum field theory coupled semiclassically to gravity. The conjecture unifies and generalizes several ideas in holography. In particular, the quantum focussing conjecture implies a bound on entropies which is similar to, but subtly different from, the Bousso bound proven in the first half. The quantum focussing conjecture implies a novel non-gravitational energy condition, the quantum null energy condition, which gives a point-wise lower bound on the null-null component of the stress tensor of quantum matter. We give a proof of this bound in the context of free and superrenormalizable bosonic quantum field theory