## Entanglement

We review properties of several entanglement characteristics and study their implications, emphasizing distinctions between continuous and discrete quantities. As an example of a continuous characteristic, we look at the entanglement entropy and show that a no-gravitational collapse condition on a pure state in quantum mechanics is sufficient to exclude faster-than-area-law entropy scaling. This observation leads to an interpretation of holography as an upper bound on the realizable entropy of a region, rather than on the dimension of its Hilbert space. Studying a class of scalar field theories in arbitrary geometry, we prove that the entanglement entropy between two regions of space is proportional to the volume of the hypersurface separating the regions, and obtain a complete asymptotic expansion for the entropy. As an example of discrete entanglement characteristics, we introduce new algebraic entanglement invariants. The resulting method of classifying entangled states works for an arbitrary finite number of finite-dimensional state subspaces, and we use it to solve a large selection of cases.