  We prove an area law for the entanglement entropy in gapped one dimensional
quantum systems. The bound on the entropy grows surprisingly rapidly with the
correlation length; we discuss this in terms of properties of quantum expanders
and present a conjecture on completely positive maps which may provide an
alternate way of arriving at an area law. We also show that, for gapped, local
systems, the bound on Von Neumann entropy implies a bound on R\'{e}nyi entropy
for sufficiently large $\alpha<1$ and implies the ability to approximate the
ground state by a matrix product state.
