An area law for one-dimensional quantum systems

We prove an area law for the entanglement entropy in gapped onedimensional 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 matrix product states 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 ' enyi entropy for su. ciently large a < 1 and implies the ability to approximate the ground state by a matrix product state.