# What is the 'area law' in the context of matrix product states?

I am trying to get into the topic of matrix product states by reading this:

A practical introduction to tensor networks: Matrix product states and projected entangled pair states. R. Orús. Ann. Phys. 349, 117 (2014), arXiv:1306.2164.

There, often, the word "area-law" is mentioned, but it's not very well explained what is meant by that... It's somehow that states in a Hilbert space are entangled with the neighbored states. (is that right?) But why? And what is meant by a local Hamiltonian?

## 1 Answer

The area law says that the entanglement of any part of a system with the rest of of the system scales like the boundary (the "surface area") of the region. E.g., in a one-dimensional chain, the entanglement of a contiguous block with the rest should be bounded by a constant, and in 2D, the entanglement of e.g. a square region with the rest should scale like the linear size of this square.

The area law is a property which is proven to be satisfied by ground states of local gapped Hamiltonians in one dimension (see arXiv:0705.2024 and arXiv:1301.1162), and the corresponding statement is believed to be true in two dimensions. However, even for systems without a gap the area law is only mildly violated (in that the entanglement does not grow like the volume).

Local Hamiltonian refers to the fact that the Hamiltonian is a sum of terms each of which only acts on a small number of closeby spins, e.g. nearest neighbors on a lattice.

• Thanks! Could you suggest some papers where this statement has been proven? (Additional question: This is only true for low-energy states, isn't it?) – QuantumMechanics Oct 15 '15 at 13:22
• @QuantumMechanics I've added some references. – Norbert Schuch Oct 15 '15 at 13:28