Originating from the works of Bekenstein and Hawking on the entropy of black holes, area laws constitute a central tool for understanding entanglement and locality properties in many-body quantum systems. Essentially, in a system that obeys an area law, the entanglement entropy of a bounded region scales like its boundary area, rather than its volume.

In 2007 Hastings proved that all 1D quantum spin systems with a constant spectral gap obey an area law in their ground state. A major open problem is whether an area law holds also in two or more dimensions.

In this talk I will present a new proof for the 1D area law, which gives an exponentially smaller bound on the entanglement entropy, and is within a polynomial factor from the best known lower bounds. The proof also implies that the ground state can be well-approximated by an MPS with sublinear bond dimension. I will talk about the main mathematical ideas behind the proof, as well as the challanges that one faces when trying to prove the 2D case.

Joint work with A. Kitaev, Z. Landau and U. Vazirani