Quantum Colloquium — The Area-Law Conjecture In Many-Body Quantum Physics

There is an inherent contradiction between the exponential complexity of quantum states and the possibility of studying quantum many-body physics. One of the important principles that makes quantum many body physics tractable is that there are large classes of physically occurring systems for which ground and low energy states are conjectured to have succinct classical descriptions. This is of course quite relevant to the issue of simulation of quantum systems. Formally this principle is encapsulated in the area-law of the entanglement entropy: the entanglement of a contiguous region with respect to the rest of the system scales like its boundary area, rather than its volume. It is conjectured that ground states of gapped spin systems — those that have a gap between the ground energy and the first excited level — satisfy an area-law, and might therefore be easier to simulate. So far, the conjecture has only been fully proven for one dimensional systems (Hastings' 2007). However, a sequence of improvements of Hastings' result in 1D has recently led to a breakthrough towards a possible proof of the 2D case. In this talk I will introduce this conjecture and give an overview of the research that led to the recent breakthroughs, which includes ideas that span Computer Science, Math and Physics.