Entanglement area law for gapped ground states in two dimensions
In plain words
In one dimension it is proven that ground states of systems with an energy gap hold entanglement only across region boundaries, which is why they are easy to compute. The same statement in two dimensions is widely believed but unproved.
Precise statement
For a 2D lattice Hamiltonian of nearest-neighbour terms with $\mid\mid h\mid\mid \le 1$, a unique ground state and spectral gap $\Delta>0$ uniform in system size, prove that every region $A$ satisfies $S(A)\le c(\Delta) \mid\text{boundary of } A\mid$ (possibly with logarithmic corrections), or construct a counterexample. Proved in 1D (Hastings 2007) and for 2D frustration-free models with a local gap (Anshu, Arad and Gosset 2022).
What would settle it
A proof of the 2D area law for general gapped local Hamiltonians, or a gapped 2D counterexample with volume-law or faster-than-boundary entanglement.