Locality of Edge States and Entanglement Spectrum from Strong Subadditivity

We consider two-dimensional statesof mattersatisfying an uniform area law for entanglement. We show that the topological entanglement entropy is equal to the minimum relative entropy distance of the edge state of the system to the set of thermal states of local models. The argument is based on strong subadditivityof quantum entropy. For states with zero topological entanglement entropy, in particular, the formula gives locality of the edge states as thermal states of local Hamiltonians. It also implies that the entanglement spectrum of a region is equal to the spectrum of a one-dimensional local thermal state on the boundary of the region. Our result gives a precise information-theoretic interpretation for topological entanglement entropy as the number of bits of information needed to describe the non-local degrees of freedom of edge states.