Entropy plays a fundamental role in quantum information theory through applications ranging from communication theory to condensed matter physics. These applications include finding the best possible communication rates over noisy channels and characterizing ground state entanglement in strongly-correlated quantum systems. In the latter, localized entanglement is often characterized by an area law for entropy. Long-range entanglement, on the other hand, can give rise to topologically ordered materials whose collective excitations are robust against local noise. In this talk, I will present a property of quantum entropy for multipartite quantum systems that resolves several open questions in quantum information theory about entanglement measures, provides new algorithmic opportunities and makes nontrivial statements about the structure of states with vanishing - but nonzero - topological entropy. I will also comment how extensions of this work could help our understanding of quantum communication over certain very noisy channels.