Observational entropy of quantum correlations and entanglement

The use of coarse graining to connect physical and information theoretic entropies has recently been given a precise formulation in terms of ``observational entropy'', describing entropy for observers with respect to a measurement. Here we consider observers with various locality restrictions, including local measurements (LO), measurements based on local operations with classical communication (LOCC), and separable measurements (SEP), with the idea that the ``entropy gap'' between the minimum locally measured observational entropy and the von Neumann entropy quantifies quantum correlations in a given state. After introducing entropy gaps for general classes of measurements and deriving their general properties, we specialize to LO, LOCC, SEP and other measurement classes related to the locality of subsystems. For those, we show that the entropy gap can be related to well-known measures of entanglement or non-classicality of the state (even though we point out that they are not entanglement monotones themselves). In particular, for bipartite pure states, all of the ``local'' entropy gaps reproduce the entanglement entropy, and for general multipartite states they are lower-bounded by the relative entropy of entanglement. The entropy gaps of the different measurement classes are ordered, and we show that in general (mixed and multipartite states) they are all different.