Layers of classicality in the compatibility of measurements

The term "Layers of classicality" in the context of quantum measurements, was introduced in [T. Heinosaari, Phys. Rev. A 93, 042118 (2016)]. The strongest layer among these consists of the sets of observables that can be broadcast and the weakest layer consists of the sets of compatible observables. There are several other layers in between those two layers. In this work, we study their physical and geometric properties and show the differences and similarities among the layers in these properties. In particular we show that: (i) none of the layers of classicality respect transitivity property, (ii) the concept like degree of broadcasting similar to degree of compatibility does not exist, (iii) there exist informationally incomplete POVMs that are not individually broadcastable, (iv) a set of broadcasting channels can be obtained through concatenation of broadcasting and non-disturbing channels, (v) unlike compatibility, other layers of classicality are not convex, in general. Finally, we discuss the relations among these layers. More specifically, we show that specific type of concatenation relations among broadcasting channels decide the layer in which a pair of observables resides.