Reduction and extremality of finite observables

We investigate the theory of finite observables, i.e., resolutions of the finite-dimensional identity by means of positive operators, that have a physical interpretation in terms of measurement schemes. We focus on extremal and rank-one observables and consider various constructions that reduce observables to simpler ones. However, these constructions do not suffice to generate all finite extremal observables, as we show by means of counter-examples.