Extremal generalized quantum measurements

A measurement on a section K   of the set of states of a finite dimensional C⁎C⁎-algebra is defined as an affine map from K to a probability simplex. Special cases of such sections are used in description of quantum networks, in particular quantum channels. Measurements on a section correspond to equivalence classes of the so-called generalized POVMs, which are called quantum testers in the case of networks. We find extremality conditions for measurements on K and characterize generalized POVMs such that the corresponding measurement is extremal. These results are applied to the set of channels. We find explicit extremality conditions for two outcome measurements on qubit channels and give an example of an extremal qubit 1-tester such that the corresponding measurement is not extremal.