  Generalized quantum instruments correspond to measurements where the input
and output are either states or more generally quantum circuits. These
measurements describe any quantum protocol including games, communications, and
algorithms. The set of generalized quantum instruments with a given input and
output structure is a convex set. Here we investigate the extremal points of
this set for the case of finite dimensional quantum systems and generalized
instruments with finitely many outcomes. We derive algebraic necessary and
sufficient conditions for extremality.
