  Similarly to quantum states, also quantum measurements can be "mixed",
corresponding to a random choice within an ensemble of measuring apparatuses.
Such mixing is equivalent to a sort of hidden variable, which produces a noise
of purely classical nature. It is then natural to ask which apparatuses are
"indecomposable", i. e. do not correspond to any random choice of apparatuses.
This problem is interesting not only for foundations, but also for
applications, since most optimization strategies give optimal apparatuses that
are indecomposable.
  Mathematically the problem is posed describing each measuring apparatus by a
positive operator-valued measure (POVM), which gives the statistics of the
outcomes for any input state. The POVM's form a convex set, and in this
language the indecomposable apparatuses are represented by extremal points--the
analogous of "pure states" in the convex set of states. Differently from the
case of states, however, indecomposable POVM's are not necessarily rank-one, e.
g. von Neumann measurements.
  In this paper we give a complete classification of indecomposable apparatuses
(for discrete spectrum), by providing different necessary and sufficient
conditions for extremality of POVM's, along with a simple general algorithm for
the decomposition of a POVM into extremals. As an interesting application,
"informationally complete" measurements are analyzed in this respect. The
convex set of POVM's is fully characterized by determining its border in terms
of simple algebraic properties of the corresponding POVM's.
