  A finite set of quantum observables (positive operator valued measures) is
called compatible if these observables are marginals of a some observable,
called a joint observable of them. For a given set of compatible observables,
their joint observable is in general not unique and it is desirable to take a
minimal joint observable in the post-processing order since a less informative
observable disturbs less the system. We address the question of the minimality
of finite-outcome joint observables and prove that any joint observable is
lower bounded by a minimal joint observable in the post-processing order. We
also give characterizations of the minimality of a joint observable that can be
checked by finite-step algorithms and apply them to the case of non-commuting
dichotomic qubit observables.
