Optimal covariant quantum measurements

We discuss symmetric quantum measurements and the associated covariant observables modelled, respectively, as instruments positive-operator-valued measures. The emphasis of this work are optimality properties measurements, namely, extremality, informational completeness, rank-1 property which contrast complementary class (rank-1) projection-valued first half concentrates solely on finite-outcome w.r.t. finite groups where we derive exhaustive characterizations for pointwise Kraus-operators necessary sufficient extremality conditions using these Kraus-operators. motivate use covariance methods by showing that with respect to contain a family representatives from both classes show even slight deviation measure can yield an extreme informationally complete observable. latter derives similar results continuous in (possibly) infinite dimensions. As example study phase space instruments, their structure, properties.