  We introduce positive operator-valued measure (POVM) generated by the
projective unitary representation of a direct product of locally compact
Abelian group $G$ with its dual $\hat G$. The method is based upon the
Pontryagin duality allowing to establish an isometrical isomorphism between the
space of Hilbert-Schmidt operators in $L^2(G)$ and the Hilbert space $L^2(\hat
G\times G)$. Any such a measure determines a pair of hybrid (containing
classical and quantum parts) quantum channels consisting of the measurement
channel and the channel transmitting an initial quantum state to the ensemble
of quantum states on the group. It is shown that the second channel can be
called a complementary channel to the measurement channel.
