On quantum channels generated by covariant positive operator-valued measures on a locally compact group

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