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

On constructing informationally complete covariant positive operator-valued measures

“…In this paper we investigated a projective unitary representation of a direct product Ĝ × G, where G is a locally compact Abelian group and Ĝ is its dual consisting of characters. Earlier it was shown [4,5] that orbits of this representation determines POVM using the Pontryagin duality. Continuing this study, we introduce a family of contractions associated with the constructed POVM.…”

“…we obtain that the Haar measure corresponding to the Pontryagin duality [4] is 1 2π dx and we are coming to the famous Glauber-Sudarshan measure…”

“…In fact, the map ρ → T r(ρU χ,g ) determines the isometrical isomorphism [4,5] T r(ρσ) = G T r(ρU χ,g )T r(σU χ,g )dν(χ)dν(g).…”

“…In the case of compact group, it is possible to obtain the measure possessing operator density with respect to the finite Haar measure on a group. If a group is locally compact, problems arise with the existence of an integral, which can be solved by using the Pontryagin duality principle [4]. This topic is closely related to quantum tomography, which assumes not one, but a series of measurements determined by a set of different POVMs.…”

“…The paper is organized as follows. At first (Section 2) we provide the necessary information about the construction of POVM by means of orbits of the projective unitary representation introduced in [4,5]. In Section 3 we define a family of contractions associated with the constructed POVM and show that these contractions are multiples of unitary operators of the representation.…”