Positive operator-valued measures that allow restoring trace class operators

The history of positive operator-valued measures (POVM) begins with the pioneering work of M.A. Naimark in 1943. One can look at such a measure as a characteristic of some contraction operator that can be expanded to a unitary and this view led to the creation of the famous Hungarian mathematical school. Another view of it is given by the quantum theory of statistical solutions developed by A.S. Holevo in the 70s of the XX century. Here the question is about the exact or approximate restoration of the quantum state (a positive operator with a unit trace in Hilbert space) based on the results of its measurement by POVM. The talk will be devoted to the question of constructing POVM in the form of an orbit of a group of automorphisms generated by a projective unitary representation of some locally compact group.