Radon–Nikodym derivatives of quantum instruments

A convenient representation for Radon–Nikodym derivatives of completely positive (c.p.) instruments on B(H) with respect to a scalar measure is suggested, similar to the Stinespring–Kraus representation for c.p. maps, but involving possibly nonclosable unbounded operators. The structure of covariant c.p. instruments is studied in detail. In particular, an exhaustive description is given to instruments covariant with respect to shifts or rotations, corresponding to “position” or “angle” measurements.