A Generalization of Gleason’s Frame Function for Quantum Measurement

The goal is to extend Gleason’s notion of a frame function, which essential in his fundamental theorem quantum measurement, more general function acting on 1-tight, so-called, Parseval frames. We refer these functions as Gleason for reason our generalization that positive operator-valued measures (POVMs) are essentially equivalent frames and POVMs arise naturally measurement theory. prove under the proper assumptions, quadratic forms, well other results analogous original theorem. Furthermore, we solve an intrinsic problem relating different lengths. use this solution weaken hypotheses finite dimensional version Busch’s theorem, itself analog mathematical characterization states.