A Generalization of Gleason's Frame Function for Quantum Measurement

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