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The goal is to extend Gleason's notion of a frame function, which isessential in his fundamental theorem in quantum measurement, to a more generalfunction acting on 1-tight, so-called, Parseval frames. We refer to thesefunctions as Gleason functions for Parseval frames. The reason for ourgeneralization is that positive operator valued measures (POVMs) areessentially equivalent to Parseval frames, and that POVMs arise naturally inquantum measurement theory. We prove that under the proper assumptions, Gleasonfunctions for Parseval frames are quadratic forms, as well as other resultsanalogous to Gleason's original theorem. Further, we solve an intrinsic problemrelating Gleason functions for Parseval frames of different lengths. We usethis solution to weaken the hypotheses in the finite dimensional version ofBusch's theorem, that itself is an analog of Gleason's mathematicalcharacterization of quantum states.

A Generalization of Gleason's Frame Function for Quantum Measurement