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We prove a Gleason-type theorem for the quantum probability rule using framefunctions defined on positive-operator-valued measures (POVMs), as opposed tothe restricted class of orthogonal projection-valued measures used in theoriginal theorem. The advantage of this method is that it works fortwo-dimensional quantum systems (qubits) and even for vector spaces overrational fields--settings where the standard theorem fails. Furthermore, unlikethe method necessary for proving the original result, the present one is ratherelementary. In the case of a qubit, we investigate similar results for framefunctions defined upon various restricted classes of POVMs. For the so-calledtrine measurements, the standard quantum probability rule is again recovered.Comment: 10 pages RevTeX, no figure

Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements