We prove a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures (POVMs), as opposed to the restricted class of orthogonal projection-valued measures used in the original theorem. The advantage of this method is that it works for two-dimensional quantum systems (qubits) and even for vector spaces over rational fields--settings where the standard theorem fails. Furthermore, unlike the method necessary for proving the original result, the present one is rather elementary. In the case of a qubit, we investigate similar results for frame functions defined upon various restricted classes of POVMs. For the so-called trine measurements, the standard quantum probability rule is again recovered.
Busch's theorem deriving the standard quantum probability rule can be regarded as a more general form of Gleason's theorem. Here we show that a further generalisation is possible by reducing the number of quantum postulates used by Busch. We do not assume that the positive measurement outcome operators are effects or that they form a probability operator measure. We derive a more general probability rule from which the standard rule can be obtained from the normal laws of probability when there is no measurement outcome information available, without the need for further quantum postulates. Our general probability rule has prediction-retrodiction symmetry and we show how it may be applied in quantum communications and in retrodictive quantum theory.
We subsequently reanalyse the mixed-triangle measurement scheme previously formulated within PDL, in which a minimal
(4,6) closure encodes a spin doublet and a Stern–Gerlach-type device is represented by a finite active surface composed of mixed triangles. We prove that, under mild symmetry constraints on the apparatus, the effective probabilities generated by this scheme satisfy our operational axioms and thus fall within the scope of the uniqueness theorem. It follows that Born’s rule for spin-
21 measurements is not merely implementable in PDL, but is uniquely enforced by the stipulated operational and combinatorial conditions. This result provides an initial link between Gleason-type derivations and the discrete relational substrate underlying PDL, and suggests the feasibility of a more general uniqueness programme for quantum probabilities within this framework.