Quantum ensembles as waves on state space

A quantum ensemble contains preparation weights that are lost in its average density operator. We show how optimal decisions read these weights as a wave on projective state space. Varying the input prior sweeps a surface of fixed Born overlap through the ensemble; the resulting response propagates according to the geometry of pure states and reconstructs their distribution in every finite dimension. Conditional preparations can fix this wave through contact, selecting the ensemble at a steering boundary. Positive propagation gives exact biased-qubit solutions and explicit parent measurements for noisy channels. When measurement records are combined, their state directions are averaged and the wave acquires a source whose growth measures the lost discrimination advantage. Preparation, inference and noise can thus be understood as operations on the same recoverable distribution, linking quantum correlations to measurement resolution.