Insolubility of the quantum measurement problem for unsharp observables

The quantum mechanical measurement problem is the difficulty of dealing with the indefiniteness of the pointer observable at the conclusion of a measurement process governed by unitary quantum dynamics. There has been hope to solve this problem by eliminating idealizations from the characterization of measurement. We state and prove two ‘insolubility theorems’ that disappoint this hope. In both the initial state of the apparatus is taken to be mixed rather than pure, and the correlation of the object observable and the pointer observable is allowed to be imperfect. In the insolubility theorem for sharp observables, which is only a modest extension of previous results, the object observable is taken to be an arbitrary projection valued measure. In the insolubility theorem for unsharp observables, which is essentially new, the object observable is taken to be a positive operator valued measure. Both theorems show that the measurement problem is not the consequence of neglecting the ever-present imperfections of actual measurements.