The Curious Quantum Mechanics of Pre- and Post-Selected Ensembles

In a provocative paper, (~) Albert, Aharonov, and D'Amato proposed an analysis of preand post-selected ensembles in quantum mechanics (PPS ensembles) which seemed to show that in the interval between preand post-selection, certain noncommuting observables all have determinate values. Moreover, these values are context dependent: The value assumed by a degenerate observable (e.g., an observable represented by a onedimensional projection operator) depends on what other observables are selected in a measurement of a complete commuting set with P (i.e., the value of the one-dimensional projection operator, 1 or 0, depends on the choice of a complete commuting set of basis vectors spanning the Hilbert space). The fallacy in this argument was pointed out by Bub and Brown. (2) They showed that the identity of a PPS ensemble depends on the measurement(s) made on the system in the ensemble in the interval between the preand post-selection. For the example proposed by Albert, Aharonov, and D'Amato, Bub and Brown demonstrated that the PPS ensembles are manifestly different. They went on to construct an example