Measurements, disturbances and the quantum three box paradox

A quantum pre- and post-selection paradox involves making measurements at two separate times on a quantum system, and making inferences about the state of the system at an intermediate time, conditional upon the observed outcomes. The inferences lead to predictions about the results of measurements performed at the intermediate time, which have been well confirmed experimentally, but which nevertheless seem paradoxical when inferences about different intermediate measurements are combined. The three box paradox is the paradigm example of such an effect, where a ball is placed in one of three boxes and is shuffled between the boxes in between two measurements of its location. By conditionalising on the outcomes of those measurements, it is inferred that between the two measurements the ball would have been found with certainty in Box 1 and with certainty in Box 2, if either box been opened on their own. Despite experimental confirmation of the predictions, and much discussion, it has remained unclear what exactly is supposed to be paradoxical or what specifically is supposed to be quantum, about these effects. In this paper I identify precisely the conditions under which the quantum three box paradox occurs, and show that these conditions are the same as arise in the derivation of the Leggett–Garg Inequality, which is supposed to demonstrate the incompatibility of quantum theory with macroscopic realism. I will argue that, as in Leggett–Garg Inequality violations, the source of the effect actually lies in the disturbance introduced by the intermediate measurement, and that the quantum nature of the effect is that no classical model of measurement disturbance can reproduce the paradox.