Measurements, disturbances and the quantum three box paradox

Abstract

Quantum pre-and post-selection (PPS) paradoxes occur when counterfactual inferences are made about different measurements that might have been performed, between two measurements that are actually performed. The 3 box paradox is the paradigm example of such a paradox, where a ball is placed in one of three boxes and it is inferred that it would have been found, with certainty, both in box 1 and in box 2 had either box been opened on their own. Precisely what is at stake in PPS paradoxes has been unclear, and classical models have been suggested which are supposed to mimic the essential features of the problem. We show that the essential difference between the classical and quantum pre-and post-selection effects lies in the fact that for a quantum PPS paradox to occur the intervening measurement, had it been performed, would need to be invasive but non-detectable. This invasiveness is required even for null result measurements. While some quasi-classical features (such as non-contextuality and macrorealism) are compatible with PPS paradoxes, it seems no fully classical model of the 3 box paradox is possible.

Exploring Quantum Foundations with Single Photons

“…In other words, for an anomalous shift to occur the final measurement must be correlated to the intermediate measurement in a way that is not only due to the initial state ψ. In particular, this implies that in an ontological model, there must be some disturbance at the ontic level for an anomalous post-selected shift to occur [118], which agrees with previous results [105]. The coin-flip model is an example for such a correlated model.…”

“…Consider the three box paradox [69,105,108,109], where a ball can be in any of three boxes, represented by the states 1⟩, 2⟩ and 3⟩. At the beginning of the game the system is prepared in the state ( 1⟩+ 2⟩+ 3⟩) √ 3 (e.g.…”

“…In the paradox scenario itself, however, the intermediate measurement could conceivably disturb the system enough to encode information about the measurement setting, thereby changing the post-selection probability. In fact, any noncontextual model for a pre-and post-selection paradox must involve measurement disturbance [69,105]. It has been shown, however, that the required form of disturbance is itself incompatible with noncontextuality [69].…”

“…conditioned on a certain outcome of a later measurement) in the state φ⟩, quantum theory can give "paradoxical" predictions intermediate measurements. Specifically, in the case of socalled logical pre-and post-selection paradoxes (where probabilities only take values 0 and 1), the paradox is in effect a conflict with Kochen-Specker noncontextual ontological models [69,106,107].Consider the three box paradox [69,105,108,109], where a ball can be in any of three boxes, represented by the states 1⟩, 2⟩ and 3⟩. At the beginning of the game the system is prepared in the state ( 1⟩+ 2⟩+ 3⟩) √ 3 (e.g.…”

“…It has been argued that such paradoxes require a measurement between pre-and post-selection that is invasive at the ontic level, but operationally undetectable (i.e. there is no difference in the statistics between the cases where the measurement is performed or not) [105]. Finally, the set of temporal quantum correlations is not equivalent to that of spatial correlations, such that, for example, monogamy of entanglement, which is a central concept for spatial entanglement, does not hold in the temporal analogue.…”