Coherence as a resource for N-box and quantum pigeonhole paradoxes

Overview of PPS Paradoxes

Pre- and post-selection (PPS) paradoxes, such as the three-box paradox and the quantum pigeonhole principle, challenge classical intuitions by suggesting that quantum systems can exhibit properties that seem to contradict classical logic. While logical PPS paradoxes (where ABL probabilities are strictly 0 or 1) are known to be linked to quantum contextuality, the non-logical variants—which involve probabilistic outcomes—have remained less understood. This paper provides a unified framework for these non-logical paradoxes, demonstrating that they are essentially manifestations of quantum coherence.

The Role of Coherence

The authors show that for an N-box or quantum pigeonhole scenario to produce a paradox, the pre-selected and post-selected states must be coherent with respect to the basis of the intermediate measurement. By defining the paradox through the violation of a classical probability bound, the researchers prove that a coherence-free subtheory cannot reproduce these paradoxical statistics. The presence of a paradox is tied to the existence of an anomalous weak value for a specific operator, which acts as a quantifiable witness for the necessary coherence.

Connecting to Macrorealism

The study links these findings to the Leggett-Garg inequality, which tests the limits of macrorealism. By reinterpreting the three-box paradox through the lens of operational non-disturbance, the authors show that the non-classicality observed in these scenarios is consistent with a theory-independent, strict form of classicality. This clarifies that the "paradoxical" nature of these experiments is not merely a curiosity but a direct consequence of the departure from incoherent, classical sub-theories.

This work provides a rigorous foundation for understanding non-logical PPS paradoxes. By identifying coherence as the essential resource, the authors bridge the gap between abstract quantum foundations and operational, measurable statistics. This helps researchers distinguish between the different types of non-classicality—such as contextuality and coherence—that drive various quantum phenomena, offering a clearer path for future investigations into the limits of classical descriptions of quantum systems.