Experimental discrimination of hypotheses in a paradoxical entanglement setup

Violations of Bell inequalities show that quantum entanglement produces nonclassical correlations between spatially separated systems. In quantum eraser experiments, interference is restored once distinguishability is removed. This highlights the essential role of distinguishability in quantum state transformations. However, it has not been recognized that a paradox can emerge using only conventional optical transformations of quantum states, and no experiment has been proposed to determine which of three mutually exclusive hypotheses outside the implicit assumptions of standard optical transformations and capable of resolving the paradox is realized in practice. Here we show that a paradox emerges in a configuration that reduces the distinguishability of an entangled state, and we propose an experiment capable of identifying which hypothesis is realized. Theory alone cannot determine which hypothesis is realized. Therefore, experimental verification is required. Regardless of the experimental outcome, the experiment would reveal an unrecognized property linking entanglement and distinguishability.