Relation between the three measurement contexts (position at t=0, momentum at t=0, and position at t=t_M)

Determine the relation between the measurement outcomes obtained in the three experimental settings—initial position at t=0, initial momentum at t=0 via a Fourier-transform plane, and intermediate position at t=t_M—for photons prepared in a superposition of the position-localized state |L> and the momentum-localized state |B> using the described Sagnac interferometer, clarifying how these results are connected across the different measurement contexts.

Background

The paper reports an experiment preparing photons in a superposition of a position-localized state |L> and a momentum-localized state |B> within a Sagnac interferometer, and measures three distributions: position at t=0, momentum at t=0 (via a Fourier transform), and position at an intermediate time t=t_M. The results violate a classical particle propagation inequality and reveal interference contributions linked to Wigner-function negativity.

While the authors decompose observed statistics into contributions from |L>, |B>, and their interference, they note an interpretational gap: it is not evident how to relate the outcomes across the three distinct measurement contexts for the same ensemble. They explicitly state that they do not know what the relation between these measurements is, highlighting an unresolved question about connecting measurement results across contexts in this setup.