Figure 3: Double-slit interferometer for arbitrary quantum particles.

The first guessing game (coloured red) involves Alice guessing which slit the quanton goes through, given that she can measure a system E₁ that has interacted with, and hence may contain information about, the quanton. In the second game (coloured blue), Bob randomly chooses the source’s vertical coordinate, and Alice tries to guess where the source was located, given some other system E₂ and given where the quanton was finally detected. Note that the source’s location determines the relative phase between the which-slit states, |0› and |1›, and we assume that Bob chooses one of two possible locations such that the relative phase is either 0 or π. Here, the state |0› (|1›) is defined as the pure state at the slit exit that one would obtain from blocking the bottom (top) slit. Our framework provides a WPDR that constrains Alice’s ability to win these complementary games. Furthermore, one can reinterpret the probability to win the second game, for the case where E₂ is trivial, in terms of the traditional fringe visibility. The latter quantifies the amplitude of intensity oscillations as one varies the detector location y. Note that varying y changes the relative path lengths from the slits to the detector and hence is analogous to applying a relative phase φ between the two paths. (This assumes that the envelope function (dashed curve) associated with the interference pattern is flat over the range of y values considered, which is often the case when L is very large.) So, the double-slit fringe visibility is equivalent to the notion captured by in equation (7), where the detector is spatially fixed but a phase is varied, and we relate to our entropic measure of wave behaviour in the Methods section.