Multipath wave–particle duality with exact quantification of which-path knowledge

Abstract

A quantum system (quanton) traverses an interferometer with N equally probable paths and interacts with another quantum system (detector) that stores path information in a set of symmetric states. In this interferometric framework, we present a systematic quantitative study of wave–particle duality based on entropic trade-off relations between quantum coherence, characterized by the relative entropy of coherence of the quanton state, and which-path knowledge, quantified by the mutual information obtained through detector-state discrimination. Unlike prior upper-bound-based formulations, we obtain an exact quantification of which-path knowledge for the discrimination strategies considered here, encompassing previously considered strategies and extending beyond them. A general optimal measurement achieving this is implemented through a state-separation map governed by a continuous parameter ξ∈[0,1], followed by minimum-error (ME) discrimination of either the successful outputs only (standard approach) or both the successful and failure outputs (concatenated approach). Our framework identifies a strategy-dependent hierarchy that generally emerges in the tightness of the resulting duality relations. At ξ=0, both approaches reduce to the ME measurement, which typically yields the tightest relation. For ξ>0, each approach yields a distinct relation that generally becomes less tight as ξ increases, with the concatenated strategy always yielding a relation at least as tight as its standard counterpart. We further characterize the saturation of these relations by identifying the sets of detector states that lead to it and show that nontrivial saturation occurs only for interferometers with a nonprime number of paths. From these saturating sets, we elucidate how the underlying structure of the quanton-detector correlations gives rise to this behavior and explains the lack of universal saturation in multipath duality relations for pure states.