Wave-particle duality of unpolarized photons

Photons in a two-path interferometer best embody wave-particle duality (WPD), which is a core concept of quantum theory. So far, the WPD relation is commonly written as V² + D² ≤ 1, where V is the interference fringe visibility, and D is path distinguishability, i.e., the distinguishability of which path a photon passed. This inequality is saturated only when the which-way marker (WWM), which embodies which-path information (WPI) via an internal degree of freedom of photons, such as polarization, is in a pure state. For mixed-state WWM, conventionally defined distinguishability underestimates the amount of WPI and thus does not saturate the WPD relation. Here, we introduce a generalized measure of distinguishability D that properly quantifies the WPI and saturates the WPD relation for all pure- and mixed-state WWM within a purification-based framework. To this end, mixed-state WWM is treated as a result of entanglement formation between the WWM and an external degree of freedom, e.g., environment, and D is defined so that it incorporates the total WPI shared between the WWM and the environment. We show that D thus defined is experimentally quantifiable, independently of V, without access to the environment. We experimentally evaluate V and D using true single photons generated in the completely mixed (unpolarized) state, and thus verify the saturated WPD relation.