Controlling wave-particle duality with entanglement between single-photon and Bell states

Wave-particle duality and entanglement are two fundamental characteristics of quantum mechanics. All previous works on experimental investigations in wave-particle properties of single photons (or single particles in general) showed that a well-defined interferometer setting determines a well-defined property of single photons. Here we take a conceptual step forward and control the wave-particle property of single photons with a Bell state. By doing so, we experimentally test the complementarity principle in a scenario in which the setting of the interferometer is not defined at any instance of the experiment, not even in principle. To achieve this goal, we establish the three-photon entangled state, i.e., the entanglement between a single photon and a two-photon Bell state, send the photon of interest $S$ into a quantum Mach-Zehnder interferometer (MZI), in which the output beam splitter of the MZI is controlled by the quantum state of the second photon $C$, which is entangled with a third photon $A$. Therefore, the individual quantum state of photon $C$ is undefined, which implements the undefined settings of the MZI for photon $S$. This is realized by using three cascaded phase-stable interferometers for three photons. There is typically no well-defined setting of the MZI and thus the very formulation of the wave-particle objectivity from local hidden variable models becomes internally inconsistent.