  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)
show 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 quantum entanglement. 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 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), who 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 properties becomes internally inconsistent.
