  We study wave-particle duality by exploring for the first time effects of a
quantum object's source. A single photon emitted from a pair of nonlocally
entangled two-level atoms is specifically analyzed. Surprisingly, duality is
found to be a conditional phenomenon depending on the photon's atomic source.
It can be tuned maximum, medium, and even minimum (completely absent) by the
atomic state purity through an exact quadratic relation that can be called
Duality Pythagorean Theorem. The analysis shows a new way of investigating
duality by accounting how the single quantum object is created. The result
sheds a new light on the fundamental understanding of the completeness of
wave-particle duality, and can be tested in various practical physical systems.
