  Bohr's complementarity and Schr\"odinger's entanglement are two prominent
physical characters of quantum systems. In this letter, we formally connect
them. It is known that complementarity relations for wave-particle duality are
saturated only for pure, single-quanton, quantum states. For mixed states, the
wave-particle quantifiers never saturate a complementarity relation and can
even reach zero for a maximally mixed state. To fully characterize a quanton,
it is not enough to consider its wave-particle aspect; we have also to regard
its quantum correlations with other systems. Here we prove that for any
complete complementarity relation involving predictability and visibility
measures that satisfy the criteria established in the literature, these
corresponding quantum correlations are entanglement monotones.
