  Wave-particle duality, an important and fundamental concept established upon
pure quantum systems, is central to the complementarity principle in quantum
mechanics. However, due to the environment effects or even the entanglement
between the quanton and the which-way detector (WWD), the quanton should be
described by a mixed quantum state but not a pure quantum state. Although there
are some attempts to clarify the complementarity principle for mixed quantum
systems, it is still unclear how the mixedness affects the complementary
relation. Here, we give a ternary complementary relation (TCR) among wave,
particle and mixedness aspects for arbitrary multi-state systems, which are
respectively quantified by the $l_1$ measure for quantum coherence, the
which-path predictability, and the linear entropy. In particular, we show how a
WWD can transform entropy into predictability and coherence. Through modifying
the POVM (positive-operator valued measure) measurement on WWD, our TCR can be
simplified as the wave-mixedness and particle-mixedness duality relations.
Beyond enclosing the wave-particle duality relation [PRL 116, 160406 (2016)],
our TCR relates to the wave-particle-entanglement complementarity relation [PRL
98, 250501 (2008); Opt. Commun. 283, 827 (2010)].
