  Wave-particle duality is an essential character of quantum systems. In the
last few years, much progress has being made towards formally quantifying these
quantum features. The properties of the quantum density matrix were shown to
lead to duality inequalities for single-quanton mixed states and to triality
identities, also known as complete complementarity relations (CCRs), for
two-quanton pure states. In this article, we establish connections between CCRs
and EPR (ir)realism, as formalized by Bilobran and Angelo in [Europhys. Lett.
112, 40005 (2015)]. Besides, we show that, for tripartite pure states
$|\psi_{ABE}\rangle$, CCRs can be applied to obtain the system-environment
entanglement of formation, $E_{f}(\rho_{AE})$, through the local information of
a system $A$ and its classical correlation with an auxiliary system $B$.
Moreover, we discuss CCRs in the context of the emergence of the pointer basis
in the decoherence process identified via the constancy of system-apparatus
classical correlations. We also obtain CCRs for purified mixed bipartite
quantum-classical states. In addition, we derive CCR for mixed-bipartite
two-qudit quantum states and discuss its interpretations.
