Interpreting quantum coherence through a quantum measurement process

Recently, there has been a renewed interest in the quantification of coherence or other coherence-like concepts within the framework of quantum resource theory. However, rigorously defined or not, the notion of coherence or decoherence has already been used by the community for decades since the advent of quantum theory. Intuitively, the definitions of coherence and decoherence should be the two sides of the same coin. Therefore, a natural question is raised: how can the conventional decoherence processes, such as the von Neumann-L\"{u}ders (projective) measurement postulation or partially dephasing channels, fit into the bigger picture of the recently established theoretic framework? Here we show that the state collapse rules of the von Neumann or L\"{u}ders-type measurements, as special cases of genuinely incoherent operations (GIO), are consistent with the resource theories of quantum coherence. New hierarchical measures of coherence are proposed for the L\"{u}ders-type measurement and their relationship with measurement-dependent discord is addressed. Moreover, utilizing the fixed point theory for $C^\ast$-algebra, we prove that GIO indeed represent a particular type of partially dephasing (phase-damping) channels which have a matrix representation based on the Schur product. By virtue of the Stinespring's dilation theorem, the physical realizations of incoherent operations are investigated in detail and we find that GIO in fact constitute the core of strictly incoherent operations (SIO) and generally incoherent operations (IO) and the unspeakable notion of coherence induced by GIO can be transferred to the theories of speakable coherence by the corresponding permutation or relabeling operators.