  A simple relation is established for the evolution equation of quantum
information processing protocols such as quantum teleportation, remote state
preparation, Bell-inequality violation and particularly dynamics of the
geometric quantum correlation measures. This relation shows that when the
system traverses the local quantum channel, various figures of merit of the
quantum correlations for different protocols demonstrate a factorization decay
behavior for dynamics. We identified the family of quantum states for different
kinds of quantum channels under the action of which the relation holds. This
relation simplifies the assessment of many quantum tasks.
