  Coherence is the most fundamental quantum feature in quantum mechanics. For a
bipartite quantum state, if a measurement is performed on one party, the other
party, based on the measurement outcomes, will collapse to a corresponding
state with some probability and hence gain the average coherence. It is shown
that the average coherence is not less than the coherence of its reduced
density matrix. In particular, it is very surprising that the extra average
coherence (and the maximal extra average coherence with all the possible
measurements taken into account) is upper bounded by the classical correlation
of the bipartite state instead of the quantum correlation. We also find the
sufficient and necessary condition for the null maximal extra average
coherence. Some examples demonstrate the relation and, moreover, show that
quantum correlation is neither sufficient nor necessary for the nonzero extra
average coherence within a given measurement. In addition, the similar
conclusions are drawn for both the basis-dependent and the basis-free coherence
measure.
