  We introduce a measure of coherence, which is extended from the coherence
rank via the standard convex roof construction, we call it the logarithmic
coherence number. This approach is parallel to the Schmidt measure in
entanglement theory, We study some interesting properties of the logarithmic
coherence number, and show that this quantifier can be considered as a proper
coherence measure. We also find that the logarithmic coherence number can be
calculated exactly for a large class of states. We give the relationship
between coherence and entanglement in bipartite system, and our results are
generalized to multipartite setting. Finally, we find that the creation of
entanglement with bipartite incoherent operations is bounded by the logarithmic
coherence number of the initial system during the process.
