Coherence measure: Logarithmic coherence number

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.