Coherence resource power of isocoherent states

We address the problem of comparing quantum states with the same amount of coherence in terms of their coherence resource power given by the preorder of incoherent operations. For any coherence measure, two states with null or maximum value of coherence are equivalent with respect to that preorder. This is no longer true for intermediate values of coherence when pure states of quantum systems with dimension greater than two are considered. In particular, we show that, for any value of coherence (except the extreme values, zero and the maximum), there are infinite incomparable pure states with that value of coherence. These results are not peculiarities of a given coherence measure, but common properties of every well-behaved coherence measure. Furthermore, we show that for qubit mixed states there exist coherence measures, such as the relative entropy of coherence, that admit incomparable isocoherent states.