LAUSR

The relative entropy of coherence is the keystone of the operational resource theory of coherence. The closest incoherent state of any generic state, as measured by relative entropy, is its diagonal part in a fixed basis, so is the relative entropy of coherence always the minimal-coherence-value measure among all known coherence measures? In this paper, we partially resolve the question within the set of genuine coherence monotones. In single qubit system, we compare the relative entropy of coherence with other three value coherence measures and find that it is really minimal. Meanwhile, we obtain some necessary and sufficient conditions for saturating the lower bound of the relative entropy of coherence. In high dimensional quantum system, the relative entropy of coherence is always the minimal-coherence-value measure for any pure state under consideration, but there exists some mixed states which violate the result.