Quantum coherence quantifiers based on the R\'{e}nyi $\alpha$-relative entropy

The resource theories of quantum coherence attract a lot of attention in recent years. Especially, the monotonicity property plays a crucial role here. In this paper we investigate the monotonicity property for the coherence measures induced by the R\'{e}nyi $\alpha$-relative entropy. We show that the R\'{e}nyi $\alpha$-relative entropy of coherence does not in general satisfy the monotonicity requirement under the subselection of measurements condition and it also does not satisfy the extension of monotonicity requirements which present in [Phys. Rev. A 93, 032136, 2016]. Due to the R\'{e}nyi $\alpha$-relative entropy of coherence can act as a coherence monotone quantity, we examine the trade-off relations between coherence and mixedness. Finally, some properties for the single qubit of R\'{e}nyi $2$-relative entropy of coherence are derived.