Remedying the strong monotonicity of the coherence measure in terms of the Tsallis relative $α$ entropy

Abstract: Coherence is the most fundamental quantum feature of the nonclassical systems. The understanding of coherence within the resource theory has been attracting increasing interest among which the quantification of coherence is an essential ingredient. A satisfactory measure should meet certain standard criteria. It seems that the most crucial criterion should be the strong monotonicity, that is, average coherence doesn't increase under the (sub-selective) incoherent operations. Recently, the Tsallis relative $\alpha $ entropy [A. E. Rastegin, Phys. Rev. A \textbf{93}, 032136 (2016)] has been tried to quantify the coherence. But it was shown to violate the strong monotonicity, even though it can unambiguously distinguish the coherent and the incoherent states with the monotonicity. Here we establish a family of coherence quantifiers which are closely related to the Tsallis relative $\alpha $ entropy. It proves that this family of quantifiers satisfy all the standard criteria and particularly cover several typical coherence measures.