Abstract
In this paper, we study the ordering states with Tsallis relative \(\alpha \)-entropies of coherence and \(l_{1}\) norm of coherence for single-qubit states. Firstly, we show that any Tsallis relative \(\alpha \)-entropies of coherence and \(l_{1}\) norm of coherence give the same ordering for single-qubit pure states. However, they do not generate the same ordering for some high-dimensional states, even though these states are pure. Secondly, we also consider three special Tsallis relative \(\alpha \)-entropies of coherence for \(\alpha =2, 1, \frac{1}{2}\) and show these three measures and \(l_{1}\) norm of coherence will not generate the same ordering for some single-qubit mixed states. Nevertheless, they may generate the same ordering if we only consider a special subset of single-qubit mixed states. Furthermore, we find that any two of these three special measures generate different ordering for single-qubit mixed states. Finally, we discuss the degree of violation of between \(l_{1}\) norm of coherence and Tsallis relative \(\alpha \)-entropies of coherence. In a sense, this degree can measure the difference between these two coherence measures in ordering states.
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Acknowledgements
This paper is supported by National Natural Science Foundation of China (Grants Nos. 11271237, 61228305, 61602291), The Higher School Doctoral Subject Foundation of Ministry of Education of China (Grant No. 20130202110001) and Fundamental Research Funds for the Central Universities (No. 2016CBY003).
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Appendix
Appendix
We provide a proof of \(\frac{\partial r}{\partial z}\ge 0\). The first equation comes from the derivation of \(r_{\frac{1}{2}}\) with respect to z. In the second equation, we use distributive law and then merge similar items. The last inequality comes from the fact \(2-z^2-t^2\ge 2\sqrt{1-\sqrt{z^2+t^2}}\).
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Zhang, FG., Shao, LH., Luo, Y. et al. Ordering states with Tsallis relative \(\alpha \)-entropies of coherence. Quantum Inf Process 16, 31 (2017). https://doi.org/10.1007/s11128-016-1488-4
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DOI: https://doi.org/10.1007/s11128-016-1488-4