Abstract
Relative entropy of coherence can be written as an entropy difference of the original state and the incoherent state closest to it when measured by relative entropy. The natural question is, if we generalize this situation to Tsallis or Rényi entropies, would it define good coherence measures? In other words, we define a difference between Tsallis entropies of the original state and the incoherent state closest to it when measured by Tsallis relative entropy. Taking Rényi entropy instead of the Tsallis entropy, leads to the well-known distance-based Rényi coherence, which means this expression defined a good coherence measure. Interestingly, we show that Tsallis entropy does not generate even a genuine coherence monotone, unless it is under a very restrictive class of operations. Additionally, we provide continuity estimate for Rényi coherence. Furthermore, we present two coherence measures based on the closest incoherent state when measures by Tsallis or Rényi relative entropy.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11128-023-03872-y/MediaObjects/11128_2023_3872_Fig1_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11128-023-03872-y/MediaObjects/11128_2023_3872_Fig2_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11128-023-03872-y/MediaObjects/11128_2023_3872_Fig3_HTML.png)
Similar content being viewed by others
Data availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
References
Aberg, J.: Catalytic coherence. Phys. Rev. Lett. 113, 150402 (2014)
Baumgratz, T., Cramer, M., Plenio, M.B.: Quantifying coherence. Phys. Rev. Lett. 113(14), 140401 (2014)
Chitambar, E., Gour, G.: Critical examination of incoherent operations and a physically consistent resource theory of quantum coherence. Phys. Rev. Lett. 117(3), 030401 (2016)
Chitambar, E., Gour, G.: Comparison of incoherent operations and measures of coherence. Phys. Rev. A 94(5), 052336 (2016)
Chitambar, E., Hsieh, M.H.: Relating the resource theories of entanglement and quantum coherence. Phys. Rev. Lett. 117(2), 020402 (2016)
Cwikliski, P., Studziski, M., Horodecki, M., Oppenheim, J.: Limitations on the evolution of quantum coherences: towards fully quantum second laws of thermodynamics. Phys. Rev. Lett. 115(21), 210403 (2015)
Dai, Y., Hu, J., Zhang, Z., Zhang, C., Dong, Y., Wang, X.: Measurement-induced entropy increment for quantifying genuine coherence. Quantum Inf. Process. 20(8), 1–12 (2021)
De Vicente, J.I., Streltsov, A.: Genuine quantum coherence. J. Phys. A Math. Theor. 50(4), 045301 (2016)
Du, S., Bai, Z., Guo, Y.: Conditions for coherence transformations under incoherent operations. Phys. Rev. A 91(5), 052120 (2015)
Glauber, R.J.: Coherent and incoherent states of the radiation field. Phys. Rev. 131(6), 2766 (1963)
Hardy, G.H., Littlewood, J.E., Pólya, G.: Some simple inequalities satisfied by convex function. Messenger Math. 58, 145–152 (1929)
Hiai, F., Mosonyi, M.: Different quantum \(f\)-divergences and the reversibility of quantum operations. Rev. Math. Phys. 29(07), 1750023 (2017)
Hu, M.L., Hu, X., Wang, J., Peng, Y., Zhang, Y.R., Fan, H.: Quantum coherence and geometric quantum discord. Phys. Rep. 762, 1–100 (2018)
Kammerlander, P., Anders, J.: Coherence and measurement in quantum thermodynamics. Sci. Rep. 6(1), 1–7 (2016)
Lostaglio, M., Jennings, D., Rudolph, T.: Description of quantum coherence in thermodynamic processes requires constraints beyond free energy. Nat. Commun. 6, 6383 (2015)
Ma, J., Yadin, B., Girolami, D., Vedral, V., Gu, M.: Converting coherence to quantum correlations. Phys. Rev. Lett. 116(16), 160407 (2016)
Marshall, A. W., Olkin, I., Arnold, B.C.: Doubly Stochastic Matrices. In Inequalities: Theory of Majorization and Its Applications, pp. 29–77. Springer, New York (2010)
Piani, M., Cianciaruso, M., Bromley, T.R., Napoli, C., Johnston, N., Adesso, G.: Robustness of asymmetry and coherence of quantum states. Phys. Rev. A 93(4), 042107 (2016)
Pinelis, I.: Modulus of continuity of the quantum f-entropy with respect to the trace distance. arXiv preprint arXiv:2107.10112 (2021)
Radhakrishnan, C., Parthasarathy, M., Jambulingam, S., Byrnes, T.: Distribution of quantum coherence in multipartite systems. Phys. Rev. Lett. 116(15), 150504 (2016)
Rana, S., Parashar, P., Lewenstein, M.: Trace-distance measure of coherence. Phys. Rev. A 93(1), 012110 (2016)
Rastegin, A.E.: Quantum-coherence quantifiers based on the Tsallis relative \(\alpha \) entropies. Phys. Rev. A 93(3), 032136 (2016)
Rebentrost, P., Mohseni, M., Aspuru-Guzik, A.: Role of quantum coherence and environmental fluctuations in chromophoric energy transport. J. Phys. Chem. B 113, 9942 (2009)
Schur, I.: Uber eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie. Sitzungsberichte der Berliner Mathematischen Gesellschaft 22(9–20), 51 (1923)
Scully, M.O., Zubairy, M.S.: Quantum Optics, Cambridge. Ch, 4, 17 (1997)
Shao, L.H., Xi, Z., Fan, H., Li, Y.: Fidelity and trace-norm distances for quantifying coherence. Phys. Rev. A 91(4), 042120 (2015)
Shao, L.H., Li, Y.M., Luo, Y., Xi, Z.J.: Quantum Coherence quantifiers based on Rényi \(\alpha \)-relative entropy. Commun. Theor. Phys. 67(6), 631 (2017)
Streltsov, A., Adesso, G., Plenio, M.B.: Colloquium: quantum coherence as a resource. Rev. Mod. Phys. 89(4), 041003 (2017)
Streltsov, A., Singh, U., Dhar, H.S., Bera, M.N., Adesso, G.: Measuring quantum coherence with entanglement. Phys. Rev. Lett. 115(2), 020403 (2015)
Tan, K.C., Kwon, H., Park, C.Y., Jeong, H.: Unified view of quantum correlations and quantum coherence. Phys. Rev. A 94(2), 022329 (2016)
Vershynina, A.: Measure of genuine coherence based of quasi-relative entropy. Quantum Inf. Process. 21(5), 1–22 (2022)
Virosztek, D.: Quantum entropies, relative entropies, and related preserver problems (2016)
Winter, A., Yang, D.: Operational resource theory of coherence. Phys. Rev. Lett. 116(12), 120404 (2016)
Witt, B., Mintert, F.: Stationary quantum coherence and transport in disordered networks. New J. Phys. 15, 093020 (2013)
Yadin, B., Ma, J., Girolami, D., Gu, M., Vedral, V.: Quantum processes which do not use coherence. Phys. Rev. X 6(4), 041028 (2016)
Yao, Y., Xiao, X., Ge, L., Sun, C.P.: Quantum coherence in multipartite systems. Phys. Rev. A 92(2), 022112 (2015)
Yu, X.D., Zhang, D.J., Xu, G.F., Tong, D.M.: Alternative framework for quantifying coherence. Phys. Rev. A 94(6), 060302 (2016)
Zhao, H., Yu, C.S.: Coherence measure in terms of the Tsallis relative \(\alpha \) entropy. Sci. Rep. 8(1), 299 (2018)
Zhu, H., Ma, Z., Cao, Z., Fei, S.M., Vedral, V.: Operational one-to-one mapping between coherence and entanglement measures. Phys. Rev. A 96(3), 032316 (2017)
Acknowledgements
A. V. was supported by NSF grant DMS-2105583.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author has no competing interests or conflict of interest to declare that are relevant to the content of this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Vershynina, A. Coherence as entropy increment for Tsallis and Rényi entropies. Quantum Inf Process 22, 127 (2023). https://doi.org/10.1007/s11128-023-03872-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11128-023-03872-y