International Journal of Theoretical Physics

, Volume 58, Issue 4, pp 1195–1201 | Cite as

Is the Relative Entropy of Coherence Always the Minimal-Coherence-Value Measure?

  • Dong-Mei GaoEmail author
  • Ting-Hua Lv


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.


Quantum coherence The relative entropy of coherence Minimal-coherence-value measure 



This work was supported by the National Natural Science Foundation of China, Grant Nos. 61771294, 61602232; Shandong Provincial Natural Science Foundation, China, Grant No. ZR2015FQ006.


  1. 1.
    Bromley, T.R., Cianciaruso, M., Adesso, G.: Frozen quantum coherence. Phys. Rev. Lett. 114, 210401 (2015)ADSCrossRefGoogle Scholar
  2. 2.
    Liu, F., Li, F., Chen, J., Xing, W.: Uncertainty-like relations of the relative entropy of coherence. Quantum Inf. Process. 15, 3459 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baumgratz, T., Cramer, M., Plenio, M.B.: Quantifying coherence. Phys. Rev. Lett. 113, 140401 (2014)ADSCrossRefGoogle Scholar
  4. 4.
    Winter, A., Yang, D.: Operational resource theory of coherence. Phys. Rev. Lett. 116, 120404 (2016)ADSCrossRefGoogle Scholar
  5. 5.
    Bagan, E., Bergou, J.A., Cottrell, S.S., Hillery, M.: Relations between coherence and path information. Phys. Rev. Lett. 116, 160406 (2016)ADSCrossRefGoogle Scholar
  6. 6.
    Rana, S., Parashar, P., Lewenstein, M.: Trace-distance measure of coherence. Phys. Rev. A 93, 012110 (2016)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Napoli, C., Bromley, T.R., Cianciaruso, M., Piani, M., Johnston, N., Adesso, G.: Robustness of coherence: an operational and observable measure of quantum coherence. Phys. Rev. Lett. 116, 150502 (2016)ADSCrossRefGoogle Scholar
  8. 8.
    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, 042107 (2016)ADSCrossRefGoogle Scholar
  9. 9.
    Yuan, X., Zhou, H., Cao, Z., Ma, X.: Intrinsic randomness as a measure of quantum coherence. Phys. Rev. A. 92, 022124 (2015)ADSCrossRefGoogle Scholar
  10. 10.
    Hu, M.L., Hu, X., Peng, Y., Zhang, Y.R., Fan, H.: Quantum coherence and quantum correlations. arXiv:1703.01852 (2017)
  11. 11.
    de Vicente1, J.I., Streltsov, A.: Genuine quantum coherence. J. Phys. A 50, 045301 (2017)Google Scholar
  12. 12.
    Streltsov, A., Singh, U., Dhar, H.S., Bera, M.N., Adesso, G.: Measuring quantum coherence with entanglement. Phys. Rev. Lett. 115, 020403 (2015)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of Mathematic and Information ScienceShandong Technology and Business UniversityYantaiChina

Personalised recommendations