# Symmetry-Like Relation of Relative Entropy Measure of Quantum Coherence

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. Lower and Upper Bounds for the Relative Entropy of Coherence

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

## 4. The Relation between ${\mathit{C}}_{\mathit{r}}(\mathit{\rho})$ and ${\mathit{C}}_{{\ell}_{\mathit{1}}}(\mathit{\rho})$

**Proposition**

**1.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Marius, N.; Selim, G. On the importance of parallelism for quantum computation and the concept of a universal computer. In Unconventional Computation; Calude, C.S., Dinneen, M.J., Păun, G., Pérez-Jímenez, M.J., Rozenberg, G., Eds.; Springer: Berlin/Heidelberg, Germany, 2005; Volume 3699, pp. 176–190. [Google Scholar]
- Paredes, B.; Verstraete, F.; Cirac, J.I. Exploiting quantum parallelism to simulate quantum random many-body systems. Phys. Rev. Lett.
**2005**, 93, 140501. [Google Scholar] [CrossRef] [PubMed][Green Version] - O’Leary, D.P.; Brennen, G.K.; Bullock, S.S. Parallelism for quantum computation with qudits. Phys. Rev. A
**2006**, 74, 032334. [Google Scholar] [CrossRef][Green Version] - Nielsen, M.A.; Chuang, I.L. Quantum Computation and Quantum Information; Cambridge University Press: Cambridge, UK, 2001. [Google Scholar]
- Rebentrost, P.; Mohseni, M.; Aspuru-Guzik, A. Role of quantum coherence and environmental fluctuations in chromophoric energy transport. J. Phys. Chem. B
**2009**, 113, 9942–9947. [Google Scholar] [CrossRef] [PubMed][Green Version] - Lloyd, S. Quantum coherence in biological systems. J. Phys. Conf. Ser.
**2011**, 302, 012037. [Google Scholar] [CrossRef] - Li, C.M.; Lambert, N.; Chen, Y.N.; Chen, G.Y.; Nori, F. Witnessing quantum coherence: From solid-state to biological systems. Sci. Rep.
**2012**, 2, 885. [Google Scholar] [CrossRef][Green Version] - Huelga, S.; Plenio, M. Vibrations, quanta and biology. Contemp. Phys.
**2013**, 54, 181–207. [Google Scholar] [CrossRef] - Plenio, M.B.; Huelga, S.F. Dephasing-assisted transport: Quantum networks and biomolecules. New J. Phys.
**2008**, 10, 113019. [Google Scholar] [CrossRef] - Levi, F.; Mintert, F. A quantitative theory of coherent delocalization. New J. Phys.
**2014**, 16, 033007. [Google Scholar] [CrossRef] - Vazquez, H.; Skouta, R.; Schneebeli, S.; Kamenetska, M.; Breslow, R.; Venkataraman, L.; Hybertsen, M.S. Probing the conductance superposition law in single-molecule circuits with parallel paths. Nat. Nanotechnol.
**2012**, 7, 663–667. [Google Scholar] [CrossRef] - Karlstrom, O.; Linke, H.; Karlstrom, G.; Wacker, A. Increasing thermoelectric performance using coherent transport. Phys. Rev. B
**2011**, 84, 113415. [Google Scholar] [CrossRef][Green Version] - Giovannetti, V.; Lloyd, S.; Maccone, L. Advances in quantum metrology. Nat. Photonics
**2011**, 5, 222–229. [Google Scholar] [CrossRef] - Giovannetti, V. Quantum-enhanced measurements: Beating the standard quantum limit. Science
**2004**, 306, 1330–1336. [Google Scholar] [CrossRef] [PubMed][Green Version] - Baumgratz, T.; Cramer, M.; Plenio, M.B. Quantifying coherence. Phys. Rev. Lett.
**2014**, 113, 140401. [Google Scholar] [CrossRef] [PubMed][Green Version] - Shao, L.H.; Xi, Z.J.; Fan, H.; Li, Y.M. Fidelity and trace norm distances for quantifying coherence. Phys. Rev. A
**2014**, 91, 042120. [Google Scholar] [CrossRef][Green Version] - 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.
**2016**, 116, 150502. [Google Scholar] [CrossRef][Green Version] - Streltsov, A.; Singh, U.; Dhar, H.S.; Bera, M.N.; Adesso, G. Measuring quantum coherence with entanglement. Phys. Rev. Lett.
**2015**, 115, 020403. [Google Scholar] [CrossRef][Green Version] - Winter, A.; Yang, D. Operational resource theory of coherence. Phys. Rev. Lett.
**2016**, 116, 120404. [Google Scholar] [CrossRef][Green Version] - Hu, M.L.; Fan, H. Relative quantum coherence, incompatibility, and quantum correlations of states. Phys. Rev. A
**2017**, 95, 052106. [Google Scholar] [CrossRef][Green Version] - Qi, X.; Gao, T.; Yan, F. Measuring coherence with entanglement concurrence. J. Phys. A Math. Theor.
**2017**, 50, 285301. [Google Scholar] [CrossRef] - Rana, S.; Parashar, P.; Lewenstein, M. Trace-distance measure of coherence. Phys. Rev. A
**2016**, 93, 012110. [Google Scholar] [CrossRef][Green Version] - Chen, B.; Fei, S.M. Notes on modified trace distance measure of coherence. Quantum Inf. Comput.
**2018**, 17, 107. [Google Scholar] [CrossRef][Green Version] - Yu, X.D.; Zhang, D.J.; Xu, G.F.; Tong, D.M. Alternative framework for quantifying coherence. Phys. Rev. A
**2016**, 94, 060302. [Google Scholar] [CrossRef][Green Version] - Yao, Y.; Xiao, X.; Ge, L.; Sun, C.P. Quantum coherence in multipartite systems. Phys. Rev. A
**2015**, 92, 022112. [Google Scholar] [CrossRef][Green Version] - Streltsov, A. Genuine quantum coherence. J. Phys. A
**2017**, 50, 045301. [Google Scholar] - Sun, Y.; Mao, Y.; Luo, S.L. From quantum coherence to quantum correlations. Europhys. Lett.
**2017**, 118, 60007. [Google Scholar] [CrossRef] - Tan, K.C.; Kwon, H.; Park, C.Y.; Jeong, H. Unified view of quantum correlations and quantum coherence. Phys. Rev. A
**2017**, 96, 069905. [Google Scholar] [CrossRef][Green Version] - Guo, Y.; Goswami, S. Discordlike correlation of bipartite coherence. Phys. Rev. A
**2017**, 95, 062340. [Google Scholar] [CrossRef][Green Version] - Xi, Z.; Li, Y.; Fan, H. Quantum coherence and correlations in quantum system. Sci. Rep.
**2015**, 5, 10922. [Google Scholar] [CrossRef][Green Version] - Liu, F.; Li, F.; Chen, J.; Xing, W. Uncertainty-like relations of the relative entropy of coherence. Quantum Inf. Comput.
**2017**, 15, 3459–3465. [Google Scholar] [CrossRef] - Radhakrishnan, C.; Parthasarathy, M.; Jambulingam, S.; Byrnes, T. Distribution of quantum coherence in multipartite systems. Phys. Rev. Lett.
**2016**, 116, 150504. [Google Scholar] [CrossRef][Green Version] - Xi, Z. Quantum coherence over the noisy quantum channels. Sci. China-Phys. Mech. Astron.
**2015**, 45, 030302. [Google Scholar] [CrossRef] - Singh, U.; Bera, M.N.; Dhar, H.S.; Pati, A.K. Maximally coherent mixed states: Complementarity between maximal coherence and mixedness. Phys. Rev. A
**2015**, 91, 052115. [Google Scholar] [CrossRef][Green Version] - Liu, C.L.; Yu, X.D.; Xu, G.F.; Tong, D.M. Ordering states with coherence measures. Quantum Inf. Comput.
**2016**, 15, 4189–4201. [Google Scholar] [CrossRef][Green Version] - Bagan, E.; Bergou, J.A.; Cottrell, S.S.; Hillery, M. Relations between coherence and path information. Phys. Rev. Lett.
**2016**, 116, 160406. [Google Scholar] [CrossRef] [PubMed][Green Version] - Cheng, S.; Hall, M.J.W. Complementarity relations for quantum coherence. Phys. Rev. A
**2015**, 92, 042101. [Google Scholar] [CrossRef][Green Version] - Ma, J.; Yadin, B.; Girolami, D.; Vedral, V.; Gu, M. Converting coherence to quantum correlations. Phys. Rev. Lett.
**2016**, 116, 160407. [Google Scholar] [CrossRef] - Rana, S.; Parashar, P.; Winter, A.J.; Lewenstein, M. Logarithmic coherence: Operational interpretation of ℓ
_{1}-norm coherence. Phys. Rev. A**2017**, 96, 052336. [Google Scholar] [CrossRef][Green Version] - Hu, M.L.; Hu, X.Y.; Wang, J.C.; Peng, Y.; Zhang, Y.R.; Fan, H. Quantum coherence and geometric quantum discord. Phys. Rep.
**2018**, 762–764, 1–100. [Google Scholar] [CrossRef][Green Version] - Guo, Z.H.; Cao, H.X. Creating quantum correlation from coherence via incoherent quantum operations. J. Phys. A Math. Theor.
**2019**, 52, 265301. [Google Scholar] [CrossRef] - Choi, M.D. Completely positive linear maps on complex matrices. Linear Algebra Appl.
**1975**, 10, 285–290. [Google Scholar] [CrossRef][Green Version]

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, C.; Guo, Z.; Cao, H. Symmetry-Like Relation of Relative Entropy Measure of Quantum Coherence. *Entropy* **2020**, *22*, 297.
https://doi.org/10.3390/e22030297

**AMA Style**

Zhang C, Guo Z, Cao H. Symmetry-Like Relation of Relative Entropy Measure of Quantum Coherence. *Entropy*. 2020; 22(3):297.
https://doi.org/10.3390/e22030297

**Chicago/Turabian Style**

Zhang, Chengyang, Zhihua Guo, and Huaixin Cao. 2020. "Symmetry-Like Relation of Relative Entropy Measure of Quantum Coherence" *Entropy* 22, no. 3: 297.
https://doi.org/10.3390/e22030297