# Quantumness and Dequantumness Power of Quantum Channels

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## Abstract

**:**

## 1. Introduction

## 2. Measure of Quantumness

## 3. Quantumness and Dequantumness Power

**Proof.**

**Example**

**1.**

## 4. Completely Dequantumness Channel and Its Relationship with Quantum Markovianity

**Proof.**

#### 4.1. Phase Damping Dynamics

#### 4.2. Amplitude Damping Dynamics

#### 4.3. Random Unitary Dynamics

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

- Schlosshauer, M. Decoherence, the measurement problem, and interpretations of quantum mechanics. Rev. Mod. Phys.
**2005**, 76, 1267. [Google Scholar] [CrossRef] - Joos, E.; Zeh, H.D.; Kiefer, C.; Giulini, D.; Kupsch, J.; Stamatescu, I.-O. Decoherence and the Appearance of a Classical World in Quantum Theory; Springer: New York, NY, USA, 2003. [Google Scholar]
- Yu, T.; Eberly, J.H. Finite-time disentanglement via spontaneous emission. Phys. Rev. Lett.
**2004**, 93, 140404. [Google Scholar] [CrossRef] [PubMed] - Maziero, J.; Céleri, L.C.; Serra, R.M.; Vedral, V. Classical and quantum correlations under decoherence. Phys. Rev. A
**2009**, 80, 044102. [Google Scholar] [CrossRef] - Bellomo, B.; Franco, R.L.; Compagno, G. Non-markovian effects of the dynamics of entanglement. Phys. Rev. Lett.
**2007**, 99, 160502. [Google Scholar] [CrossRef] - Mazzola, L.; Piilo, J.; Maniscalco, S. Sudden transition between classical and quantum decoherence. Phys. Rev. Lett.
**2010**, 104, 200401. [Google Scholar] [CrossRef] [PubMed] - Zanardi, P.; Zalka, C.; Faoro, L. Entangling power of quantum evolutions. Phys. Rev. A
**2000**, 62, 030301(R). [Google Scholar] [CrossRef] - Luo, S.; Fu, S.; Li, N. Decorrelating capabilities of operations with application to decoherence. Phys. Rev. A
**2010**, 82, 052122. [Google Scholar] [CrossRef] - Wang, L.; Yu, C. The roles of a quantum channel on a quantum state. Int. J. Theor. Phys.
**2014**, 53, 715–726. [Google Scholar] [CrossRef] - Mani, A.; Karimipour, V. Cohering and De-cohering power of quantum channels. Phys. Rev. A
**2015**, 92, 032331. [Google Scholar] [CrossRef] - García-Díaz, M.; Egloff, D.; Plenio, M.B. A note on coherence power of N-dimensional unitary operators. arXiv
**2015**, arXiv:1510.06683. [Google Scholar] [CrossRef] - Zanardi, P.; Styliaris, G.; Venuti, L.C. Coherence-generating power of quantum unitary maps and beyond. Phys. Rev. A
**2017**, 95, 052306. [Google Scholar] [CrossRef] - Zanardi, P.; Styliaris, G.; Venuti, L.C. Measures of coherence-generating power for quantum unital operations. Phys. Rev. A
**2017**, 95, 052307. [Google Scholar] [CrossRef] - Bu, K.; Kumar, A.; Zhang, L.; Wu, J. Cohering power of quantum operations. Phys. Lett. A
**2017**, 381, 1670–1676. [Google Scholar] [CrossRef] - Li, N.; Luo, S.; Mao, Y. Quantumness-generating capability of quantum dynamics. Quantum Inf. Process.
**2018**, 17, 74. [Google Scholar] [CrossRef] - Long, G.L.; Zhou, Y.F.; Jin, J.Q.; Sun, Y.; Lee, H.W. Density matrix in quantum mechanics and distinctness of ensembles having the same compressed density matrix. Found. Phys.
**2006**, 36, 1217. [Google Scholar] [CrossRef] - Fuchs, C.A. Just two nonorthogonal quantum states. arXiv
**1998**, arXiv:9810032v1. [Google Scholar] - Fuchs, C.A.; Sasaki, M. The quantumness of a set of quantum states. arXiv
**2003**, arXiv:0302108v1. [Google Scholar] - Horodecki, M.; Horodecki, P.; Horodecki, R.; Piani, M. Quantumness of ensemble from no-broadcasting principle. Int. J. Quantum Inf.
**2006**, 4, 105. [Google Scholar] [CrossRef] - Oreshkov, O.; Calsamiglia, J. Distinguishability measures between ensembles of quantum states. Phys. Rev. A
**2009**, 79, 032336. [Google Scholar] [CrossRef] - Zhu, X.; Pang, S.; Wu, S.; Liu, Q. The classicality and quantumness of a quantum ensemble. Phys. Lett. A
**2011**, 375, 1855. [Google Scholar] [CrossRef] - Ma, T.; Zhao, M.J.; Wang, Y.K.; Fei, S.M. Non-commutativity and local indistinguishability of quantum states. Sci. Rep.
**2014**, 4, 6336. [Google Scholar] [CrossRef] [PubMed] - Piani, M.; Narasimhachar, V.; Calsamiglia, J. Quantumness of correlations, quantumness of ensembles and quantum data hiding. New J. Phys.
**2014**, 16, 113001. [Google Scholar] [CrossRef] - Luo, S.; Li, N.; Fu, S. Quantumness of quantum ensemble. Theor. Math. Phys.
**2011**, 169, 1724. [Google Scholar] [CrossRef] - Li, N.; Luo, S.; Mao, Y. Quantifying quantumness of ensembles. Phys. Rev. A
**2017**, 96, 022132. [Google Scholar] [CrossRef] - Luo, S.; Li, N.; Sun, W. How quantum is a quantum ensemble. Quantum Inf. Process.
**2010**, 9, 711. [Google Scholar] [CrossRef] - Mao, Y.; Song, H. Quantumness of ensembles via coherence. Phys. Lett. A
**2019**, 383, 2698. [Google Scholar] [CrossRef] - Naikoo, J.; Banerjee, S. A study of coherence based measure of quantumness in (non) Markovian channels. Quantum Inf. Process.
**2020**, 19, 29. [Google Scholar] [CrossRef] - Shahbeigi, F.; Akhtarshenas, S.J. Quantumness of quantum channels. Phys. Rev. A
**2018**, 98, 042313. [Google Scholar] [CrossRef] - Iyengar, P.; Chandan, G.N.; Srikanth, R. Quantifying quantumness via commutators: An application to quantum walk. arXiv
**2013**, arXiv:1312.1329. [Google Scholar] - Ferro, L.; Fazio, R.; Illuminate, F.; Marmo, G.; Vedral, V.; Pascazio, S. Measuring quantumness: From theory to observability in interferometric setups. Eur. Phys. J. D
**2018**, 72, 1. [Google Scholar] [CrossRef] - Naikoo, J.; Banerjee, S.; Srikanth, R. Quantumness of channels. Quantum Inf. Process.
**2021**, 20, 32. [Google Scholar] [CrossRef] - Hall, M.J.; Cresser, J.D.; Li, L.; Andersson, E. Canonical form of master equations and characterization of non-Markovianity. Phys. Rev. A
**2014**, 89, 042120. [Google Scholar] [CrossRef] - Wolf, M.M.; Eisert, J.; Cubitt, T.S.; Cirac, J.I. Assessing non-Markovian quantum dynamics. Phys. Rev. Lett.
**2008**, 101, 150402. [Google Scholar] [CrossRef] [PubMed] - Rivas, Á.; Huelga, S.F.; Plenio, M.B. Entanglement and non-Markovianity of quantum evolutions. Phys. Rev. Lett.
**2010**, 105, 050403. [Google Scholar] [CrossRef] - Hou, S.C.; Yi, X.X.; Yu, S.X.; Oh, C.H. Alternative non-Markovianity measure by divisibility of dynamical maps. Phys. Rev. A
**2011**, 83, 062115. [Google Scholar] [CrossRef] - Breuer, H.-P.; Laine, E.-M.; Piilo, J. Measure for the degree of non-Markovian behavior of quantum processes in open systems. Phys. Rev. Lett.
**2009**, 103, 210401. [Google Scholar] [CrossRef] - Breuer, H.-P. Foundations and measures of quantum non-Markovianity. J. Phys. B
**2012**, 45, 154001. [Google Scholar] [CrossRef] - Rajagopal, A.K.; Usha Devi, A.R.; Rendell, R.W. Kraus representation of quantum evolution and fidelity as manifestations of Markovian and non-Markovian forms. Phys. Rev. A
**2010**, 82, 042107. [Google Scholar] [CrossRef] - Alipour, S.; Mani, A.; Rezakhani, A.T. Quantum discord and non-Markovianity of quantum dynamics. Phys. Rev. A
**2012**, 85, 052108. [Google Scholar] [CrossRef] - Luo, S.; Fu, S.; Song, H. Quantifying non-Markovianity via correlations. Phys. Rev. A
**2012**, 86, 044101. [Google Scholar] [CrossRef] - Lu, X.-M.; Wang, X.; Sun, C.P. Quantum Fisher information flow and non-Markovian processes of open systems. Phys. Rev. A
**2010**, 82, 042103. [Google Scholar] [CrossRef] - Song, H.; Luo, S.; Hong, Y. Quantum non-Markovianity based on the Fisher-information matrix. Phys. Rev. A
**2015**, 91, 042110. [Google Scholar] [CrossRef] - Naikoo, J.; Dutta, S.; Banerjee, S. Facets of quantum information under non-Markovian evolution. Phys. Rev. A
**2019**, 99, 042128. [Google Scholar] [CrossRef] - Song, H.; Mao, Y. Dynamics of Rényi entropy and applications in detecting quantum non-Markovianity. Phys. Rev. A
**2017**, 96, 032115. [Google Scholar] [CrossRef] - He, Z.; Mao, Y.; Zeng, H.-S.; Li, Y.; Wang, Q.; Yao, C. Non-Markovianity measure based on the relative entropy of coherence in an extended space. Phys. Rev. A
**2017**, 96, 022106. [Google Scholar] [CrossRef] - Chruściński, D.; Wudarski, F.A. Non-Markovianity degree for random unitary evolution. Phys. Rev. A
**2015**, 91, 012104. [Google Scholar] [CrossRef] - Chruściński, D.; Wudarski, F.A. Non-Markovian random unitary qubit dynamics. Phys. Lett. A
**2013**, 377, 1425. [Google Scholar] [CrossRef] - Li, N.; Luo, S.; Song, H. Monotonicity of quantumness of ensembles under commutativity-preserving channels. Phys. Rev. A
**2019**, 99, 052114. [Google Scholar] [CrossRef]

Channel | Completely Dequantumness | Markovianity | Relationship |
---|---|---|---|

Phase Damping | ${\gamma}_{t}\ge 0$ | ${\gamma}_{t}\ge 0$ | CDQ ⇔Markovianity |

Amplitude Damping | $|{G}_{t}|\le \frac{1}{4}$ and $\frac{d|{G}_{t}|}{dt}\le 0$ | $\frac{d|{G}_{t}|}{dt}\le 0$ | CDQ ${}_{\nLeftarrow}^{\Rightarrow}$ Markovianity |

Random Unitary | ${\gamma}_{1}\left(t\right)+{\gamma}_{2}\left(t\right)+{\gamma}_{3}\left(t\right)+{\gamma}_{j}\left(t\right)\ge 0$ | ${\gamma}_{i}\left(t\right)+{\gamma}_{j}\left(t\right)\ge 0,\phantom{\rule{3.33333pt}{0ex}}i\ne j$ | CDQ ${}_{\Leftarrow}^{\nRightarrow}$ Markovianity |

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Song, H.; Li, N.
Quantumness and Dequantumness Power of Quantum Channels. *Entropy* **2022**, *24*, 1146.
https://doi.org/10.3390/e24081146

**AMA Style**

Song H, Li N.
Quantumness and Dequantumness Power of Quantum Channels. *Entropy*. 2022; 24(8):1146.
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**Chicago/Turabian Style**

Song, Hongting, and Nan Li.
2022. "Quantumness and Dequantumness Power of Quantum Channels" *Entropy* 24, no. 8: 1146.
https://doi.org/10.3390/e24081146