# Coherence Trapping in Open Two-Qubit Dynamics

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## Abstract

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## 1. Introduction

## 2. Open Quantum System and Coherence

## 3. Asymptotic Dynamics of Quantum Coherence in Open Quantum System Systems

## 4. Coherence and Total Quantum Correlation

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Dynamics of the coherence for initially separable states is displayed versus the time t for different values of the parameter m. The dashed blue ($m=0.5$) and dash-dotted red ($m=0.1$) lines correspond to an initially separable pure state $W\left(0\right)=|0\rangle \langle 0|$; the solid green ($m=0.5$) and dotted black ($m=0.1$) lines correspond to an initially separable mixed state, $W\left(0\right)=1/2\left(\right)open="("\; close=")">|2\rangle \langle 2|+|3\rangle \langle 3|$. The influence of the environment on the measure of coherence ${C}_{L}$ is obtained to be similar, and that the dissipation can be able to enhance the amount of the quantum coherence during the dynamics; then, the function ${C}_{L}$ reaches a constant value and is preserved even in the regime of long times. Furthermore, the revival rate of the function ${C}_{L}$ depends on the choice of the parameter m.

**Figure 2.**Dynamics of the coherence for initially entangled quantum states is displayed versus the time t for different values of the parameter m. The dashed blue ($m=0.5$) and dash-dotted red ($m=0.1$) lines correspond to an initially maximally pure state, $|\mathsf{\Psi}\rangle =1/\sqrt{2}\left(\right)open="("\; close=")">|01\rangle +|10\rangle $; the solid green ($m=0.5$) and dotted black ($m=0.1$) lines correspond to an initial Horodecki state, $W\left(0\right)=a|\mathsf{\Psi}\rangle \langle \mathsf{\Psi}|+(1-a)\left(\right)open="("\; close=")">|0\rangle \langle 0|$, with $a=0.4$. The influence of the environment on the function ${C}_{L}$ is obtained to be similar for initially entangled states, so the function ${C}_{L}$ tends to attain a constant value and is preserved even in the regime of long times. The dependence of the quantum coherence on the initial conditions clearly demonstrates that an appropriate choice of system parameters can lead to the enhancement and preservation of quantum coherence during the dynamics.

**Figure 3.**Dynamics of the total quantum correlation and coherence for initially separable states is displayed versus the time t with $m=0.1$. The dashed blue (total quantum correlation) and dash-dotted red (quantum coherence) lines correspond to an initially separable pure state $W\left(0\right)=|0\rangle \langle 0|$; the solid green (total quantum correlation) and dotted black (quantum coherence) lines correspond to an initially separable mixed state, $W\left(0\right)=1/2\left(\right)open="("\; close=")">|2\rangle \langle 2|+|3\rangle \langle 3|$. The asymptotic behavior of the functions ${C}_{L}$ and ${U}_{T}$ is shown to be comparable according to the various parameters that are involved in the state of two qubits, and the quantifiers exhibit the same behavior and maintain their value even over extended periods of time.

**Figure 4.**Dynamics of the total quantum correlation and coherence for initially entangled states is displayed versus the time with $m=0.1$. The dashed blue (total quantum correlation) and dash-dotted red (quantum coherence) lines correspond to an initially maximally pure state, $|\mathsf{\Psi}\rangle =1/\sqrt{2}\left(\right)open="("\; close=")">|01\rangle +|10\rangle $; the solid green (total quantum correlation) and dotted black (quantum coherence) lines correspond to an initial Horodecki state, $W\left(0\right)=a|\mathsf{\Psi}\rangle \langle \mathsf{\Psi}|+(1-a)\left(\right)open="("\; close=")">|0\rangle \langle 0|$, with $a=0.4$. The asymptotic behavior of the functions ${C}_{L}$ and ${U}_{T}$ is shown to be comparable according to the various parameters that are involved in the state of two qubits, and the quantifiers exhibit the same behavior and maintain their value even over extended periods of time.

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**MDPI and ACS Style**

Algarni, M.; Berrada, K.; Abdel-Khalek, S.; Eleuch, H.
Coherence Trapping in Open Two-Qubit Dynamics. *Symmetry* **2021**, *13*, 2445.
https://doi.org/10.3390/sym13122445

**AMA Style**

Algarni M, Berrada K, Abdel-Khalek S, Eleuch H.
Coherence Trapping in Open Two-Qubit Dynamics. *Symmetry*. 2021; 13(12):2445.
https://doi.org/10.3390/sym13122445

**Chicago/Turabian Style**

Algarni, Mariam, Kamal Berrada, Sayed Abdel-Khalek, and Hichem Eleuch.
2021. "Coherence Trapping in Open Two-Qubit Dynamics" *Symmetry* 13, no. 12: 2445.
https://doi.org/10.3390/sym13122445