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The Bipartite and Tripartite Entanglement in $\mathcal{PT}$ -Symmetric System

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## Abstract

**:**

## 1. Introduction

## 2. The Model

- $\widehat{\rho}(t=0)=|100\rangle \langle 100|$;
- $\widehat{\rho}(t=0)=|010\rangle \langle 010|$;
- $\widehat{\rho}(t=0)=|001\rangle \langle 001|$,

## 3. The Entanglement

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**The time evolution of bipartite negativities ${N}_{ij}$: (

**a**) $\widehat{\rho}(t=0)=|100\rangle \langle 100|$, $\gamma =0.01\beta $; (

**b**) $\widehat{\rho}(t=0)=|100\rangle \langle 100|$, $\gamma =0.8\beta $; (

**c**) $\widehat{\rho}(t=0)=|010\rangle \langle 010|$, $\gamma =0.01\beta $; (

**d**) $\widehat{\rho}(t=0)=|010\rangle \langle 010|$, $\gamma =0.8\beta $; (

**e**) $\widehat{\rho}(t=0)=|001\rangle \langle 001|$, $\gamma =0.01\beta $; (

**f**) $\widehat{\rho}(t=0)=|001\rangle \langle 001|$, $\gamma =0.8\beta $.

**Figure 3.**Maximal values of the bipartite negativities ${N}_{ij}$ versus $\gamma /\beta $ for $\omega =5\beta $ and for various initial states $\widehat{\rho}(t=0)$: (

**a**) $|100\rangle \langle 100|$, (

**b**) $|010\rangle \langle 010|$, (

**c**) $|001\rangle \langle 001|$.

**Figure 4.**Steady-state solutions for bipartite negativities ${N}_{ij}$ versus the value of the ratio $\gamma /\beta $ for $\omega =5\beta $.

**Figure 5.**The time evolution of bipartite (${N}_{i-jk}$) and tripartite (${N}_{ijk}$) negativities: (

**a**) $\widehat{\rho}(t=0)=|100\rangle \langle 100|$, $\gamma =0.01\beta $; (

**b**) $\widehat{\rho}(t=0)=|100\rangle \langle 100|$, $\gamma =0.8\beta $; (

**c**) $\widehat{\rho}(t=0)=|010\rangle \langle 010|$, $\gamma =0.01\beta $; (

**d**) $\widehat{\rho}(t=0)=|010\rangle \langle 010|$, $\gamma =0.8\beta $; (

**e**) $\widehat{\rho}(t=0)=|001\rangle \langle 001|$, $\gamma =0.01\beta $; (

**f**) $\widehat{\rho}(t=0)=|001\rangle \langle 001|$, $\gamma =0.8\beta $.

**Figure 6.**Maximal values of bipartite (${N}_{i-jk}$) and tripartite (${N}_{ijk}$) negativities versus $\gamma /\beta $ for $\omega =5\beta $ and for various initial states $\widehat{\rho}(t=0)$: (

**a**) $|100\rangle \langle 100|$, (

**b**) $|010\rangle \langle 010|$, (

**c**) $|001\rangle \langle 001|$.

**Figure 7.**Steady-state solutions for the negativities (full tripartite ${N}_{123}$ and bipartite ${N}_{i-jk}$) versus the value of the ratio $\gamma /\beta $ for $\omega =5\beta $.

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**MDPI and ACS Style**

Le Duc, V.; Nowotarski, M.; Kalaga, J.K.
The Bipartite and Tripartite Entanglement in *Symmetry* **2021**, *13*, 203.
https://doi.org/10.3390/sym13020203

**AMA Style**

Le Duc V, Nowotarski M, Kalaga JK.
The Bipartite and Tripartite Entanglement in *Symmetry*. 2021; 13(2):203.
https://doi.org/10.3390/sym13020203

**Chicago/Turabian Style**

Le Duc, Vinh, Mateusz Nowotarski, and Joanna K. Kalaga.
2021. "The Bipartite and Tripartite Entanglement in *Symmetry* 13, no. 2: 203.
https://doi.org/10.3390/sym13020203