# Entanglement Dynamics Governed by Time-Dependent Quantum Generators

## Abstract

**:**

## 1. Introduction

## 2. Partially Commutative Open Quantum Systems

**Theorem 1**

## 3. Two-Qubit Entangled States

#### 3.1. Example 1: Evolution of $\mathsf{\Phi}\left(\varphi \right)$

#### 3.2. Example 2: Evolution of $\left|\Psi \right(\varphi )\rangle $

## 4. Three-Qubit Entangled States

#### 4.1. Example 1: Evolution of the GHZ State

#### 4.2. Example 2: Evolution of the W State

## 5. Two-Qutrit Entangled States

## 6. Non-Markovian Evolution of Two-Qubit Entangled States

## 7. Discussion and Outlook

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Plots present the probability of finding a two-qubit system in one of the possible states. The initial state is represented by (15).

**Figure 2.**Trajectories of the phase factor, ${\rho}_{14}^{AB}\left(t\right)$, presented on the complex plane for two values of $\mathcal{E}$.

**Figure 3.**Concurrence, $C\left[{\rho}^{AB}\left(t\right)\right]$, of the two-qubit density matrix with the initial state (15) for three values of $\omega $.

**Figure 4.**Plots present the probability of finding a two-qubit system in one of the possible states. The initial state is represented by (19).

**Figure 5.**Concurrence, $C\left[{\rho}^{AB}\left(t\right)\right]$, of the two-qubit density matrix with the initial state (19) for three values of $\omega $.

**Figure 6.**Plots present the probability of finding a three-qubit system in one of the possible states. The initial state is represented by (23).

**Figure 7.**Trajectories of the phase factor, ${\rho}_{18}^{ABC}\left(t\right)$, presented on the complex plane for two values of $\mathcal{E}$.

**Figure 8.**Plots present the Bures distance, $\mathcal{D}\left({\rho}^{ABC}\left(t\right)\right)$, for three values of $\omega $. Initially, the system was represented by the GHZ state.

**Figure 9.**Plots present the probability of finding a three-qubit system in one of the possible states. The initial state is represented by (28).

**Figure 10.**Plots present the Bures distance, $\mathcal{D}\left({\rho}^{ABC}\left(t\right)\right)$, for three values of $\omega $. Initially, the system was represented by the W state.

**Figure 11.**Plots present the probability of finding a two-qutrit system in one of the possible states. The initial state is represented by (33).

**Figure 12.**Plots present the Bures distance, $\mathcal{D}\left({\rho}^{AB}\left(t\right)\right)$, for three values of $\omega $.

**Figure 14.**Concurrence, $C\left[{\rho}^{AB}\left(t\right)\right]$, for three values of $\omega $ and the initial state: ${\rho}^{AB}\left(0\right)=|{\mathsf{\Phi}}^{+}\rangle \phantom{\rule{-0.166667em}{0ex}}\langle {\mathsf{\Phi}}^{+}|$.

**Figure 15.**Plots present the probability of finding the system ${\rho}^{AB}\left(t\right)$ in one of the possible states, assuming $\omega =2$ (arb. units).

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Czerwinski, A.
Entanglement Dynamics Governed by Time-Dependent Quantum Generators. *Axioms* **2022**, *11*, 589.
https://doi.org/10.3390/axioms11110589

**AMA Style**

Czerwinski A.
Entanglement Dynamics Governed by Time-Dependent Quantum Generators. *Axioms*. 2022; 11(11):589.
https://doi.org/10.3390/axioms11110589

**Chicago/Turabian Style**

Czerwinski, Artur.
2022. "Entanglement Dynamics Governed by Time-Dependent Quantum Generators" *Axioms* 11, no. 11: 589.
https://doi.org/10.3390/axioms11110589