A Numerical Study of Quantum Entropy and Information in the Wigner–Fokker–Planck Equation for Open Quantum Systems
Abstract
:1. Introduction
2. Materials and Methods
2.1. Monte-Carlo Solver: Euler–Maruyama Method
2.2. Pseudo-Code and Methodology Description
Algorithm Specifics for the Monte Carlo Solver of Wigner–Fokker–Planck
- Parameter Definitions
- L: Length of the domain in position space.
- and : Diagonal matrix elements of the covariance matrix of the Wigner function at the initial time.
- : Time step for the simulation.
- : Mean value vector for the initial distribution.
- and : Components of the mean value vector for the position and momentum, respectively.
- and : Diffusion coefficients for position and momentum, respectively.
- D: Diffusion matrix (assumed having zero off-diagonal terms).
- : Friction coefficient.
- Array Definitions
- q: Array to store position values for each particle at each time step.
- p: Array to store momentum values for each particle at each time step.
- Initial Conditions
- Gaussian Sampling: Initialize the position and momentum of each particle at the first time step using normal random number generation with mean components and standard deviations , .
- Time Evolution Loop
- Use nested loops to iterate over each time step i and each particle j.
- Generate random noise using a multivariate normal distribution with mean and as covariance matrix for position and momentum.
- Update the position and momentum of each particle using the Euler–Maruyama stochastic method.
- Simulation Output
- After the completion of the time evolution loop, the arrays q and p contain the simulated trajectories of position and momentum for each particle over time.
Algorithm 1 Euler–Maruyama for the Wigner–Fokker–Planck equation (harmonic potential) |
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3. Results
3.1. Wigner–Fokker–Planck Model: Gaussian States under Harmonic Potential
3.1.1. Husimi Transform of a Gaussian State (Gaussian Wigner Function)
3.1.2. Husimi Transform of Steady-State Solution for Harmonic Benchmark Problem
3.1.3. Husimi Transform of Harmonic Groundstate
3.1.4. Wehrl Entropy of a Gaussian State through Its Husimi Transform
3.1.5. Numerical Results of Entropy Behavior
4. Discussion
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Weighted L2-Distance between Wigner Function and Steady-State Solution for Harmonic Potential
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Edrisi, A.; Patwa, H.; Morales Escalante, J.A. A Numerical Study of Quantum Entropy and Information in the Wigner–Fokker–Planck Equation for Open Quantum Systems. Entropy 2024, 26, 263. https://doi.org/10.3390/e26030263
Edrisi A, Patwa H, Morales Escalante JA. A Numerical Study of Quantum Entropy and Information in the Wigner–Fokker–Planck Equation for Open Quantum Systems. Entropy. 2024; 26(3):263. https://doi.org/10.3390/e26030263
Chicago/Turabian StyleEdrisi, Arash, Hamza Patwa, and Jose A. Morales Escalante. 2024. "A Numerical Study of Quantum Entropy and Information in the Wigner–Fokker–Planck Equation for Open Quantum Systems" Entropy 26, no. 3: 263. https://doi.org/10.3390/e26030263
APA StyleEdrisi, A., Patwa, H., & Morales Escalante, J. A. (2024). A Numerical Study of Quantum Entropy and Information in the Wigner–Fokker–Planck Equation for Open Quantum Systems. Entropy, 26(3), 263. https://doi.org/10.3390/e26030263