# Differential Parametric Formalism for the Evolution of Gaussian States: Nonunitary Evolution and Invariant States

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## Abstract

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## 1. Introduction

## 2. One-Dimensional Quantum Quadratic Hamiltonian and Its Linear Invariant Operators

#### 2.1. Dynamics of Non-Pure States

#### Example

#### 2.2. Invariant States

## 3. Multidimensional Quadratic System

## 4. Nonunitary Evolution for Gaussian Subsystems

#### 4.1. Nonunitary Evolution on a Bipartite System

#### 4.2. Invariant and Quasi-Invariant States

#### 4.3. Frequency Converter

#### 4.4. Parametric Amplifier

## 5. Gaussian States and Their Evolution in the Tomographic-Probability Representation

## 6. Two-Mode Gaussian States in the Tomographic-Probability Representation

## 7. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Correspondence between the Gaussian Density Matrix Parameters and the Covariance Matrix

## Appendix B. Matrix M for Bipartite System

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**Figure 1.**(

**a**) Mean values $\langle \widehat{p}\rangle \left(t\right)$ (black) and $\langle \widehat{q}\rangle \left(t\right)$ (gray) for the dynamics of Hamiltonian (18) and the state with initial conditions $\langle \widehat{p}\rangle \left(0\right)=0$ and $\langle \widehat{q}\rangle \left(0\right)=1$. (

**b**) Covariances ${\sigma}_{pp}\left(t\right)$ (black), ${\sigma}_{qq}\left(t\right)$ (gray), and ${\sigma}_{pq}\left(t\right)$ (dashed) for the initial state with ${\sigma}_{pp}\left(0\right)={\sigma}_{qq}=1$ and ${\sigma}_{pq}\left(0\right)=1/\sqrt{2}$. In both cases, we took frequencies $\omega =2$ and $\nu =1$.

**Figure 2.**Time evolution for the covariances (

**a**) ${\sigma}_{{p}_{1}{p}_{1}}$ (black), ${\sigma}_{{q}_{1}{q}_{1}}$ (dashed), and ${\sigma}_{{p}_{2}{p}_{2}}={\sigma}_{{q}_{2}{q}_{2}}$ (gray) and (

**b**) the covariances ${\sigma}_{{p}_{1}{p}_{2}}$ (black), ${\sigma}_{{p}_{1}{q}_{2}}$ (black dashed), ${\sigma}_{{p}_{2}{q}_{1}}$ (black dot-dashed), and ${\sigma}_{{q}_{1}{q}_{2}}$ (gray), (

**c**) $det{\sigma}_{1}=det{\sigma}_{2}$ for the subsystems (black) and the time dependence of the mean value $\langle \widehat{H}\rangle \left(t\right)$ (gray). For all the plots, the initial values are ${\sigma}_{{p}_{1}{p}_{1}}\left(0\right)=1$, ${\sigma}_{{q}_{1}{q}_{1}}\left(0\right)=1/4$, and ${\sigma}_{{p}_{2}{p}_{2}}\left(0\right)={\sigma}_{{q}_{2}{q}_{2}}\left(0\right)=1/2$. All the other initial covariances are equal to zero. The frequencies used are ${\omega}_{1}=2$, ${\omega}_{2}=1$, $\omega =7$, and $\kappa =\sqrt{10}$.

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**MDPI and ACS Style**

López-Saldívar, J.A.; Man’ko, M.A.; Man’ko, V.I.
Differential Parametric Formalism for the Evolution of Gaussian States: Nonunitary Evolution and Invariant States. *Entropy* **2020**, *22*, 586.
https://doi.org/10.3390/e22050586

**AMA Style**

López-Saldívar JA, Man’ko MA, Man’ko VI.
Differential Parametric Formalism for the Evolution of Gaussian States: Nonunitary Evolution and Invariant States. *Entropy*. 2020; 22(5):586.
https://doi.org/10.3390/e22050586

**Chicago/Turabian Style**

López-Saldívar, Julio A., Margarita A. Man’ko, and Vladimir I. Man’ko.
2020. "Differential Parametric Formalism for the Evolution of Gaussian States: Nonunitary Evolution and Invariant States" *Entropy* 22, no. 5: 586.
https://doi.org/10.3390/e22050586