# Implementation of Two-Mode Gaussian States Whose Covariance Matrix Has the Standard Form

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Gaussian Unitaries and Gaussian States

#### 2.1. Degrees of Freedom

#### 2.2. Gaussian States

#### 2.3. Gaussian States in the Two-Mode

## 3. Implementation with Primitive Components

#### Implementation in the Two-Mode

## 4. Evaluation of the Covariance Matrix

#### 4.1. The Symplectic Matrix

#### 4.2. The Covariance Matrix (cm)

**Remark**

**1.**

## 5. The Standard Form of the Covariance Matrix

- For every two-mode Gaussian state having the ordinary CM $\mathbf{V}$, it is possible to obtain the corresponding standard form ${\mathbf{V}}_{sf}$ from $\mathbf{V}$ with a local symplectic transformation ${\mathbf{S}}_{l}$.
- The standard form ${\mathbf{V}}_{sf}$ contains all the relevant information on the Gaussian state, so that the transformation $\mathbf{V}\u27f6{\mathbf{V}}_{sf}$ may be considered as the removal of the redundancy in $\mathbf{V}$.

#### 5.1. Properties of Symplectic Invariants

#### Meaning of the CM Entries According to Probability Theory

- ${q}_{1},{p}_{1}$ are uncorrelated with the same variance ${\sigma}_{{q}_{1}}^{2}={\sigma}_{{p}_{1}}^{2}=a$;
- ${q}_{2},{p}_{2}$ are uncorrelated with the same variance ${\sigma}_{{q}_{2}}^{2}={\sigma}_{{p}_{2}}^{2}=b$;
- ${q}_{1},{q}_{2}$ have cross–covariance ${v}_{{q}_{1}{q}_{2}}={c}_{+}$ and then normalized covariance${c}_{{q}_{1}{q}_{2}}=\frac{{c}_{+}}{\sqrt{ab}}\phantom{\rule{1.em}{0ex}}\to \phantom{\rule{1.em}{0ex}}0\le \left|{c}_{+}\right|\le \sqrt{ab}$;
- ${p}_{1},{p}_{2}$ have cross–covariance ${v}_{{p}_{1}{p}_{2}}={c}_{-}$ and then normalized covariance${c}_{{p}_{1}{p}_{2}}=\frac{{c}_{-}}{\sqrt{ab}}\phantom{\rule{1.em}{0ex}}\to \phantom{\rule{1.em}{0ex}}0\le \left|{c}_{-}\right|\le \sqrt{ab}$;
- $({q}_{1},{p}_{1})$, $({q}_{1},{p}_{2})$, $({q}_{2},{p}_{1})$ and $({q}_{2},{p}_{2})$ are uncorrelated pairs.

#### 5.2. The Correlations $(A,B,{C}_{\pm})$ from the Ordinary Cm $\mathbf{V}$

**Proposition**

**1.**

#### 5.3. The Standard Form Ii (Sf–Ii)

**Proposition**

**2.**

## 6. Gallery of Covariance Matrices and Classification

**standard Gaussian state**. Before we discuss the forms of interest encountered in the solution of our problem, we believe it will be convenient to discuss several forms of CMs, which are collected in Table 1.

- Full SF: is the class obtained by imposing the conditions $a\ne b$, $|{c}_{-}|\ne |{c}_{+}|$.
- Lateral–symmetric SF: is the class in which $a\ne b$, ${c}_{-}={c}_{+}$.
- Lateral–antiymmetric SF: is the class in which $a\ne b$, ${c}_{=}-{c}_{+}$.

- standard variables: $(a,b,{c}_{+},{c}_{-})$
- standard II variables: $({a}_{1},{a}_{2},{b}_{1},{b}_{2},{c}_{1},{c}_{2})$
- physical variables: $({n}_{1},{n}_{2},s,\mu ,{r}_{1},{r}_{2},{\gamma}_{11},{\gamma}_{12},p,\u03f5)$

## 7. Two Fundamental Cases

#### 7.1. EPR State with Noise

#### 7.2. Cases Obtained by Setting All the Phases to Zero

#### 7.3. Physical Variables from the Standard Variables II

#### 7.4. Physical Variables from the Standard Variables

#### 7.4.1. Thermal Photon Numbers

#### 7.4.2. Squeeze Parameters

#### 7.4.3. BS Parameters

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Solution of the System (39) to (42)

- it is not a limitation to assume that in the beam-splitters $p,q,c,s\ge 0$
- ${({q}^{2}-{p}^{2})}^{2}=1-{\left(2pq\right)}^{2}$
- ${cosh}^{2}(\Delta r)-{sinh}^{2}(\Delta r)=1$

## Appendix B. Possible Approaches for the Use of This Theory

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**Figure 4.**Scheme with primitive components for the generation of a general two-mode Gaussian state, starting from two thermal states, ${\mathcal{N}}_{1}$ and ${\mathcal{N}}_{2}$. The architecture does not contain the irrelevant initial rotations ${\psi}_{11}$ and ${\psi}_{12}$ and the final displacements ${\alpha}_{1}$ and ${\alpha}_{2}$, which do not influence the covariance matrix.

**Figure 5.**The local symplectic matrices ${\mathbf{S}}_{1}$ and ${\mathbf{S}}_{2}$ that provide the transformation of the SF-II to the SF: ${\mathbf{V}}_{sf,II}\to {\mathbf{V}}_{sf,I}$.

**Figure 6.**

**Left**: The standard variables $(a,b,{c}_{+},{c}_{-})$ as a function of ${r}_{2}$, for $s=0.3$, ${n}_{1}=3.1$, ${n}_{2}=2.1$.

**Right**: The standard variables $(a,b,{c}_{+},{c}_{-})$ as a function of s, for ${r}_{2}=0.7$, ${n}_{1}=3.1$, ${n}_{2}=2.1$.

**Figure 7.**The physical variables $({n}_{1},{n}_{2})$ (

**left**) and $({r}_{1},{r}_{2})$ (

**right**) as functions of a for $b=2.62$, ${c}_{+}=1.29$, ${c}_{-}=1.36$.

**Figure 8.**The physical variables $(p,s)$ (as functions of a) for $b=2.62$, ${c}_{+}=1.29$, ${c}_{-}=1.36$.

**Figure 9.**The physical variables $({n}_{1},{n}_{2})$ (

**left**) and $({r}_{1},{r}_{2})$ (

**right**) as functions of ${c}_{+}$ for $a=2.5$, $b=2.8$, ${c}_{-}=1.35$.

Type | Covariance Matrix | Degrees of Fredom |
---|---|---|

general | $\mathbf{V}=\left[\begin{array}{cccc}{a}_{11}& {a}_{12}& {c}_{11}& {c}_{12}\\ {a}_{12}& {a}_{22}& {c}_{12}& {c}_{22}\\ {c}_{11}& {c}_{12}& {b}_{11}& {b}_{12}\\ {c}_{12}& {c}_{22}& {b}_{12}& {b}_{22}\end{array}\right]$ | 10 real variables |

standard form II | ${\mathbf{V}}_{sf}^{II}=\left[\begin{array}{cccc}{a}_{1}& 0& {c}_{1}& 0\\ 0& {a}_{2}& 0& {c}_{2}\\ {c}_{1}& 0& {b}_{1}& 0\\ 0& {c}_{2}& 0& {b}_{2}\end{array}\right]$ | 6 real variables |

standard form (SF) | ${\mathbf{V}}_{sf}=\left[\begin{array}{cccc}a& 0& {c}_{+}& 0\\ 0& a& 0& {c}_{-}\\ {c}_{+}& 0& b& 0\\ 0& {c}_{-}& 0& b\end{array}\right]$ | 4 real variables |

SF lateral symmetric | ${\mathbf{V}}_{sf}^{LS}=\left[\begin{array}{cccc}a& 0& c& 0\\ 0& a& 0& c\\ c& 0& b& 0\\ 0& c& 0& b\end{array}\right]$ | 3 real variables |

SF lateral antisymmetric | ${\mathbf{V}}_{sf}^{LA}=\left[\begin{array}{cccc}a& 0& c& 0\\ 0& a& 0& -c\\ c& 0& b& 0\\ 0& -c& 0& b\end{array}\right]$ | 3 real variables |

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Cariolaro, G.; Corvaja, R.
Implementation of Two-Mode Gaussian States Whose Covariance Matrix Has the Standard Form. *Symmetry* **2022**, *14*, 1485.
https://doi.org/10.3390/sym14071485

**AMA Style**

Cariolaro G, Corvaja R.
Implementation of Two-Mode Gaussian States Whose Covariance Matrix Has the Standard Form. *Symmetry*. 2022; 14(7):1485.
https://doi.org/10.3390/sym14071485

**Chicago/Turabian Style**

Cariolaro, Gianfranco, and Roberto Corvaja.
2022. "Implementation of Two-Mode Gaussian States Whose Covariance Matrix Has the Standard Form" *Symmetry* 14, no. 7: 1485.
https://doi.org/10.3390/sym14071485