# Gaussian States: Evaluation of the Covariance Matrix from the Implementation with Primitive Component

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Gaussian Unitaries

#### 2.1. Gaussian Unitary in Terms of Fundamental Unitaries (FGUs)

- N-mode displacement operator$$D\left(\mathbf{\alpha}\right):={\mathrm{e}}^{{\mathbf{\alpha}}^{\mathrm{T}}{\mathbf{a}}_{*}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}-{\mathbf{\alpha}}^{*}\phantom{\rule{0.166667em}{0ex}}\mathbf{a}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}\phantom{\rule{2.em}{0ex}}\mathbf{\alpha}={[{\mathbf{\alpha}}_{1},\dots ,{a}_{n}]}^{\mathrm{T}}\phantom{\rule{0.166667em}{0ex}}\in {\mathbb{C}}^{N}$$
- N-mode rotation operator$$R\left(\mathbf{\varphi}\right):={\mathrm{e}}^{\phantom{\rule{0.166667em}{0ex}}i\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\mathbf{a}}^{*}\mathbf{\varphi}\phantom{\rule{0.166667em}{0ex}}\mathbf{a}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\mathbf{\varphi}\phantom{\rule{1.em}{0ex}}N\times N\phantom{\rule{4.pt}{0ex}}\mathrm{Hermitian}\phantom{\rule{4.pt}{0ex}}\mathrm{matrix}$$
- N-mode squeeze operator$$S\left(\mathbf{z}\right):={\mathrm{e}}^{\frac{1}{2}\left[\phantom{\rule{0.166667em}{0ex}}({\mathbf{a}}^{*}\phantom{\rule{0.166667em}{0ex}}\mathbf{z}\phantom{\rule{0.166667em}{0ex}}{\mathbf{a}}_{*}-{\mathbf{a}}^{\mathrm{T}}\phantom{\rule{0.166667em}{0ex}}{\mathbf{z}}^{*}\phantom{\rule{0.166667em}{0ex}}a)\right]}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mathbf{z}\phantom{\rule{1.em}{0ex}}N\times N\phantom{\rule{4.pt}{0ex}}\mathrm{symmetric}\phantom{\rule{4.pt}{0ex}}\mathrm{matrix}$$

**Theorem**

**1.**

#### 2.2. Gaussian Unitaries Decomposed according to the Block–Messiah Reduction

**Theorem**

**2.**

#### Specification and Degree of Freedom

## 3. Gaussian States

#### 3.1. Specification and Degree of Freedom

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

## 4. Implementation with Primitive Components

#### 4.1. Primitive Components

#### 4.2. Implementation in the Two Mode

**Proposition**

**1.**

#### 4.3. Strategy for the Evaluation

**Example**

**1.**

## 5. Evaluation of Symplectic Matrices

#### The Symplectic Matrix (SM)

## 6. Evaluation of the Covariance Matrix (CM)

#### 6.1. Standard Form of the Covariance Matrix

**Evaluation of “invariants” $(\mathbf{a},\mathbf{b},{\mathbf{c}}_{\pm}$) from the ordinary CM $\mathbf{V}$.**The invariants can be obtained from the blocks of the ordinary (nonstandard) covariance matrix (see (30)).

**Proposition**

**2.**

#### 6.2. Physical Analysis of the Global Architectures

## 7. Examples

## 8. Separability and Entanglement

## 9. Conclusions

- The structural approach is
**completely radical free**compared to the algebraic approach and requires several matrix operations leading to results that contain the radicals of radicals. Note that the key to avoiding radicals is the following: the quantities which exhibit radicals in the algebraic approach become independent variables (data) in the structural approach. - The structural approach is completely general, while the algebraic approach exhibits several degeneracies (mainly coincident eigenvalues) concerning some very important cases (see EPR states). These cases should be treated separately with ad hoc procedures. Such a distinction is not required in the structural approach.
- In the structural approach, all the variables have a precise physical meaning, related to the corresponding components of the architecture, i.e., squeezers, beam splitters, phase shifters, and one can choose the specific variables to achieve the desired properties of the covariance matrix, for example, the entanglement.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Algebraic Approach

#### Appendix A.1. Polar Decomposition of the Squeeze Matrix **z**

#### Appendix A.2. Evaluation of the Matrices cosh(**r**) and sinh(**r**)

#### Appendix A.3. Evaluation of the Phase Matrix **ϕ**

#### Appendix A.4. Evaluation of Matrices cos**ϕ** and sin**ϕ**

## Appendix B

**Proof of Proposition**

**2.**

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**Figure 5.**Implementation with primitive components of the most general two-mode Gaussian unitary, according to the Bloch–Messiah reduction and the Takagi factorization of the squeeze matrix.

**Figure 6.**Generation with primitive components of a general two-mode Gaussian state, starting from two thermal states ${\mathcal{N}}_{1}$ and ${\mathcal{N}}_{2}$. The two initial rotations ${\psi}_{11}$ and ${\psi}_{12}$ and the final displacements ${\alpha}_{1}$, ${\alpha}_{2}$ are not introduced because they do not influence the covariance matrix.

thermal states | ${\nu}_{1}$ | ${\nu}_{2}$ | ||

phases | $\mu =0$ | ${\gamma}_{11}=0$ | ${\gamma}_{12}=0$ | $\u03f5=0$ |

squeezers | ${r}_{1}$ | ${r}_{2}$ | ||

beam splitters | p | q | s | c |

thermal states | ${\nu}_{1}$ | ${\nu}_{2}$ | ||

phases | $\mu =0$ | ${\gamma}_{11}=0$ | ${\gamma}_{12}=0$ | $\u03f5=0$ |

squeezers | ${r}_{1}={{r}}_{{0}}$ | ${r}_{2}={{r}}_{{0}}$ | ||

beam splitters | $p=1/\sqrt{2}$ | $q=1/\sqrt{2}$ | s | c |

states | ${\nu}_{1}$ | ${\nu}_{2}$ | ||

phases | $\mu =\pi /2$ | ${\gamma}_{11}=-\pi /4$ | ${\gamma}_{12}=\pi /4$ | $\u03f5=-\pi /2$ |

squeezers | ${r}_{1}={{r}}_{{0}}$ | ${r}_{2}={{r}}_{{0}}$ | ||

beam splitters | $s=1/\sqrt{2}$ | $c=1/\sqrt{2}$ | $p=1/\sqrt{2}$ | $q=1/\sqrt{2}$ |

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Cariolaro, G.; Corvaja, R.; Miatto, F.
Gaussian States: Evaluation of the Covariance Matrix from the Implementation with Primitive Component. *Symmetry* **2022**, *14*, 1286.
https://doi.org/10.3390/sym14071286

**AMA Style**

Cariolaro G, Corvaja R, Miatto F.
Gaussian States: Evaluation of the Covariance Matrix from the Implementation with Primitive Component. *Symmetry*. 2022; 14(7):1286.
https://doi.org/10.3390/sym14071286

**Chicago/Turabian Style**

Cariolaro, Gianfranco, Roberto Corvaja, and Filippo Miatto.
2022. "Gaussian States: Evaluation of the Covariance Matrix from the Implementation with Primitive Component" *Symmetry* 14, no. 7: 1286.
https://doi.org/10.3390/sym14071286