# Nonclassicality by Local Gaussian Unitary Operations for Gaussian States

^{1}

^{2}

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Gaussian States and Gaussian Unitary Operations

**Lemma**

**1**

**Lemma**

**2**

**.**If ${\rho}_{AB}\in S({H}_{A}\otimes {H}_{B})$ is an $(n+m)$-mode Gaussian state, then ${\rho}_{AB}$ is a product state, that is, ${\rho}_{AB}={\sigma}_{A}\otimes {\sigma}_{B}$ for some ${\sigma}_{A}\in \mathcal{S}({H}_{A})$ and ${\sigma}_{B}\in \mathcal{S}({H}_{B})$, if and only if $\Gamma ={\Gamma}_{A}\oplus {\Gamma}_{B}$, where Γ, ${\Gamma}_{A}$ and ${\Gamma}_{B}$ are the CMs of ${\rho}_{AB}$, ${\sigma}_{A}$ and ${\sigma}_{B}$, respectively.

**Lemma**

**3**

**.**Assume that ρ is any n-mode Gaussian state with CM Γ and displacement vector $\mathbf{d}$, and ${U}_{\mathbf{S},\mathbf{m}}$ is a Gaussian unitary operator. Then, the characteristic function of the Gaussian state $\sigma =U\rho {U}^{\u2020}$ is of the form $exp(-\frac{1}{4}{z}^{\mathrm{T}}{\Gamma}_{\sigma}z+i{\mathbf{d}}_{\sigma}^{\mathrm{T}}z)$, where ${\Gamma}_{\sigma}=\mathbf{S}\Gamma {\mathbf{S}}^{\mathrm{T}}$ and ${\mathbf{d}}_{\sigma}=\mathbf{m}+\mathbf{S}\mathbf{d}$.

## 3. Quantum Correlation Introduced by Gaussian Unitary Operations

**Definition**

**1.**

**Remark**

**1.**

**Proposition**

**1**(Local Gaussian unitary invariance)

**.**

**Proof**

**of**

**Proposition**

**1.**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2.**

**Theorem**

**3.**

**Proof**

**of**

**Theorem**

**3.**

## 4. Comparison with Other Quantum Correlations

**Proposition**

**2.**

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**(

**a**) $\mathcal{N}({\rho}_{AB})$ for separable SSTSs as a function of $\mu $ and ${\overline{v}}_{-}$; (

**b**) from top to bottom, ${\overline{v}}_{-}$ = 1.0, 1.2, 1.5, 2.0.

**Figure 2.**(

**a**) $\mathcal{N}({\rho}_{AB})$ for entangled SSTS as a function of $\mu $ and ${\overline{v}}_{-}$; (

**b**) from top to bottom, ${\overline{v}}_{-}=0.1,0.2,0.5,0.8$.

**Figure 3.**$\mathcal{N}({\rho}_{AB})$ for SSTS as a function of $\overline{n}$ and r. (

**a**) from top to bottom $\overline{n}=0,0.5,1,2,3$; (

**b**) from top to bottom $r=0.5,1,5,10,20.$

**Figure 5.**Comparison with ${D}_{G}({\rho}_{AB})$ for nonsymmetric STS. (

**a**) and (

**b**) are correspond to nonsymmetric STS with $r=0.5,5$, respectively.

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Wang, Y.; Qi, X.; Hou, J.
Nonclassicality by Local Gaussian Unitary Operations for Gaussian States. *Entropy* **2018**, *20*, 266.
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**AMA Style**

Wang Y, Qi X, Hou J.
Nonclassicality by Local Gaussian Unitary Operations for Gaussian States. *Entropy*. 2018; 20(4):266.
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**Chicago/Turabian Style**

Wang, Yangyang, Xiaofei Qi, and Jinchuan Hou.
2018. "Nonclassicality by Local Gaussian Unitary Operations for Gaussian States" *Entropy* 20, no. 4: 266.
https://doi.org/10.3390/e20040266