# Continuous-Variable Entanglement Swapping

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Formalism Review

^{†}respectively. These operators obey the commutation relations $[{\widehat{a}}_{i},{\widehat{a}}_{j}^{\u2020}]={\delta}_{ij}$ and $[{\widehat{a}}_{i},{\widehat{a}}_{j}]=[{\widehat{a}}_{i}^{\u2020},{\widehat{a}}_{j}^{\u2020}]=0$. The space ${\mathscr{H}}_{k}$ is spanned by the Fock basis ${\left\{|{n}_{k}\rangle \right\}}_{n=0}^{\infty}$ of eigenstates of the number operator ${\widehat{n}}_{k}={\widehat{a}}_{k}^{\u2020}{\widehat{a}}_{k}$. These eigenstates have the property that $\widehat{n}|n\rangle =n|n\rangle $, $\widehat{a}|n\rangle =\sqrt{n|}n-1\rangle $, ${\widehat{a}}^{\u2020}|n\rangle =\sqrt{n+1}|n+1\rangle $, as well as the fact that the vacuum state |0⟩ is annihilated by â|0⟩ = 0. In the absence of any interactions, these modes evolve according to the Hamiltonian $H={\displaystyle {\sum}_{k=1}^{N}({\widehat{a}}_{k}^{\u2020}{\widehat{a}}_{k}+1/2)}$. We can define two quadrature operators ${\widehat{q}}_{k}={\widehat{a}}_{k}+{\widehat{a}}_{k}^{\u2020}$ and ${\widehat{p}}_{k}=i({\widehat{a}}_{k}^{\u2020}-{\widehat{a}}_{k})$ which act in a similar fashion to the position and momentum operators in the quantum harmonic oscillator. These commutation relations can be compactly written by defining

_{S,d}generated from a quadratic Hamiltonian there exists a symplectic operator S and vector d which generate the mapping $\widehat{\mathrm{x}}\to S\widehat{\mathrm{x}}+\mathrm{d}$. This transforms the moments of a Gaussian state as $\overline{\mathrm{x}}\to S\overline{\mathrm{x}}+\mathrm{d}$ and V → SV S

^{T}.

## 3. Entanglement Swapping

#### 3.1. Gaussian

_{12}and σ

_{34}we find that by choosing the gains on both remaining modes in an optimal fashion we obtain the new covariance matrix [19]

^{2}/(a + b) and where we have chosen c

_{+}= −c

_{−}= c for simplicity.

#### 3.2. Non-Gaussian

## 4. Summary

## Conflicts of Interest

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Marshall, K.; Weedbrook, C.
Continuous-Variable Entanglement Swapping. *Entropy* **2015**, *17*, 3152-3159.
https://doi.org/10.3390/e17053152

**AMA Style**

Marshall K, Weedbrook C.
Continuous-Variable Entanglement Swapping. *Entropy*. 2015; 17(5):3152-3159.
https://doi.org/10.3390/e17053152

**Chicago/Turabian Style**

Marshall, Kevin, and Christian Weedbrook.
2015. "Continuous-Variable Entanglement Swapping" *Entropy* 17, no. 5: 3152-3159.
https://doi.org/10.3390/e17053152