# Continuous Variables Graph States Shaped as Complex Networks: Optimization and Manipulation

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## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Background: Cluster States and Complex Networks

#### 2.2. Improving the Overall Quality of a Complex Cluster

#### 2.3. Concentrating the Squeezing

#### 2.4. Creating a Quantum Channel between Nodes by Manipulating Existing Networks

## 3. Discussion

## 4. Materials and Methods

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Graphs whose results for the creation of an EPR channel between Alice (green) and Bob (blue) are shown in this Appendix. (

**a**) 6-mode grid cluster; (

**b**) 6-mode fully connected cluster.

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**Figure 1.**Rewiring of a regular 48-node network for the construction of a “small world” network as shown in [35].

**Figure 2.**Comparison between two models of complex networks, both with an average degree of $\langle k\rangle \sim 3.9$. The size of the dots increases with the number of links. (

**a**) Barabási–Albert model with ${m}_{BA}=2$, with a maximum node degree of k = 22; (

**b**) Erdős–Rényi model with ${p}_{ER}=4/49$, with a maximum node degree of k = 8.

**Figure 3.**Realistic implementation of complex cluster states. (

**a**) implementation of a cluster via a linear optics transformation acting on a series of squeezed modes; (

**b**) list of realistic squeezing values of the input modes for the implementation of a 48-mode cluster.

**Figure 4.**Plot of the mean squeezing value of the nullifiers of the cluster as a function of its average degree $\langle k\rangle $ for the different topologies of complex graphs. In the legend, “BA” = Barabási–Albert, “ER” = Erdős–Rényi, “WS p = 0” = Watts–Strogatz with ${p}_{WS}=0$, “WS p = 0.25” = Watts–Strogatz with ${p}_{WS}=0.25$ and “WS p = 0.5” = Watts–Strogatz with ${p}_{WS}=0.5$.

**Figure 5.**(

**a**) a quantum network is created; (

**b**) the resource is distributed to two spatially separated teams, Alice and Bob; (

**c**) Alice performs a linear optics operation ${U}_{A}$ on her set of nodes and Bob performs a linear optics operation ${U}_{B}$ on his set of nodes to create a quantum channel out of two given nodes; (

**d**) the quantum channel is established.

**Figure 6.**Graphs analyzed with the aim of creating an EPR channel between Alice (green) and Bob (blue) (or eventually between nodes of the same team).

**Table 1.**Mean $\mu $ and standard deviation $\sigma $ of the values ${\mu}_{j}$ of Equation (4) evaluated on $N=100$ Barabási–Albert (a) and Erdős–Rényi (b) graphs with different characterizing parameters and consequently different average degrees $\langle k\rangle $, optimized using the function $f({\Delta}^{2}{\delta}_{i})={\sum}_{i}{\Delta}^{2}{\overline{\delta}}_{i}$. Without the optimization protocol, $\mu $ takes the value of −3.48 dB for the Erdős–Rényi model, independently of the value of ${p}_{ER}$, and it oscillates between −3.48 dB and −3.72 dB for the Barabási–Albert model.

(a) Barabási–Albert | (b) Erdős–Rényi | ||||||
---|---|---|---|---|---|---|---|

${\mathit{m}}_{\mathit{B}\mathit{A}}$ | $\mathit{\mu}$ (dB) | $\mathbf{[}\mathit{\mu}\mathbf{\pm}\mathit{\sigma}\mathbf{]}$ (dB) | $\mathbf{\langle}\mathit{k}\mathbf{\rangle}$ | ${\mathit{p}}_{\mathit{E}\mathit{R}}$ | $\mathit{\mu}$ (dB) | $\mathbf{[}\mathit{\mu}\mathbf{\pm}\mathit{\sigma}\mathbf{]}$ (dB) | $\overline{\mathbf{\langle}{\mathit{k}}_{\mathit{j}}\mathbf{\rangle}}$ |

1 | −4.70 | [−4.73,−4.67] | 1.96 | 0.2 | −5.50 | [−5.54,−5.46] | 9.35 |

5 | −5.55 | [−5.58, −5.53] | 9.38 | 0.4 | −5.80 | [−5.83, −5.76] | 18.83 |

10 | −5.82 | [−5.84, −5.80] | 17.71 | 0.6 | −6.02 | [−6.04, −6.00] | 28.29 |

20 | −6.15 | [−6,16, −6.14] | 31.25 | 0.8 | −6.22 | [−6.23, −6.21] | 37.58 |

47 | −6.33 | [−6.33, −6.33] | 47 | 1 | −6.33 | [−6.33, −6.33] | 47 |

**Table 2.**Mean $\mu $ and standard deviation $\sigma $ of the values ${\mu}_{j}$ of Equation (4) evaluated on $N=100$ Watts–Strogatz graphs with different parameter ${p}_{WS}$ and different $\langle k\rangle $, optimized using the function $f({\Delta}^{2}{\delta}_{i})={\sum}_{i}{\Delta}^{2}{\overline{\delta}}_{i}$. Without the optimization protocol, $\mu $ takes the value of $-3.48$ dB, independently of the value of ${p}_{WS}$ or $\langle k\rangle $.

(a) $\mathbf{\langle}\mathit{k}\mathbf{\rangle}\mathbf{=}\mathbf{4}$ | (b) $\mathbf{\langle}\mathit{k}\mathbf{\rangle}\mathbf{=}\mathbf{8}$ | ||||
---|---|---|---|---|---|

${\mathit{p}}_{\mathit{W}\mathit{S}}$ | $\mathit{\mu}$ (dB) | $\mathbf{[}\mathit{\mu}\mathbf{\pm}\mathit{\sigma}\mathbf{]}$ (dB) | ${\mathit{p}}_{\mathit{W}\mathit{S}}$ | $\mathit{\mu}$ (dB) | $\mathbf{[}\mathit{\mu}\mathbf{\pm}\mathit{\sigma}\mathbf{]}$ (dB) |

0 | −5.19 | $[-5.19,-5.19]$ | 0 | −5.79 | $[-5.79,-5.79]$ |

0.1 | −5.16 | $[-5.17,-5.14]$ | 0.1 | −5.69 | $[-5.71,-5.66]$ |

0.4 | −5.10 | $[-5.12,-5.07]$ | 0.4 | −5.49 | $[-5.51,-5.46]$ |

0.7 | −5.09 | $[-5.11,-5.07]$ | 0.7 | −5.43 | $[-5.46,-5.40]$ |

1 | −5.09 | $[-5.12,-5.06]$ | 1 | −5.43 | $[-5.46,-5.41]$ |

**Table 3.**Means ${\mu}_{12}$, ${\mu}_{13}$ and $\mu $ of the nullifiers of the nodes 12 and 13 and of the value ${\mu}_{j}$ of Equation (5) evaluated on $N=100$ Barabási–Albert graphs with different parameter ${m}_{BA}$, optimized using the function $f({\Delta}^{2}{\widehat{\delta}}_{i})={\sum}_{i}{A}_{i}{\Delta}^{2}\overline{{\delta}_{i}}$, where ${A}_{i}={10}^{5}$ if $i=12,13$ and ${A}_{i}=1$ otherwise.

${\mathit{m}}_{\mathit{B}\mathit{A}}$ | ${\mathit{\mu}}_{\mathbf{12}}$ (dB) | ${\mathit{\mu}}_{\mathbf{13}}$ (dB) | $\mathit{\mu}$ (dB) |
---|---|---|---|

1 | −6.51 | −6.51 | −4.61 |

5 | −6.51 | −6.51 | −5.48 |

10 | −6.51 | −6.51 | −5.76 |

20 | −6.51 | −6.51 | −6.10 |

47 | −6.51 | −6.51 | −6.32 |

**Table 4.**Results on the possibility to create a quantum communication channel for the graphs of Figure 6, between nodes belonging to different teams A and B and between nodes belonging to the same team.

Graph | Between A and B | Same Team |
---|---|---|

6-node grid | Yes | No |

8-node grid | No | No |

10-node grid | Yes | No |

“X” | Yes | No |

“Y” | Yes | No |

Fully-connected | No | Yes |

“Z” | No | No |

Dual-rail | No | No |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Sansavini, F.; Parigi, V.
Continuous Variables Graph States Shaped as Complex Networks: Optimization and Manipulation. *Entropy* **2020**, *22*, 26.
https://doi.org/10.3390/e22010026

**AMA Style**

Sansavini F, Parigi V.
Continuous Variables Graph States Shaped as Complex Networks: Optimization and Manipulation. *Entropy*. 2020; 22(1):26.
https://doi.org/10.3390/e22010026

**Chicago/Turabian Style**

Sansavini, Francesca, and Valentina Parigi.
2020. "Continuous Variables Graph States Shaped as Complex Networks: Optimization and Manipulation" *Entropy* 22, no. 1: 26.
https://doi.org/10.3390/e22010026