# End-to-End Entanglement Generation Strategies: Capacity Bounds and Impact on Quantum Key Distribution

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## Abstract

**:**

## 1. Introduction

- Offering a brief introduction on the evolution towards the Quantum Internet. Specifically, the three main research and innovation avenues are mentioned: quantum transmission, networking protocols and management-control paradigms;
- Providing an overview of current strategies for end-to-end entanglement generation, with the discussion of capacity upper bounds and their impact of secret key rate in QKD systems;
- Presenting the simulation of a simple quantum network model with a certain number of QRs.

## 2. Toward the Quantum Internet

#### 2.1. Network Protocols

#### 2.2. Management and Control

## 3. End-To-End Entanglement Generation

#### 3.1. Entanglement as a Resource

#### 3.2. Quantum Repeater Networks

#### 3.3. Capacity of a QR Chain

- $T(n,k,{L}_{tot})$ is the time to generate a Bell state over the total distance ${L}_{tot}$, using an n-nested QR configuration with k rounds of purification for each nesting level; this means that the number of intermediate QR nodes is ${2}^{n-1}$, and the distance between nodes is $L={L}_{tot}/{2}^{n}$;
- $M(k,n)$ is the number of quantum memories used; discounting R with M provides a fairer comparison when different purification protocols are used [12].

**Theorem**

**1.**

**Proof.**

#### 3.4. Secret Key Rate for QKD

## 4. Simulation-Based Performance Evaluation

#### 4.1. NetSquid

#### 4.2. Simulation Model of a Quantum Repeater Chain

#### 4.3. Results

- c = $2\times {10}^{8}$ m/s
- ${p}_{di}=\{0.006,0.012\}$
- ${p}_{dl}=\{0.015,0.025\}$
- ${T}_{1}=\{\infty ,2.68\phantom{\rule{4.pt}{0ex}}\mathrm{ms}\}$
- ${T}_{2}=\{1.46\phantom{\rule{4.pt}{0ex}}\mathrm{s},1\phantom{\rule{4.pt}{0ex}}\mathrm{ms}\}$

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations

BSM | Bell-State Measurement |

QKD | Quantum Key Distribution |

QR | Quantum Repeater |

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**Figure 1.**Operations of a QR network with hierarchical BSMs: in the initial entanglement distribution, multiple Bell states are established across each pair of adjacent repeater nodes; then, purification and swapping iterations are executed, until the end-to-end entangled state has been generated.

**Figure 2.**Capacity upper bounds for QR chains with entanglement swapping and k rounds of purification, using Deutsch’s protocol.

**Figure 3.**Capacity of a QR chain with simultaneous BSMs, using M parallel channels, and a time-multiplexing block length m. Here, we assume $\alpha =0.15$ dB/km, $\tau =50$ ns, $\mu =0.405$, and $q=0.255$.

**Figure 4.**Capacity of a QR chain with simultaneous BSMs, using M parallel channels, and a time-multiplexing block length m. We assume a 3-link QR chain with ${\alpha}_{1}=0.25$ dB/km, ${\alpha}_{2}=0.15$ dB/km, ${\alpha}_{3}=0.25$ dB/km, $\tau =50$ ns, $\mu =0.405$, and $q=0.255$. In (

**a**), the three links have lengths ${L}_{1}=25$ km, ${L}_{2}=40$ km, and ${L}_{3}=30$ km, respectively. In (

**b**), the lengths of all the three links are doubled.

**Figure 5.**Fidelity vs. number of nodes, for different total lengths; (

**left**) ${p}_{di}=0.006$, ${p}_{dl}=0.015$; (

**right**) ${p}_{di}=0.012$, ${p}_{dl}=0.025$. In both cases, ${T}_{1}=\infty $ and ${T}_{2}=1.46$ s. No purification is applied.

**Figure 6.**Fidelity vs. number of nodes, for different total lengths; (

**left**) ${p}_{di}=0.006$, ${p}_{dl}=0.015$; (

**right**) ${p}_{di}=0.012$, ${p}_{dl}=0.025$. In both cases, ${T}_{1}=2.68$ ms and ${T}_{2}=1$ ms. No purification is applied.

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**MDPI and ACS Style**

Manzalini, A.; Amoretti, M.
End-to-End Entanglement Generation Strategies: Capacity Bounds and Impact on Quantum Key Distribution. *Quantum Rep.* **2022**, *4*, 251-263.
https://doi.org/10.3390/quantum4030017

**AMA Style**

Manzalini A, Amoretti M.
End-to-End Entanglement Generation Strategies: Capacity Bounds and Impact on Quantum Key Distribution. *Quantum Reports*. 2022; 4(3):251-263.
https://doi.org/10.3390/quantum4030017

**Chicago/Turabian Style**

Manzalini, Antonio, and Michele Amoretti.
2022. "End-to-End Entanglement Generation Strategies: Capacity Bounds and Impact on Quantum Key Distribution" *Quantum Reports* 4, no. 3: 251-263.
https://doi.org/10.3390/quantum4030017