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

^{1}

^{2}

^{*}

## 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 |

## References

- Pederzolli, F.; Faticanti, F.; Siracusa, D. Optimal Design of Practical Quantum Key Distribution Backbones for Securing CoreTransport Networks. Quantum Rep.
**2020**, 2, 114–125. [Google Scholar] [CrossRef] [Green Version] - Bacco, D.; Vagniluca, I.; Da Lio, B.; Biagi, N.; Della Frera, A.; Calonico, D.; Toninelli, C.; Cataliotti, F.S.; Bellini, M.; Oxenløwe, L.K.; et al. Field trial of a three-state quantum key distribution scheme in the Florence metropolitan area. EPJ Quantum Technol.
**2019**, 6, 5. [Google Scholar] [CrossRef] [Green Version] - Honjo, T.; Nam, S.W.; Takesue, H.; Zhang, Q.; Kamada, H.; Nishida, Y.; Tadanaga, O.; Asobe, M.; Baek, B.; Hadfield, R.; et al. Long-distance entanglement-based quantum key distribution over optical fiber. Opt. Express
**2008**, 16, 19118–19126. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Herrero-Collantes, M.; Garcia-Escartin, J.C. Quantum random number generators. Rev. Mod. Phys.
**2017**, 189, 015004. [Google Scholar] [CrossRef] [Green Version] - Nielsen, M.; Chuang, I.L. Quantum Computation and Quantum Information, 2nd ed.; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]
- Kandala, A.; Temme, K.; Corcoles, A.D.; Mezzacapo, A.; Chow, J.M.; Gambetta, J.M. Error mitigation extends the computational reach of a noisy quantum processor. Nature
**2019**, 567, 491–495. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Garlatti, E.; Guidi, T.; Ansbro, S.; Santini, P.; Amoretti, G.; Ollivier, J.; Mutka, H.; Timco, G.; Vitorica-Yrezabal, I.J.; Whitehead, G.F.S.; et al. Portraying entanglement between molecular qubits with four-dimensional inelastic neutron scattering. Nat. Commun.
**2017**, 8, 1–7. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Georgescu, I.M.; Ashab, S.; Nori, F. Quantum Simulation. Rev. Mod. Phys.
**2014**, 85, 153. [Google Scholar] [CrossRef] [Green Version] - Crippa, L.; Tacchino, F.; Chizzini, M.; Alta, A.; Grossi, M.; Chiesa, A.; Santini, P.; Tavernelli, I.; Carretta, S. Simulating Static and Dynamic Properties of Magnetic Molecules with Prototype Quantum Computers. Magnetochemistry
**2021**, 7, 117. [Google Scholar] [CrossRef] - Degen, C.L.; Reinhard, F.; Cappelaro, P. Quantum sensing. Rev. Mod. Phys.
**2017**, 89, 035002. [Google Scholar] [CrossRef] [Green Version] - Wehner, S.; Elkouss, D.; Hanson, R. Quantum internet: A vision for the road ahead. Science
**2018**, 362, eaam9288. [Google Scholar] [CrossRef] [Green Version] - Rohde, P.P. The Quantum Internet—The Second Quantum Revolution; Cambridge University Press: Cambridge, UK, 2021. [Google Scholar]
- Kozlowski, W.; Wehner, S.; Van Meter, R.; Rijsman, B.; Cacciapuoti, A.S.; Caleffi, M.; Nagayama, S. Architectural Principles for a Quantum Internet. Internet Engineering Task Force, Internet-Draft draft-irtfqirg-principles-03. 2021. Available online: https://www.ietf.org/id/draft-irtf-qirg-principles-10.html (accessed on 29 June 2022).
- Caleffi, M.; Cacciapuoti, A.S.; Cataliotti, F.S.; Gherardini, S.; Tafuri, F.; Bianchi, G. The Quantum Internet: Networking Challenges in Distributed Quantum Computing. IEEE Netw.
**2021**, 34, 137–143. [Google Scholar] - Amoretti, M.; Carretta, S. Entanglement verification in quantum networks with tampered nodes. IEEE J. Sel. Areas Commun.
**2020**, 38, 598–604. [Google Scholar] [CrossRef] - Ferrari, D.; Cacciapuoti, A.S.; Amoretti, M.; Caleffi, M. Compiler Design for Distributed Quantum Computing. IEEE Trans. Quantum Eng.
**2021**, 2, 1–20. [Google Scholar] [CrossRef] - Cross, A.W.; Smith, G.; Smolin, J.A. Quantum learning robust against noise. Phys. Rev. A
**2015**, 92, 012327. [Google Scholar] [CrossRef] [Green Version] - Ferrari, D.; Amoretti, M. Efficient and effective quantum compiling for entanglement-based machine learning on IBM Q devices. Int. J. Quantum Inf.
**2018**, 16, 1840006. [Google Scholar] [CrossRef] - Raussendorf, R.; Briegel, H. A one-way quantum computer. Phys. Rev. Lett.
**2001**, 86, 5188. [Google Scholar] [CrossRef] - Kashefi, E.; Pappa, A. Multiparty Delegated Quantum Computing. Cryptography
**2017**, 1, 12. [Google Scholar] [CrossRef] [Green Version] - Amoretti, M. Private Set Intersection with Delegated Blind Quantum Computing. In Proceedings of the GlobeCom 2021, Madrid, Spain, 7–11 December 2021. [Google Scholar]
- Gisin, N.; Thew, R. Quantum Communication. Nat. Photonics
**2007**, 1, 165–171. [Google Scholar] [CrossRef] [Green Version] - Pirola, S.; Laurenza, R.; Ottaviani, C.; Banchi, L. Fundamental Limits of Repeaterless Quantum Communications. Nat. Commun.
**2017**, 8, 15043. [Google Scholar] - Bratzik, S.; Abruzzo, S.; Kampermann, H.; Bruß, D. Quantum repeaters and quantum key distribution: The impact of entanglement distillation on the secret-key rate. Phys. Rev. A
**2013**, 86, 062335. [Google Scholar] [CrossRef] [Green Version] - Deutsch, D.; Ekert, A.; Jozsa, R.; Macchiavello, C.; Popescu, S.; Sanpera, A. Quantum privacy amplification and the security of quantum cryptography over noisy channels. Proc. R. Soc. Lond. A
**1996**, 400, 2818. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Pant, M.; Krovi, H.; Englund, D.; Guha, S. Rate-distance tradeoff and resource costs for all-optical quantum repeaters. Phys. Rev. A
**2017**, 95, 012304. [Google Scholar] [CrossRef] [Green Version] - Munro, W.J.; Harrison, K.A.; Stephens, A.M.; Devitt, S.J.; Nemoto, K. From quantum multiplexing to high-performance quantum networking. Nat. Photonics
**2010**, 4, 792–796. [Google Scholar] [CrossRef] [Green Version] - Schwartz, I.; Cogan, D.; Schmidgall, E.R.; Don, Y.; Gantz, L.; Kenneth, O.; Lindner, N.H.; Gershoni, D. Deterministic generation of a cluster state of entangled photons. Science
**2016**, 354, 434–437. [Google Scholar] [CrossRef] [Green Version] - Rudolph, T. Why I am optimistic about the silicon-photonic route to quantum computing. APL Photonics
**2017**, 2, 030901. [Google Scholar] [CrossRef] [Green Version] - Dhara, P.; Patil, A.; Krovi, H.; Guha, S. Subexponential rate versus distance with time-multiplexed quantum repeaters. Phis. Rev. A
**2021**, 104, 052612. [Google Scholar] [CrossRef] - Bennett, C.H.; Brassard, G. Quantum Cryptography: Public key distribution and coin tossing. Theor. Comput. Sci.
**2014**, 560, 7–11. [Google Scholar] [CrossRef] - Coopmans, T.; Knegjens, R.; Dahlberg, A.; Maier, D.; Nijsten, L.; de Oliveira Filho, J.; Papendrecht, M.; Rabbie, J.; Rozpędek, F.; Skrzypczyk, M.; et al. NetSquid, a NETwork Simulator for Quantum Information using Discrete events. Commun. Phys.
**2021**, 4, 1–15. [Google Scholar] [CrossRef]

**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.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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