pith. sign in

arxiv: 2604.03155 · v1 · pith:VXQRXWMLnew · submitted 2026-04-03 · 🪐 quant-ph

Routing Entanglement in Complex Quantum Networks Using GHZ States

Xin-An Chen , Caitao Zhan , Joaquin Chung , Jeffrey Larson This is my paper

Pith reviewed 2026-05-13 19:10 UTC · model grok-4.3

classification 🪐 quant-ph
keywords entanglement routingGHZ statesquantum networksBell state measurementshybrid routingnetwork topology
0
0 comments X

The pith

Hybrid GHZ-BSM routing outperforms pure BSM routing for entanglement distribution in square grid quantum networks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies how to route entanglement across quantum networks when links suffer loss and when measurements succeed only probabilistically. Standard Bell-state measurements (BSM) have been the default; a recent proposal replaces them with Greenberger-Horne-Zeilinger (GHZ) measurements that can connect more parties at once. Because GHZ success probabilities are lower, simply swapping in GHZ measurements everywhere reduces the overall rate. The authors therefore introduce a hybrid strategy that chooses GHZ or BSM measurements locally according to the immediate neighborhood. In regular square-grid graphs this hybrid choice already yields higher end-to-end rates than BSM alone. In irregular topologies the same local rule is no longer sufficient.

Core claim

A hybrid routing policy that mixes GHZ and BSM measurements on a per-link basis improves entanglement distribution rates over conventional BSM routing in square-grid networks; the improvement disappears in Waxman, scale-free, and real-world topologies unless global information is added.

What carries the argument

The hybrid GHZ-BSM routing strategy, which selects between a GHZ measurement (connecting three or more nodes) and a BSM (connecting two nodes) on the basis of local link success probabilities and immediate connectivity.

If this is right

  • Entanglement rates in square grids rise when the hybrid policy is applied instead of BSM-only routing.
  • Pure GHZ routing lowers rates relative to BSM routing in every topology examined.
  • Irregular networks require routing decisions that incorporate global topology information beyond local hybrid rules.
  • Real-world topologies such as SURFnet behave more like scale-free than grid graphs under the hybrid policy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Topology regularity appears to be the decisive factor that lets a simple local hybrid rule succeed.
  • Dynamic policies that recompute measurement choices from fresh global state information could close the gap in irregular networks.
  • The same hybrid logic may extend to repeater chains whose loss profiles vary along the path.

Load-bearing premise

The success probability of every GHZ measurement is fixed in advance, identical for every link, and unaffected by hardware details or distance.

What would settle it

Measure the achieved entanglement rate in a laboratory square-grid network of at least 4 by 4 nodes when the hybrid policy is used versus when only BSMs are used, with the actual, experimentally determined GHZ success probabilities.

Figures

Figures reproduced from arXiv: 2604.03155 by Caitao Zhan, Jeffrey Larson, Joaquin Chung, Xin-An Chen.

Figure 1
Figure 1. Figure 1: (a) Example of a physical Waxman network with 30 nodes on a 100 km [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Example of a virtual Waxman network. (b) Example of a virtual scale-free network. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the output states after a Bell state measurement [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) Illustration of our proposed GHZ state protocol on a square grid. The blue box [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Performance evaluation for square grid networks. (a) Average rate [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Performance evaluation for Waxman networks with [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Performance evaluation for scale-free networks with [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Performance evaluation on the SURFnet topology. (a) Average rate [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
read the original abstract

Distributing entanglement to distant parties in a network is a central task in quantum information processing and quantum networking. The sensitivity of entangled states to loss necessitates the use of entanglement routing strategies. Recently, a routing strategy using Greenberger-Horne-Zeilinger (GHZ) measurements instead of Bell state measurements (BSM) has been proposed. We further this direction of research by explicitly considering the varying measurement success probabilities of GHZ measurements. Moreover, we extend the analysis beyond square grid networks to complex network models such as Waxman networks and scale-free networks, as well as SURFnet, a real-world network topology in the Netherlands. Taking into account the varying success probabilities, naive application of GHZ routing achieves rates much lower than the conventional BSM routing. Instead, we propose a hybrid GHZ-BSM routing strategy. The hybrid GHZ-BSM routing strategy outperforms BSM routing in square grid networks. In other networks, however, more sophisticated adaptations using global information are required.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates entanglement routing strategies in quantum networks, comparing GHZ measurements to conventional Bell state measurements (BSM). It reports that naive GHZ routing yields lower rates than BSM due to varying success probabilities, but proposes a hybrid GHZ-BSM strategy that outperforms pure BSM routing in square grid networks. For complex topologies including Waxman, scale-free, and SURFnet models, the work concludes that local hybrid rules are insufficient and global information is required for effective routing.

Significance. If the simulation results hold under scrutiny, the paper provides useful guidance on when hybrid local routing can improve entanglement distribution rates in regular grids and identifies the need for global optimization in irregular networks. This extends prior work on GHZ-based routing and could inform hardware-aware protocol design, though the absence of detailed methods limits immediate applicability.

major comments (2)
  1. [Simulation setup] Simulation setup (abstract and methods): The reported hybrid advantage in square grids and the need for global information in other networks rest on direct simulations whose full methods, error bars, baseline comparisons, and exact modeling of success probabilities are not visible; this undermines verification of the central performance claims.
  2. [Results] Hybrid routing rule (results section): The outperformance of the hybrid GHZ-BSM strategy assumes fixed, known-in-advance, and uniform GHZ measurement success probabilities across all links; the paper's own observation that naive GHZ routing underperforms BSM indicates the hybrid gain is sensitive to this assumption, yet no robustness analysis against link-dependent variations (as expected from realistic loss) is provided.
minor comments (2)
  1. [Abstract] The abstract would benefit from stating the quantitative rate improvements and the precise definition of the hybrid decision threshold.
  2. [Network models] Notation for success probabilities should be introduced consistently when first used in the network model description.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which highlight important aspects of verifiability and model assumptions. We will revise the manuscript to expand the methods and add robustness checks, thereby strengthening the central claims without altering the core findings.

read point-by-point responses

  1. Referee: [Simulation setup] Simulation setup (abstract and methods): The reported hybrid advantage in square grids and the need for global information in other networks rest on direct simulations whose full methods, error bars, baseline comparisons, and exact modeling of success probabilities are not visible; this undermines verification of the central performance claims.

    Authors: We agree that the current methods description is insufficient for full verification. In the revised manuscript we will add a dedicated subsection detailing the simulation protocol, including the precise formulas used for GHZ and BSM success probabilities, the Monte-Carlo sampling procedure, the number of independent runs used to compute error bars, and explicit baseline comparisons against pure BSM routing. These additions will make the reported performance differences directly reproducible. revision: yes

  2. Referee: [Results] Hybrid routing rule (results section): The outperformance of the hybrid GHZ-BSM strategy assumes fixed, known-in-advance, and uniform GHZ measurement success probabilities across all links; the paper's own observation that naive GHZ routing underperforms BSM indicates the hybrid gain is sensitive to this assumption, yet no robustness analysis against link-dependent variations (as expected from realistic loss) is provided.

    Authors: The hybrid rule was derived under the uniform-probability model stated in the paper, which already incorporates the fact that GHZ success probabilities are lower than BSM ones (hence the need for the hybrid switch). We acknowledge that link-dependent variations due to distance-dependent loss could affect the gain. We will therefore insert a new robustness subsection that samples link-specific success probabilities drawn from realistic loss distributions and reports the fraction of parameter regimes in which the hybrid strategy still outperforms pure BSM routing. This will quantify the sensitivity while preserving the original uniform-case results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results from direct simulation of stated probabilities

full rationale

The paper's central claims rest on numerical simulations of entanglement routing rates across network topologies (square grids, Waxman, scale-free, SURFnet) using explicitly stated, fixed, and uniform GHZ and BSM measurement success probabilities. No equations reduce reported rates or outperformance metrics to quantities defined by the authors' own fitted parameters, self-referential definitions, or load-bearing self-citations. The hybrid GHZ-BSM strategy is proposed and evaluated via direct comparison to BSM routing on the simulation outputs, rendering the derivation chain self-contained and independent of the target results.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard quantum-network loss models and the assumption that GHZ success probabilities can be treated as known constants; no new entities are postulated.

free parameters (1)
  • GHZ measurement success probability
    Varying success probabilities are invoked to explain why naive GHZ routing underperforms; specific numerical values are not supplied in the abstract.
axioms (1)
  • domain assumption Entanglement routing rates are determined by the product of link transmission probabilities and local measurement success probabilities.
    Standard modeling assumption in quantum networking literature.

pith-pipeline@v0.9.0 · 5469 in / 1390 out tokens · 68842 ms · 2026-05-13T19:10:30.975469+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

  1. [1]

    Quantum cryptography based on bell’s theorem,

    A. K. Ekert, “Quantum cryptography based on Bell’s theorem,”Physical Review Letters, vol. 67, no. 6, p. 661, 1991. [Online]. Available: https://doi.org/10.1103/PhysRevLett.67.661

    work page doi:10.1103/physrevlett.67.661 1991

  2. [2]

    Distributed quantum sensing,

    Z. Zhang and Q. Zhuang, “Distributed quantum sensing,”Quantum Science and Technology, vol. 6, no. 4, p. 043001, 2021. [Online]. Available: https://doi.org/10.1088/2058-9565/abd4c3 11

    work page doi:10.1088/2058-9565/abd4c3 2021

  3. [3]

    Distributed quantum computing: A survey,

    M. Caleffi, M. Amoretti, D. Ferrari, J. Illiano, A. Manzalini, and A. S. Cacciapuoti, “Distributed quantum computing: a survey,”Computer Networks, vol. 254, p. 110672, 2024. [Online]. Available: https://doi.org/10.1016/j.comnet.2024.110672

    work page doi:10.1016/j.comnet.2024.110672 2024

  4. [4]

    Quantum repeaters: From quantum networks to the quantum internet,

    K. Azuma, S. E. Economou, D. Elkouss, P. Hilaire, L. Jiang, H.-K. Lo, and I. Tzitrin, “Quantum repeaters: From quantum networks to the quantum internet,” Reviews of Modern Physics, vol. 95, no. 4, p. 045006, 2023. [Online]. Available: https://doi.org/10.1103/RevModPhys.95.045006

    work page doi:10.1103/revmodphys.95.045006 2023

  5. [5]

    Entanglement routing in quantum networks: A comprehensive survey,

    A. Abane, M. Cubeddu, V. S. Mai, and A. Battou, “Entanglement routing in quantum networks: A comprehensive survey,”IEEE Transactions on Quantum Engineering, 2025. [Online]. Available: https://doi.org/10.1109/TQE.2025.3541123

    work page doi:10.1109/tqe.2025.3541123 2025

  6. [6]

    Routing entanglement in the quantum internet,

    M. Pant, H. Krovi, D. Towsley, L. Tassiulas, L. Jiang, P. Basu, D. Englund, and S. Guha, “Routing entanglement in the quantum internet,”npj Quantum Information, vol. 5, no. 1, p. 25, 2019. [Online]. Available: https://doi.org/10.1038/s41534-019-0139-x

    work page doi:10.1038/s41534-019-0139-x 2019

  7. [7]

    Patil, M

    A. Patil, M. Pant, D. Englund, D. Towsley, and S. Guha, “Entanglement generation in a quantum network at distance-independent rate,”npj Quantum Information, vol. 8, no. 1, p. 51, 2022. [Online]. Available: https://doi.org/10.1038/s41534-022-00536-0

    work page doi:10.1038/s41534-022-00536-0 2022

  8. [8]

    Grimmett,What is Percolation?Berlin, Heidelberg: Springer Berlin Heidelberg, 1999, pp

    G. Grimmett,What is Percolation?Berlin, Heidelberg: Springer Berlin Heidelberg, 1999, pp. 1–31. [Online]. Available: https://doi.org/10.1007/978-3-662-03981-6 1

    work page doi:10.1007/978-3-662-03981-6 1999

  9. [9]

    Routing of multipoint connections,

    B. M. Waxman, “Routing of multipoint connections,”IEEE Journal on Selected Areas in Communications, vol. 6, no. 9, pp. 1617–1622, 1988. [Online]. Available: https://doi.org/10.1109/49.12889

    work page doi:10.1109/49.12889 1988

  10. [10]

    Barab \' a si and R

    A.-L. Barab´ asi and R. Albert, “Emergence of scaling in random networks,”Science, vol. 286, no. 5439, pp. 509–512, 1999. [Online]. Available: https://doi.org/10.1126/science.286.5439.509

    work page doi:10.1126/science.286.5439.509 1999

  11. [11]

    Modeling the Internet’s large-scale topology,

    S.-H. Yook, H. Jeong, and A.-L. Barab´ asi, “Modeling the Internet’s large-scale topology,” Proceedings of the National Academy of Sciences, vol. 99, no. 21, pp. 13 382–13 386, 2002. [Online]. Available: https://doi.org/10.1073/pnas.172501399

    work page doi:10.1073/pnas.172501399 2002

  12. [12]

    The Internet Topology Zoo

    “The Internet Topology Zoo.” [Online]. Available: https://github.com/mroughan/ InternetTopologyZoo

    work page

  13. [13]

    On the geographic location of internet resources,

    A. Lakhina, J. Byers, M. Crovella, and I. Matta, “On the geographic location of internet resources,”IEEE Journal on Selected Areas in Communications, vol. 21, no. 6, pp. 934–948,

    work page

  14. [14]

    Available: https://doi.org/10.1109/JSAC.2003.814667

    [Online]. Available: https://doi.org/10.1109/JSAC.2003.814667

    work page doi:10.1109/jsac.2003.814667 2003

  15. [15]

    InterTubes: A study of the US long-haul fiber-optic infrastructure,

    R. Durairajan, P. Barford, J. Sommers, and W. Willinger, “InterTubes: A study of the US long-haul fiber-optic infrastructure,” inProceedings of the 2015 ACM Conference on Special Interest Group on Data Communication, 2015, pp. 565–578. [Online]. Available: https://dx.doi.org/10.1145/2785956.2787499

    work page doi:10.1145/2785956.2787499 2015

  16. [16]

    End-to-end capacities of a quantum communication network,

    S. Pirandola, “End-to-end capacities of a quantum communication network,”Communications Physics, vol. 2, no. 1, p. 51, 2019. [Online]. Available: https://doi.org/10.1038/ s42005-019-0147-3 12

    work page 2019

  17. [17]

    Entanglement percolation in quantum networks,

    A. Ac´ ın, J. I. Cirac, and M. Lewenstein, “Entanglement percolation in quantum networks,”Nature Physics, vol. 3, no. 4, pp. 256–259, 2007. [Online]. Available: https://doi.org/10.1038/nphys549

    work page doi:10.1038/nphys549 2007

  18. [18]

    Statistical properties of the quantum internet,

    S. Brito, A. Canabarro, R. Chaves, and D. Cavalcanti, “Statistical properties of the quantum internet,”Physical Review Letters, vol. 124, no. 21, p. 210501, 2020. [Online]. Available: https://doi.org/10.1103/PhysRevLett.124.210501

    work page doi:10.1103/physrevlett.124.210501 2020

  19. [19]

    Quantum communication capacity transition of complex quantum networks,

    Q. Zhuang and B. Zhang, “Quantum communication capacity transition of complex quantum networks,”Physical Review A, vol. 104, no. 2, p. 022608, 2021. [Online]. Available: https://doi.org/10.1103/PhysRevA.104.022608

    work page doi:10.1103/physreva.104.022608 2021

  20. [20]

    Practical routing and criticality in large-scale quantum communication networks,

    C. Harney and S. Pirandola, “Practical routing and criticality in large-scale quantum communication networks,”Physical Review Research, vol. 7, no. 4, Nov. 2025. [Online]. Available: http://dx.doi.org/10.1103/vy37-28jc

    work page doi:10.1103/vy37-28jc 2025

  21. [21]

    Fusion-based quantum computation,

    S. Bartolucci, P. Birchall, H. Bombin, H. Cable, C. Dawson, M. Gimeno-Segovia, E. Johnston, K. Kieling, N. Nickerson, M. Pantet al., “Fusion-based quantum computation,”Nature Communications, vol. 14, no. 1, p. 912, 2023. [Online]. Available: https://doi.org/10.1038/s41467-023-36493-1

    work page doi:10.1038/s41467-023-36493-1 2023

  22. [22]

    Almudever, and Sebastian Feld

    E. Kaur and S. Guha, “Distribution of entanglement in two-dimensional square grid network,” in2023 IEEE International Conference on Quantum Computing and Engineering (QCE), vol. 01, 2023, pp. 1154–1164. [Online]. Available: https: //doi.org/10.1109/QCE57702.2023.00130

    work page doi:10.1109/qce57702.2023.00130 2023

  23. [23]

    Maximum efficiency of a linear-optical Bell-state analyzer,

    J. Calsamiglia and N. L¨ utkenhaus, “Maximum efficiency of a linear-optical Bell-state analyzer,”Applied Physics B, vol. 72, no. 1, pp. 67–71, 2001. [Online]. Available: https://doi.org/10.1007/s003400000484

    work page doi:10.1007/s003400000484 2001

  24. [24]

    Arbitrarily complete Bell-state measurement using only linear optical elements,

    W. P. Grice, “Arbitrarily complete Bell-state measurement using only linear optical elements,” Physical Review A—Atomic, Molecular, and Optical Physics, vol. 84, no. 4, p. 042331, 2011. [Online]. Available: https://doi.org/10.1103/PhysRevA.84.042331

    work page doi:10.1103/physreva.84.042331 2011

  25. [25]

    Estimating the parameters of the Waxman random graph,

    M. Roughan, J. Tuke, and E. Parsonage, “Estimating the parameters of the Waxman random graph,” inInternational Workshop on Algorithms and Models for the Web-Graph. Springer, 2019, pp. 71–86. [Online]. Available: https://doi.org/10.1007/978-3-030-25070-6 6

    work page doi:10.1007/978-3-030-25070-6 2019

  26. [26]

    Average path length in complex networks: Patterns and predictions,

    R. D. Smith, “Average path length in complex networks: Patterns and predictions,”arXiv preprint arXiv:0710.2947, 2007. [Online]. Available: https://doi.org/10.48550/arXiv.0710. 2947

    work page doi:10.48550/arxiv.0710 2007

  27. [27]

    Paths in the simple random graph and the Waxman graph,

    P. Van Mieghem, “Paths in the simple random graph and the Waxman graph,”Probability in the Engineering and Informational Sciences, vol. 15, no. 4, pp. 535–555, 2001. [Online]. Available: https://doi.org/10.1017/S0269964801154070 13

    work page doi:10.1017/s0269964801154070 2001