- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Entanglement-efficiency trade-offs in the fusion-based generation of photonic Greenberger-Horne-Zeilinger-like states

A. A. Melkozerov1,2,*, M. Yu. Saygin2,3, and S. S Straupe1,2,3

  • *Contact author: melkozerov.aa17@physics.msu.ru
PDF

Phys. Rev. Research 8, 013153 – Published 10 February, 2026

DOI: https://doi.org/10.1103/t375-mck9

Abstract

Probabilistic entangling measurements are key operations in linear-optical quantum technologies, enabling the generation and manipulation of high-dimensional quantum states. While prior research has focused predominantly on specific entangled states, notably graph states and Greenberger-Horne-Zeilinger (GHZ) states, broader classes of states with variable entanglement remain underexplored. In this work, we present a linear-optical approach for generating and fusing GHZ-like states, which generalize standard GHZ states to include variable entanglement degrees. We introduce two schemes based on modified fusion gates that allow flexible control over generation efficiency and the entanglement of the output states. These results offer a promising pathway toward resource-efficient entangled-state generation for scalable quantum computing and communication.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (51)

  1. A. Farouk, J. Batle, M. Elhoseny, M. Naseri, M. Lone, A. Fedorov, M. Alkhambashi, S. H. Ahmed, and M. Abdel-Aty, Robust general N user authentication scheme in a centralized quantum communication network via generalized GHZ states, Front. Phys. 13, 130306 (2017).
  2. J. Wang, Q. Zhang, and C.-J. Tang, Multiparty controlled quantum secure direct communication using Greenberger–Horne–Zeilinger state, Opt. Commun. 266, 732 (2006).
  3. T. Gao, F. L. Yan, and Z. X. Wang, Deterministic secure direct communication using GHZ states and swapping quantum entanglement, J. Phys. A: Math. Gen. 38, 5761 (2005).
  4. G. Tóth and I. Apellaniz, Quantum metrology from a quantum information science perspective, J. Phys. A: Math. Theor. 47, 424006 (2014).
  5. P. Hyllus, W. Laskowski, R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, L. Pezzé, and A. Smerzi, Fisher information and multiparticle entanglement, Phys. Rev. A 85, 022321 (2012).
  6. M. Gimeno-Segovia, P. Shadbolt, D. E. Browne, and T. Rudolph, From three-photon Greenberger-Horne-Zeilinger states to ballistic universal quantum computation, Phys. Rev. Lett. 115, 020502 (2015).
  7. S. Bartolucci, P. Birchall, H. Bombín, H. Cable, C. Dawson, M. Gimeno-Segovia, E. Johnston, K. Kieling, N. Nickerson, M. Pant, F. Pastawski, T. Rudolph, and C. Sparrow, Fusion-based quantum computation, Nat. Commun. 14, 912 (2023).
  8. M. Grassl, A. Klappenecker, and M. Rotteler, Graphs, quadratic forms, and quantum codes, in Proceedings IEEE International Symposium on Information Theory, ISIT-02 (IEEE, Lausanne, Switzerland, 2004), p. 45.
  9. S. Bartolucci, P. M. Birchall, M. Gimeno-Segovia, E. Johnston, K. Kieling, M. Pant, T. Rudolph, J. Smith, C. Sparrow, and M. D. Vidrighin, Creation of entangled photonic states using linear optics, arXiv:2106.13825.
  10. I. Forbes, F. Ghafari, E. C. R. Deacon, S. P. Singh, E. Lavie, P. Yard, R. D. Shaw, A. Laing, and N. Tischler, Heralded generation of entanglement with photons, Rep. Prog. Phys. 88, 086002 (2025).
  11. J. Carolan, C. Harrold, C. Sparrow, E. Martín-López, N. J. Russell, J. W. Silverstone, P. J. Shadbolt, N. Matsuda, M. Oguma, M. Itoh, G. D. Marshall, M. G. Thompson, J. C. F. Matthews, T. Hashimoto, J. L. O’Brien, and A. Laing, Universal linear optics, Science 349, 711 (2015).
  12. M. Varnava, D. E. Browne, and T. Rudolph, How good must single photon sources and detectors be for efficient linear optical quantum computation? Phys. Rev. Lett. 100, 060502 (2008).
  13. H. A. Zaidi, C. Dawson, P. van Loock, and T. Rudolph, Near-deterministic creation of universal cluster states with probabilistic Bell measurements and three-qubit resource states, Phys. Rev. A 91, 042301 (2015).
  14. D. E. Browne and T. Rudolph, Resource-efficient linear optical quantum computation, Phys. Rev. Lett. 95, 010501 (2005).
  15. K. Kieling, T. Rudolph, and J. Eisert, Percolation, renormalization, and quantum computing with nondeterministic gates, Phys. Rev. Lett. 99, 130501 (2007).
  16. F. Schmidt and P. van Loock, Generalized fusions of photonic quantum states using static linear optics, arXiv:2410.20261.
  17. S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Phys. Rev. A 70, 052328 (2004).
  18. M. Hein, J. Eisert, and H. J. Briegel, Multiparty entanglement in graph states, Phys. Rev. A 69, 062311 (2004).
  19. M. F. Mor-Ruiz, J. Miguel-Ramiro, J. Wallnöfer, T. Coopmans, and W. Dür, Merging-based quantum repeater, arXiv:2502.04450.
  20. J. Calsamiglia and N. Lütkenhaus, Maximum efficiency of a linear-optical Bell-state analyzer, Appl. Phys. B 72, 67 (2001).
  21. N. Rimock, K. Cohen, and Y. Oz, Generalized type-II fusion of cluster states, Phys. Rev. A 113, 012603 (2026).
  22. F. Ewert and P. van Loock, 3/4-efficient Bell measurement with passive linear optics and unentangled ancillae, Phys. Rev. Lett. 113, 140403 (2014).
  23. W. P. Grice, Arbitrarily complete Bell-state measurement using only linear optical elements, Phys. Rev. A 84, 042331 (2011).
  24. D. Gross, J. Eisert, N. Schuch, and D. Perez-Garcia, Measurement-based quantum computation beyond the one-way model, Phys. Rev. A 76, 052315 (2007).
  25. X. Ni, O. Buerschaper, and M. Van den Nest, A non-commuting stabilizer formalism, J. Math. Phys. 56, 052201 (2015).
  26. M. A. Webster, B. J. Brown, and S. D. Bartlett, The XP stabiliser formalism: A generalisation of the Pauli stabiliser formalism with arbitrary phases, Quantum 6, 815 (2022).
  27. H. Jeong and N. Ba An, Greenberger-Horne-Zeilinger-type and W-type entangled coherent states: Generation and Bell-type inequality tests without photon counting, Phys. Rev. A 74, 022104 (2006).
  28. J. Modławska and A. Grudka, Nonmaximally entangled states can be better for multiple linear optical teleportation, Phys. Rev. Lett. 100, 110503 (2008).
  29. A. Pathak and A. Banerjee, Efficient quantum circuits for perfect and controlled teleportation of n-qubit non-maximally entangled states of generalized Bell-type, Int. J. Quantum Inf. 09, 389 (2011).
  30. H. Zakaryan, K.-R. Revis, and Z. Raissi, Non-symmetric GHZ states: Weighted hypergraph and controlled-unitary graph representations, Phys. Rev. A 112, 032438 (2025).
  31. L. Hartmann, J. Calsamiglia, W. Dür, and H. J. Briegel, Weighted graph states and applications to spin chains, lattices and gases, J. Phys. B: At., Mol. Opt. Phys. 40, S1 (2007).
  32. A. D. K. Plato, O. C. Dahlsten, and M. B. Plenio, Random circuits by measurements on weighted graph states, Phys. Rev. A 78, 042332 (2008).
  33. S. Anders, H. J. Briegel, and W. Dür, A variational method based on weighted graph states, New J. Phys. 9, 361 (2007).
  34. I. Bengtsson and K. Zyczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement (Cambridge University Press, Cambridge, 2006).
  35. M. Gimeno-Segovia, Towards Practical Linear Optical Quantum Computing, Ph.D. thesis, Imperial College London, 2016.
  36. S. Stanisic, N. Linden, A. Montanaro, and P. S. Turner, Generating entanglement with linear optics, Phys. Rev. A 96, 043861 (2017).
  37. F. V. Gubarev, I. V. Dyakonov, M. Yu. Saygin, G. I. Struchalin, S. S. Straupe, and S. P. Kulik, Improved heralded schemes to generate entangled states from single photons, Phys. Rev. A 102, 012604 (2020).
  38. S. A. Fldzhyan, M. Yu. Saygin, and S. P. Kulik, Compact linear optical scheme for Bell state generation, Phys. Rev. Res. 3, 043031 (2021).
  39. D. Bhatti and S. Barz, Heralding higher-dimensional Bell and Greenberger-Horne-Zeilinger states using multiport splitters, New J. Phys. 27, 033006 (2025).
  40. S. A. Fldzhyan, M. Yu. Saygin, and S. P. Kulik, Programmable Heralded linear optical generation of two-qubit states, Phys. Rev. Appl. 20, 054030 (2023).
  41. S. Bartolucci, P. Birchall, D. Bonneau, H. Cable, M. Gimeno-Segovia, K. Kieling, N. Nickerson, T. Rudolph, and C. Sparrow, Switch networks for photonic fusion-based quantum computing, arXiv:2109.13760.
  42. N. N. Skryabin, et al., Programmable entangled qubit states on a linear-optical platform, arXiv:2410.15697.
  43. P. Kok and B. W. Lovett, Introduction to Optical Quantum Information Processing (Cambridge University Press, Cambridge, 2010).
  44. J.-P. Li, X. Gu, J. Qin, D. Wu, X. You, H. Wang, C. Schneider, S. Höfling, Y.-H. Huo, C.-Y. Lu, N.-L. Liu, Li Li, and J.-W. Pan, Heralded nondestructive quantum entangling gate with single-photon sources, Phys. Rev. Lett. 126, 140501 (2021).
  45. S. A. Fldzhyan, S. S. Straupe, and M. Yu. Saygin, Resource-efficient linear-optical generation of GHZ-like states, arXiv:2509.14794.
  46. M. L. Chan, T. J. Bell, L. A. Pettersson, S. X. Chen, P. Yard, A. S. Sørensen, and S. Paesani, Tailoring fusion-based photonic quantum computing schemes to quantum emitters, PRX Quantum 6, 020304 (2025).
  47. N. H. Lindner and T. Rudolph, Proposal for pulsed on-demand sources of photonic cluster state strings, Phys. Rev. Lett. 103, 113602 (2009).
  48. P. Thomas, L. Ruscio, O. Morin, and G. Rempe, Efficient generation of entangled multiphoton graph states from a single atom, Nature (London), 608, 677 (2022).
  49. Z.-C. Shi, J.-H. Wang, C. Zhang, J. Song, and Y. Xia, Universal composite pulses for robust quantum state engineering in four-level systems, Phys. Rev. A 109, 022441 (2024).
  50. D. Cogan, O. Kenneth, N. H. Lindner, G. Peniakov, C. Hopfmann, D. Dalacu, P. J. Poole, P. Hawrylak, and D. Gershoni, Depolarization of electronic spin qubits confined in semiconductor quantum dots, Phys. Rev. X 8, 041050 (2018).
  51. M. Pant, D. Towsley, D. Englund, and S. Guha, Percolation thresholds for photonic quantum computing, Nat. Commun. 10, 1070 (2019).

Outline

Information

Phys. Rev. Research 8, 013153– Published 10 February, 2026
  • Received 26 August 2025
  • Accepted 21 December 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation