Efficient High-Dimensional Entangled State Analyzer with Linear Optics

Niv Bharos; Liubov Markovich; Johannes Borregaard

doi:10.22331/q-2025-04-18-1711

Efficient High-Dimensional Entangled State Analyzer with Linear Optics

Niv Bharos¹, Liubov Markovich^2,3, and Johannes Borregaard^1,4

¹QuTech, Delft University of Technology, Delft 2628CJ, Zuid-Holland, The Netherlands
²Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
³$\langle \text{aQa}^\text{L} \rangle$ Applied Quantum Algorithms Leiden, The Netherlands
⁴Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

Published:	2025-04-18, volume 9, page 1711
Editor:	Aaron Goldberg
Eprint:	arXiv:2401.15066v3
Doi:	https://doi.org/10.22331/q-2025-04-18-1711
Citation:	Quantum 9, 1711 (2025).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

The use of higher-dimensional photonic encodings (qudits) instead of two-dimensional encodings (qubits) can improve the loss tolerance and reduce the computational resources of photonic-based quantum information processing. To harness this potential, efficient schemes for entangling operations such as the high-dimensional generalization of a linear optics Bell measurement will be required. We show how an efficient high-dimensional entangled state analyzer can be implemented with a linear optics interferometer and auxiliary photonic states. The degree of entanglement of the auxiliary state is much less than in previous protocols as quantified by an exponentially smaller Schmidt rank. In addition, the auxiliary state only occupies a single spatial mode, allowing it to be generated deterministically from a single quantum emitter coupled to a small qubit register. The reduced complexity of the auxiliary states results in a high robustness to imperfections and we show that auxiliary states with fidelities above 0.9 for qudit dimensions 4 can be generated in the presence of qubit error rates on the order of 10%. This paves the way for experimental demonstrations with current hardware.

Featured image: Overview of a $d$-dimensional entangled state analyzer (ESA) of time-bin encoded qudits in spatial modes $a$ and $b$. The protocol uses $d-2$ auxiliary photons in one input mode $x_0$ to the quantum Fourier transform (QFT), the other input modes $x_1, x_2, \ldots, x_{d-3}$ contain vacuum as indicated by $\emptyset$. The QFT of the spatial modes can be implemented with standard linear optical elements. Finally, the output modes are measured with $d$ single photon detectors labeled $0, 1, \ldots d-1$. The condition for a successful ESA is to measure all photons in different time-bins and the resulting entangled state projection is determined by the specific detection pattern of the detectors.

Popular summary

Photonic qudits offer several advantages compared to qubits, e.g. to achieve higher fidelities for entanglement generation, improve the loss tolerance and lower circuit depths in photonic-based quantum computing. To harness this potential, we need an efficient generalization of the linear optics Bell state measurement: the entangled state analyzer (ESA). We show how an efficient ESA can be implemented with a linear optics interferometer and auxiliary photonic states. The degree of entanglement of the auxiliary state is much less than in previous protocols as quantified by an exponentially smaller Schmidt rank. In addition, we show how the auxiliary state can be realised with a single quantum emitter coupled to a small qubit register and we show that auxiliary states with fidelities above $0.9$ for qudit dimensions $4$ can be generated in the presence of qubit error rates on the order of $10\%$. This paves the way for experimental demonstrations with current hardware.

► BibTeX data

@article{Bharos2025efficienthigh,
  doi = {10.22331/q-2025-04-18-1711},
  url = {https://doi.org/10.22331/q-2025-04-18-1711},
  title = {Efficient {H}igh-{D}imensional {E}ntangled {S}tate {A}nalyzer with {L}inear {O}ptics},
  author = {Bharos, Niv and Markovich, Liubov and Borregaard, Johannes},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1711},
  month = apr,
  year = {2025}
}

► References

[1] H. J. Kimble. The quantum internet. Nature, 2008. https://doi.org/10.1038/nature07127.
https://doi.org/10.1038/nature07127

[2] Sergei Slussarenko and Geoff J. Pryde. Photonic quantum information processing: A concise review. Applied Physics Reviews, 2019. https://doi.org/10.1063/1.5115814.
https://doi.org/10.1063/1.5115814

[3] D. L. Moehring, P. Maunz, S. Olmschenk, K. C. Younge, D. N. Matsukevich, L. M. Duan, and C. Monroe. Entanglement of single-atom quantum bits at a distance. Nature, 2007. https://doi.org/10.1038/nature06118.
https://doi.org/10.1038/nature06118

[4] B. Hensen, H. Bernien, A. E. Dréau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L. Vermeulen, R. N. Schouten, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson. Loophole-free bell inequality violation using electron spins separated by 1.3 kilometres. Nature, 2015. https://doi.org/10.1038/nature15759.
https://doi.org/10.1038/nature15759

[5] Tim van Leent, Matthias Bock, Florian Fertig, Robert Garthoff, Sebastian Eppelt, Yiru Zhou, Pooja Malik, Matthias Seubert, Tobias Bauer, Wenjamin Rosenfeld, Wei Zhang, Christoph Becher, and Harald Weinfurter. Entangling single atoms over 33 km telecom fibre. Nature, 2022. https://doi.org/10.1038/s41586-022-04764-4.
https://doi.org/10.1038/s41586-022-04764-4

[6] Sara Bartolucci, Patrick Birchall, Hector Bombin, Hugo Cable, Chris Dawson, Mercedes Gimeno-Segovia, Eric Johnston, Konrad Kieling, Naomi Nickerson, Mihir Pant, et al. Fusion-based quantum computation. Nature Communications, 2023. https://doi.org/10.1038/s41467-023-36493-1.
https://doi.org/10.1038/s41467-023-36493-1

[7] Daniel E. Browne and Terry Rudolph. Resource-efficient linear optical quantum computation. Phys. Rev. Lett., 2005. https://doi.org/10.1103/PhysRevLett.95.010501.
https://doi.org/10.1103/PhysRevLett.95.010501

[8] D. Istrati, Y. Pilnyak, J. C. Loredo, C. Antón, N. Somaschi, P. Hilaire, H. Ollivier, M. Esmann, L. Cohen, L. Vidro, C. Millet, A. Lemaı̂tre, I. Sagnes, A. Harouri, L. Lanco, P. Senellart, and H. S. Eisenberg. Sequential generation of linear cluster states from a single photon emitter. Nature Communications, 2020. https://doi.org/10.1038/s41467-020-19341-4.
https://doi.org/10.1038/s41467-020-19341-4

[9] E. N. Knall, C. M. Knaut, R. Bekenstein, D. R. Assumpcao, P. L. Stroganov, W. Gong, Y. Q. Huan, P.-J. Stas, B. Machielse, M. Chalupnik, D. Levonian, A. Suleymanzade, R. Riedinger, H. Park, M. Lončar, M. K. Bhaskar, and M. D. Lukin. Efficient source of shaped single photons based on an integrated diamond nanophotonic system. Phys. Rev. Lett., 2022. https://doi.org/10.1103/PhysRevLett.129.053603.
https://doi.org/10.1103/PhysRevLett.129.053603

[10] Massimo Borghi, Noemi Tagliavacche, Federico Andrea Sabattoli, Houssein El Dirani, Laurene Youssef, Camille Petit-Etienne, Erwine Pargon, J E Sipe, Marco Liscidini, Corrado Sciancalepore, Matteo Galli, and Daniele Bajoni. Reconfigurable silicon photonic chip for the generation of frequency-bin-entangled qudits. Phys. Rev. Appl., Jun 2023. https://doi.org/10.1103/PhysRevApplied.19.064026.
https://doi.org/10.1103/PhysRevApplied.19.064026

[11] Nicolò Lo Piparo, William J. Munro, and Kae Nemoto. Quantum multiplexing. Phys. Rev. A, 2019. https://doi.org/10.1103/PhysRevA.99.022337.
https://doi.org/10.1103/PhysRevA.99.022337

[12] Yunzhe Zheng, Hemant Sharma, and Johannes Borregaard. Entanglement distribution with minimal memory requirements using time-bin photonic qudits. PRX Quantum, 2022. https://doi.org/10.1103/PRXQuantum.3.040319.
https://doi.org/10.1103/PRXQuantum.3.040319

[13] Zhihao Xie, Yijie Liu, Xiang Mo, Tao Li, and Zhenhua Li. Quantum entanglement creation for distant quantum memories via time-bin multiplexing. Physical Review A, 2021. https://doi.org/10.1103/PhysRevA.104.062409.
https://doi.org/10.1103/PhysRevA.104.062409

[14] Hui Zhou, Tao Li, Keyu Xia, et al. Parallel and heralded multiqubit entanglement generation for quantum networks. Physical Review A, 2023. https://doi.org/10.1103/PhysRevA.107.022428.
https://doi.org/10.1103/PhysRevA.107.022428

[15] Meng-Ying Yang, Peng Zhao, Lan Zhou, Wei Zhong, and Yu-Bo Sheng. Feasible high-dimensional measurement-device-independent quantum key distribution. Laser Physics Letters, 2021. https://doi.org/10.1088/1612-202X/ac091b.
https://doi.org/10.1088/1612-202X/ac091b

[16] Manuel Erhard, Mario Krenn, and Anton Zeilinger. Advances in high-dimensional quantum entanglement. Nature Reviews Physics, 2020. https://doi.org/10.1038/s42254-020-0193-5.
https://doi.org/10.1038/s42254-020-0193-5

[17] Nicolas J Cerf, Mohamed Bourennane, Anders Karlsson, and Nicolas Gisin. Security of quantum key distribution using d-level systems. Physical review letters, 2002. https://doi.org/10.1103/PhysRevLett.88.127902.
https://doi.org/10.1103/PhysRevLett.88.127902

[18] Helle Bechmann-Pasquinucci and Asher Peres. Quantum cryptography with 3-state systems. Physical Review Letters, 2000. https://doi.org/10.1103/PhysRevLett.85.3313.
https://doi.org/10.1103/PhysRevLett.85.3313

[19] Thomas J Bell, Jacob F F Bulmer, Alex E Jones, Stefano Paesani, Dara P S McCutcheon, and Anthony Laing. Protocol for generation of high-dimensional entanglement from an array of non-interacting photon emitters. New Journal of Physics, 2022. https://dx.doi.org/10.1088/1367-2630/ac475d.
https://doi.org/10.1088/1367-2630/ac475d

[20] D L Zhou, B Zeng, Z Xu, and C P Sun. Quantum computation based on d-level cluster state. Physical Review A, 2003. https://doi.org/10.1103/PhysRevA.68.062303.
https://doi.org/10.1103/PhysRevA.68.062303

[21] Poolad Imany, Jose A. Jaramillo-Villegas, Mohammed S. Alshaykh, Joseph M. Lukens, Ogaga D. Odele, Alexandria J. Moore, Daniel E. Leaird, Minghao Qi, and Andrew M. Weiner. High-dimensional optical quantum logic in large operational spaces. npj Quantum Information, 2019. https://doi.org/10.1038/s41534-019-0173-8.
https://doi.org/10.1038/s41534-019-0173-8

[22] Stefano Paesani, Jacob F F Bulmer, Alex E Jones, Raffaele Santagati, and Anthony Laing. Scheme for universal high-dimensional quantum computation with linear optics. Physical Review Letters, 2021. https://doi.org/10.1103/PhysRevLett.126.230504.
https://doi.org/10.1103/PhysRevLett.126.230504

[23] Michael Meth, Jan F. Haase, Jinglei Zhang, Claire Edmunds, Lukas Postler, Alex Steiner, Andrew J. Jena, Luca Dellantonio, Rainer Blatt, Peter Zoller, Thomas Monz, Philipp Schindler, Christine Muschik, and Martin Ringbauer. Simulating 2d lattice gauge theories on a qudit quantum computer. 2023. https://doi.org/10.48550/arXiv.2310.12110.
https://doi.org/10.48550/arXiv.2310.12110

[24] Daniel González-Cuadra, Torsten V Zache, Jose Carrasco, Barbara Kraus, and Peter Zoller. Hardware efficient quantum simulation of non-abelian gauge theories with qudits on rydberg platforms. Physical Review Letters, 2022. https://doi.org/10.1103/PhysRevLett.129.160501.
https://doi.org/10.1103/PhysRevLett.129.160501

[25] Márton Karácsony, László Oroszlány, and Zoltán Zimborás. Efficient qudit based scheme for photonic quantum computing. 2023. https://doi.org/10.21468/SciPostPhysCore.7.2.032.
https://doi.org/10.21468/SciPostPhysCore.7.2.032

[26] Dagomir Kaszlikowski, Piotr Gnaciński, Marek Żukowski, Wieslaw Miklaszewski, and Anton Zeilinger. Violations of local realism by two entangled n-dimensional systems are stronger than for two qubits. Physical Review Letters, 2000. https://doi.org/10.1103/PhysRevLett.85.4418.
https://doi.org/10.1103/PhysRevLett.85.4418

[27] Daniel Collins, Nicolas Gisin, Noah Linden, Serge Massar, and Sandu Popescu. Bell inequalities for arbitrarily high-dimensional systems. Physical review letters, 2002. https://doi.org/10.1103/PhysRevLett.88.040404.
https://doi.org/10.1103/PhysRevLett.88.040404

[28] Michael Kues, Christian Reimer, Piotr Roztocki, Luis Romero Cortés, Stefania Sciara, Benjamin Wetzel, Yanbing Zhang, Alfonso Cino, Sai T Chu, Brent E Little, et al. On-chip generation of high-dimensional entangled quantum states and their coherent control. Nature, 2017. https://doi.org/10.1038/nature22986.
https://doi.org/10.1038/nature22986

[29] Adetunmise C Dada, Jonathan Leach, Gerald S Buller, Miles J Padgett, and Erika Andersson. Experimental high-dimensional two-photon entanglement and violations of generalized bell inequalities. Nature Physics, 2011. https://doi.org/10.1038/nphys1996.
https://doi.org/10.1038/nphys1996

[30] Christoph Schaeff, Robert Polster, Marcus Huber, Sven Ramelow, and Anton Zeilinger. Experimental access to higher-dimensional entangled quantum systems using integrated optics. Optica, 2015. https://doi.org/10.1364/OPTICA.2.000523.
https://doi.org/10.1364/OPTICA.2.000523

[31] Ji-Gang Ren, Ping Xu, Hai-Lin Yong, Liang Zhang, Sheng-Kai Liao, Juan Yin, Wei-Yue Liu, Wen-Qi Cai, Meng Yang, Li Li, et al. Ground-to-satellite quantum teleportation. Nature, 2017. https://doi.org/10.1038/nature23675.
https://doi.org/10.1038/nature23675

[32] Mario Krenn, Marcus Huber, Robert Fickler, Radek Lapkiewicz, Sven Ramelow, and Anton Zeilinger. Generation and confirmation of a (100$\times$ 100)-dimensional entangled quantum system. Proceedings of the National Academy of Sciences, 2014. https://doi.org/10.1073/pnas.1402365111.
https://doi.org/10.1073/pnas.1402365111

[33] G Molina-Terriza, A Vaziri, J Řeháček, Z Hradil, and A Zeilinger. Triggered qutrits for quantum communication protocols. Physical review letters, 2004. https://doi.org/10.1103/PhysRevLett.92.167903.
https://doi.org/10.1103/PhysRevLett.92.167903

[34] Xiao-Min Hu, Wen-Bo Xing, Bi-Heng Liu, Yun-Feng Huang, Chuan-Feng Li, Guang-Can Guo, Paul Erker, and Marcus Huber. Efficient generation of high-dimensional entanglement through multipath down-conversion. Physical Review Letters, 2020a. https://doi.org/10.1103/PhysRevLett.125.090503.
https://doi.org/10.1103/PhysRevLett.125.090503

[35] D Giovannini, J Romero, Jonathan Leach, A Dudley, A Forbes, and Miles J Padgett. Characterization of high-dimensional entangled systems via mutually unbiased measurements. Physical review letters, 2013. https://doi.org/10.1103/PhysRevLett.110.143601.
https://doi.org/10.1103/PhysRevLett.110.143601

[36] Megan Agnew, Jonathan Leach, Melanie McLaren, F Stef Roux, and Robert W Boyd. Tomography of the quantum state of photons entangled in high dimensions. Physical Review A, 2011. https://doi.org/10.1103/PhysRevA.84.062101.
https://doi.org/10.1103/PhysRevA.84.062101

[37] Manuel Erhard, Mehul Malik, Mario Krenn, and Anton Zeilinger. Experimental greenberger–horne–zeilinger entanglement beyond qubits. Nature Photonics, 2018. https://doi.org/10.1038/s41566-018-0257-6.
https://doi.org/10.1038/s41566-018-0257-6

[38] Yuchen Wang, Zixuan Hu, Barry C Sanders, and Sabre Kais. Qudits and high-dimensional quantum computing. Frontiers in Physics, 2020. https://doi.org/10.3389/fphy.2020.589504.
https://doi.org/10.3389/fphy.2020.589504

[39] Emanuel Knill, Raymond Laflamme, and Gerald J Milburn. A scheme for efficient quantum computation with linear optics. nature, 2001. https://doi.org/10.1038/35051009.
https://doi.org/10.1038/35051009

[40] Pieter Kok, W. J. Munro, Kae Nemoto, T. C. Ralph, Jonathan P. Dowling, and G. J. Milburn. Linear optical quantum computing with photonic qubits. Rev. Mod. Phys., 2007. https://doi.org/10.1103/RevModPhys.79.135.
https://doi.org/10.1103/RevModPhys.79.135

[41] John Calsamiglia and Norbert Lütkenhaus. Maximum efficiency of a linear-optical bell-state analyzer. Applied Physics B, 2001. https://doi.org/10.1007/s003400000484.
https://doi.org/10.1007/s003400000484

[42] Xiao-Min Hu, Chao Zhang, Bi-Heng Liu, Yu Cai, Xiang-Jun Ye, Yu Guo, Wen-Bo Xing, Cen-Xiao Huang, Yun-Feng Huang, Chuan-Feng Li, et al. Experimental high-dimensional quantum teleportation. Physical Review Letters, 2020b. https://doi.org/10.1103/PhysRevLett.125.230501.
https://doi.org/10.1103/PhysRevLett.125.230501

[43] Yi-Han Luo, Han-Sen Zhong, Manuel Erhard, Xi-Lin Wang, Li-Chao Peng, Mario Krenn, Xiao Jiang, Li Li, Nai-Le Liu, Chao-Yang Lu, et al. Quantum teleportation in high dimensions. Physical review letters, 2019. https://doi.org/10.1103/PhysRevLett.123.070505.
https://doi.org/10.1103/PhysRevLett.123.070505

[44] John Calsamiglia. Generalized measurements by linear elements. Physical Review A, 2002. https://doi.org/10.1103/PhysRevA.65.030301.
https://doi.org/10.1103/PhysRevA.65.030301

[45] Chenyu Zhang, J F Chen, Chaohan Cui, Jonathan P Dowling, Z Y Ou, and Tim Byrnes. Quantum teleportation of photonic qudits using linear optics. Physical Review A, 2019. https://doi.org/10.1103/PhysRevA.100.032330.
https://doi.org/10.1103/PhysRevA.100.032330

[46] Yijian Meng, Ming Lai Chan, Rasmus B. Nielsen, Martin H. Appel, Zhe Liu, Ying Wang, Nikolai Bart, Andreas D. Wieck, Arne Ludwig, Leonardo Midolo, Alexey Tiranov, Anders S. Sørensen, and Peter Lodahl. Deterministic photon source of genuine three-qubit entanglement. 2023. https://doi.org/10.1038/s41467-024-52086-y.
https://doi.org/10.1038/s41467-024-52086-y

[47] M. Pompili, S. L. N. Hermans, S. Baier, H. K. C. Beukers, P. C. Humphreys, R. N. Schouten, R. F. L. Vermeulen, M. J. Tiggelman, L. dos Santos Martins, B. Dirkse, S. Wehner, and R. Hanson. Realization of a multinode quantum network of remote solid-state qubits. Science, 2021. https://doi.org/10.1126/science.abg1919.
https://doi.org/10.1126/science.abg1919

[48] Philip Thomas, Leonardo Ruscio, Olivier Morin, and Gerhard Rempe. Efficient generation of entangled multiphoton graph states from a single atom. Nature, 2022. https://doi.org/10.1038/s41586-022-04987-5.
https://doi.org/10.1038/s41586-022-04987-5

[49] Michael Reck, Anton Zeilinger, Herbert J Bernstein, and Philip Bertani. Experimental realization of any discrete unitary operator. Physical review letters, 1994. https://doi.org/10.1103/PhysRevLett.73.58.
https://doi.org/10.1103/PhysRevLett.73.58

[50] Ronen Barak and Yacob Ben-Aryeh. Quantum fast fourier transform and quantum computation by linear optics. JOSA B, 2007. https://doi.org/10.1364/JOSAB.24.000231.
https://doi.org/10.1364/JOSAB.24.000231

[51] Zahra Raissi, Edwin Barnes, and Sophia E. Economou. Deterministic generation of qudit photonic graph states from quantum emitters. PRX Quantum, 2024. https://doi.org/10.1103/PRXQuantum.5.020346.
https://doi.org/10.1103/PRXQuantum.5.020346

[52] Martin Ringbauer, Michael Meth, Lukas Postler, Roman Stricker, Rainer Blatt, Philipp Schindler, and Thomas Monz. A universal qudit quantum processor with trapped ions. Nature Physics, 2022. https://doi.org/10.1038/s41567-022-01658-0.
https://doi.org/10.1038/s41567-022-01658-0

Cited by

[1] Tao LI, Xueqi WANG, and Zhihao XIE, "Research progress of nonlocal quantum entanglement preparation based on quantum multiplexing", Acta Physica Sinica 74 16, 160302 (2025).

[2] Luca Bianchi, Carlo Marconi, and Davide Bacco, "Bell state measurements in quantum optics: a review of recent progress and open challenges", Quantum Science and Technology 11 2, 023001 (2026).

[3] Gözde Üstün, Eleanor G. Rieffel, Simon J. Devitt, and Jason Saied, "Fusion for high-dimensional linear-optical quantum computing with improved success probability", Physical Review Applied 24 4, 044024 (2025).

[4] Gözde Üstün and Simon J. Devitt, "Comparing schemes for creating qudit graph states from 16- and 128-dimensional Hilbert space using donors in silicon", Physical Review Research 8 1, 013343 (2026).

[5] Baghdasar Baghdasaryan, Kaushik Joarder, and Fabian Steinlechner, "Entanglement swapping in high dimensions with only linear optics", Optics Express 34 15, 27876 (2026).

[6] Yanan Liu, Zhaowei Han, Zhihui Li, and Yuehua Li, "An authenticated quantum dialog scheme based on d-dimensional Bell states", Quantum Information Processing 25 5, 167 (2026).

[7] Kazufumi Tanji, Hikaru Shimizu, and Masahiro Takeoka, "High-rate qutrit entanglement swapping with photon-number basis", Physical Review Research 8 1, 013316 (2026).

[8] Stefan Häussler and Peter van Loock, "Quantum repeaters based on stationary and flying Gottesman-Kitaev-Preskill qudits", Physical Review Research 8 3, 033101 (2026).

[9] Tomohiro Yamazaki and Koji Azuma, "Linear-Optical Fusion Boosted by High-Dimensional Entanglement", Physical Review Letters 134 20, 200801 (2025).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-09 09:56:54) and SAO/NASA ADS (last updated successfully 2026-08-08 21:37:35). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch ADS cited-by data during last attempt 2026-08-09 09:56:54: Cannot retrieve data from ADS due to rate limitations.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.