Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons
1Department of physics, University of Ottawa, Advanced Research Complex, 25 Templeton, Ottawa ON Canada, K1N 6N5
2National Research Council of Canada, 100 Sussex Drive, Ottawa ON Canada, K1A 0R6
3Max-Planck-Institut für die Physik des Lichts, Staudtstraße 2, 91058 Erlangen, Germany
4Institute of Optics, University of Rochester, Rochester, NY 14627, USA
5Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore
6Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore.
7MajuLab, CNRS-UNS-NUS-NTU International Joint Unit, UMI 3654, Singapore.
8Departamento de Óptica, Facultad de Física, Universidad Complutense, 28040 Madrid, Spain
9Department of Physics, Institute for Advanced Studies in Basic Sciences, 45137-66731 Zanjan, Iran.
| Published: | 2018-12-04, volume 2, page 111 |
| Eprint: | arXiv:1802.05773v3 |
| Doi: | https://doi.org/10.22331/q-2018-12-04-111 |
| Citation: | Quantum 2, 111 (2018). |
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
Quantum key distribution is on the verge of real world applications, where perfectly secure information can be distributed among multiple parties. Several quantum cryptographic protocols have been theoretically proposed and independently realized in different experimental conditions. Here, we develop an experimental platform based on high-dimensional orbital angular momentum states of single photons that enables implementation of multiple quantum key distribution protocols with a single experimental apparatus. Our versatile approach allows us to experimentally survey different classes of quantum key distribution techniques, such as the 1984 Bennett & Brassard (BB84), tomographic protocols including the six-state and the Singapore protocol, and to investigate, for the first time, a recently introduced differential phase shift (Chau15) protocol using twisted photons. This enables us to experimentally compare the performance of these techniques and discuss their benefits and deficiencies in terms of noise tolerance in different dimensions.

► BibTeX data
► References
[1] Gisin, N., Ribordy, G., Tittel, W. & Zbinden, H., Quantum cryptography, Rev. Mod. Phys. 74, 145 (2002).
https://doi.org/10.1103/RevModPhys.74.145
[2] Bennett, C. H. & Brassard, G., Quantum cryptography: Public key distribution and coin tossing, Proceedings of the ieee international conference on computers, systems, and signal processing, bangalore, india, 1984 (1984).
[3] Scarani, V. et al., The security of practical quantum key distribution, Rev. Mod. Phys. 81, 1301 (2009).
https://doi.org/10.1103/RevModPhys.81.1301
[4] Wiesner, S. Conjugate coding, ACM Sigact News 15, 78–88 (1983).
https://doi.org/10.1145/1008908.1008920
[5] Hillery, M., Bužek, V. & Berthiaume, A. Quantum secret sharing, Phys. Rev. A 59, 1829 (1999).
https://doi.org/10.1103/PhysRevA.59.1829
[6] Scarani, V., Iblisdir, S., Gisin, N. & Acin, A. Quantum cloning, Rev. Mod. Phys. 77, 1225 (2005).
https://doi.org/10.1103/RevModPhys.77.1225
[7] Simon, C. et al. Quantum memories, Eur. Phys. J. D 58, 1–22 (2010).
https://doi.org/10.1140/epjd/e2010-00103-y
[8] Werner, M. & Milburn, G. Eavesdropping using quantum-nondemolition measurements, Phys. Rev. A 47, 639 (1993).
https://doi.org/10.1103/PhysRevA.47.639
[9] Bennett, C. H., Brassard, G., Crépeau, C. & Maurer, U. M. Generalized privacy amplification, IEEE T. Inform. Theory 41, 1915–1923 (1995).
https://doi.org/10.1109/18.476316
[10] Bechmann-Pasquinucci, H. & Tittel, W. Quantum cryptography using larger alphabets, Phys. Rev. A 61, 062308 (2000).
https://doi.org/10.1103/PhysRevA.61.062308
[11] Cerf, N. J., Bourennane, M., Karlsson, A. & Gisin, N. Security of quantum key distribution using d-level systems, Phys. Rev. Lett. 88, 127902 (2002).
https://doi.org/10.1103/PhysRevLett.88.127902
[12] Allen, L., Beijersbergen, M. W., Spreeuw, R. & Woerdman, J. Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes, Phys. Rev. A 45, 8185 (1992).
https://doi.org/10.1103/PhysRevA.45.8185
[13] Heckenberg, N., McDuff, R., Smith, C. & White, A. Generation of optical phase singularities by computer-generated holograms, Opt. Lett. 17, 221–223 (1992).
https://doi.org/10.1364/OL.17.000221
[14] Bolduc, E., Bent, N., Santamato, E., Karimi, E. & Boyd, R. W. Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram, Opt. Lett. 38, 3546–3549 (2013).
https://doi.org/10.1364/OL.38.003546
[15] Forbes, A., Dudley, A. & McLaren, M. Creation and detection of optical modes with spatial light modulators, Advances in Optics and Photonics 8, 200–227 (2016).
https://doi.org/10.1364/AOP.8.000200
[16] Gröblacher, S., Jennewein, T., Vaziri, A., Weihs, G. & Zeilinger, A. Experimental quantum cryptography with qutrits, New J. Phys. 8, 75 (2006).
https://doi.org/10.1088/1367-2630/8/5/075
[17] Mafu, M. et al. Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases, Phys. Rev. A 88, 032305 (2013).
https://doi.org/10.1103/PhysRevA.88.032305
[18] Mirhosseini, M. et al. High-dimensional quantum cryptography with twisted light, New J. Phys. 17, 033033 (2015).
https://doi.org/10.1088/1367-2630/17/3/033033
[19] D'ambrosio, V. et al. Complete experimental toolbox for alignment-free quantum communication, Nat. Commun. 3, 961 (2012).
https://doi.org/10.1038/ncomms1951
[20] Vallone, G. et al. Free-space quantum key distribution by rotation-invariant twisted photons, Phys. Rev. Lett. 113, 060503 (2014).
https://doi.org/10.1103/PhysRevLett.113.060503
[21] Krenn, M., Handsteiner, J., Fink, M., Fickler, R. & Zeilinger, A. Twisted photon entanglement through turbulent air across Vienna, PNAS 112, 14197–14201 (2015).
https://doi.org/10.1073/pnas.1517574112
[22] Sit, A. et al. High-dimensional intracity quantum cryptography with structured photons, Optica 4, 1006–1010 (2017).
https://doi.org/10.1364/OPTICA.4.001006
[23] Cardano, F. et al. Quantum walks and wavepacket dynamics on a lattice with twisted photons, Science Adv. 1, e1500087 (2015).
https://doi.org/10.1126/sciadv.1500087
[24] Cardano, F. et al. Statistical moments of quantum-walk dynamics reveal topological quantum transitions, Nat. Commun. 7, 11439 (2016).
https://doi.org/10.1038/ncomms11439
[25] Cardano, F. et al. Detection of zak phases and topological invariants in a chiral quantum walk of twisted photons, Nature Commun. 8, 15516 (2017).
https://doi.org/10.1038/ncomms15516
[26] Babazadeh, A. et al. High-dimensional single-photon quantum gates: concepts and experiments, Phys. Rev. Lett. 119, 180510 (2017).
https://doi.org/10.1103/PhysRevLett.119.180510
[27] Erhard, M., Fickler, R., Krenn, M. & Zeilinger, A. Twisted photons: New quantum perspectives in high dimensions, Light Sci. Appl. (2018).
https://doi.org/10.1038/lsa.2017.146
[28] Bruß, D. Optimal eavesdropping in quantum cryptography with six states, Phys. Rev. Lett. 81, 3018 (1998).
https://doi.org/10.1103/PhysRevLett.81.3018
[29] Liang, Y. C., Kaszlikowski, D., Englert, B.-G., Kwek, L. C. & Oh, C. H. Tomographic quantum cryptography, Phys. Rev. A 68, 022324 (2003).
https://doi.org/10.1103/PhysRevA.68.022324
[30] Englert, B.-G. et al. Efficient and robust quantum key distribution with minimal state tomography, arXiv preprint quant-ph/0412075 (2008).
arXiv:quant-ph/0412075v4
[31] Durt, T., Englert, B.-G., Bengtsson, I. & Życzkowski, K. On mutually unbiased bases, Int. J. Quantum Inf. 8, 535–640 (2010).
https://doi.org/10.1142/S0219749910006502
[32] Renes, J. M., Blume-Kohout, R., Scott, A. J. & Caves, C. M. Symmetric informationally complete quantum measurements, J. Math. Phys. 45, 2171–2180 (2004).
https://doi.org/10.1063/1.1737053
[33] Chuang, I. L. & Nielsen, M. A. Prescription for experimental determination of the dynamics of a quantum black box, J. Mod. Opt. 44, 2455–2467 (1997).
https://doi.org/10.1080/09500349708231894
[34] Lo, H.-K., Chau, H. F., & Ardehali, M. Efficient Quantum Key Distribution Scheme and a Proof of Its Unconditional Security, Journal of Cryptology 18, 133–165 (2005).
https://doi.org/10.1007/s00145-004-0142-y
[35] Brádler, K., Mirhosseini, M., Fickler, R., Broadbent, A. & Boyd, R. Finite-key security analysis for multilevel quantum key distribution, New Journal of Physics 18, 073030 (2016).
https://doi.org/10.1088/1367-2630/18/7/073030
[36] Ding, Y. et al. High-dimensional quantum key distribution based on multicore fiber using silicon photonic integrated circuits, npj Quantum Information 3, 25 (2017).
https://doi.org/10.1038/s41534-017-0026-2
[37] Genovese, M. & Traina, P. Review on qudits production and their application to quantum communication and studies on local realism, Advanced Science Letters 1, 153–160 (2008).
https://doi.org/10.1166/asl.2008.014
[38] Bouchard, F., Fickler, R., Boyd, R. W. & Karimi, E. High-dimensional quantum cloning and applications to quantum hacking, Science Adv. 3, e1601915 (2017).
https://doi.org/10.1126/sciadv.1601915
[39] Sheridan, L. & Scarani, V. Security proof for quantum key distribution using qudit systems, Phys. Rev. A 82, 030301 (2010).
https://doi.org/10.1103/PhysRevA.82.030301
[40] D'ambrosio, V. et al. Test of mutually unbiased bases for six-dimensional photonic quantum systems, Sci. Rep. 3, 2726 (2013).
https://doi.org/10.1038/srep02726
[41] Ekert, A. K. Quantum cryptography based on bell's theorem, Phys. Rev. Lett. 67, 661 (1991).
https://doi.org/10.1103/PhysRevLett.67.661
[42] Bent, N. et al. Experimental realization of quantum tomography of photonic qudits via symmetric informationally complete positive operator-valued measures, Phys. Rev. X 5, 041006 (2015).
https://doi.org/10.1103/PhysRevX.5.041006
[43] Inoue, K., Waks, E. & Yamamoto, Y. Differential phase shift quantum key distribution, Phys. Rev. Lett. 89, 037902 (2002).
https://doi.org/10.1103/PhysRevLett.89.037902
[44] Sasaki, T., Yamamoto, Y. & Koashi, M. Practical quantum key distribution protocol without monitoring signal disturbance, Nature 509, 475 (2014).
https://doi.org/10.1038/nature13303
[45] Bouchard F., Sit A., Heshami K., Fickler R. & Karimi E. Round-robin differential phase-shift quantum key distribution with twisted photons, Phys. Rev. A 98, 010301(R) (2018).
https://doi.org/10.1103/PhysRevA.98.010301
[46] Chau, H. Quantum key distribution using qudits that each encode one bit of raw key, Phys. Rev. A 92, 062324 (2015).
https://doi.org/10.1103/PhysRevA.92.062324
[47] Chau, H., Wang, Q. & Wong, C. Experimentally feasible quantum-key-distribution scheme using qubit-like qudits and its comparison with existing qubit-and qudit-based protocols, Phys. Rev. A 95, 022311 (2017).
https://doi.org/10.1103/PhysRevA.95.022311
[48] Mair, A., Vaziri, A., Weihs, G. & Zeilinger, A. Entanglement of the orbital angular momentum states of photons, Nature 412, 313–316 (2001).
https://doi.org/10.1038/35085529
[49] Qassim, H. et al. Limitations to the determination of a laguerre–gauss spectrum via projective, phase-flattening measurement, J. Opt. Soc. Am. B 31, A20–A23 (2014).
https://doi.org/10.1364/JOSAB.31.000A20
[50] Waks, E. et al. Security aspects of quantum key distribution with sub-Poisson light, Phys. Rev. A 66, 042315 (2002).
https://doi.org/10.1103/PhysRevA.66.042315
[51] Schiavon, M. et al. Heralded single-photon sources for quantum-key-distribution applications, Phys. Rev. A 93, 012331 (2016).
https://doi.org/10.1103/PhysRevA.93.012331
[52] Wang, S. et al. Proof-of-principle experimental realization of a qubit-like qudit-based quantum key distribution scheme, Quantum Sci. Technol. 3, 025006 (2018).
https://doi.org/10.1088/2058-9565/aaace4
[53] Bouchard, F., Sit, A., Hufnagel, F., Abbas, A., Zhang, Y., Heshami, K., Fickler, R., Marquardt, C., Leuchs, G., Boyd, R. W. & Karimi, E. Quantum cryptography with twisted photons through an outdoor underwater channel, Opt. Express 26, 22563–22573 (2018).
https://doi.org/10.1364/OE.26.022563
[54] Scott, A. J. & Grassl, M. Symmetric informationally complete positive-operator-valued measures: A new computer study, J. Math. Phys. 51, 042203 (2010).
https://doi.org/10.1063/1.3374022
[55] Ndagano, B. et al. Characterizing quantum channels with non-separable states of classical light, Nat. Phys. 13, 397 (2017).
https://doi.org/10.1038/nphys4003
[56] Bongioanni, I., Sansoni, L., Sciarrino, F., Vallone, G., & Mataloni, P., Experimental quantum process tomography of non-trace-preserving maps, Phys. Rev. A 82, 042307 (2010).
https://doi.org/10.1103/PhysRevA.82.042307
[57] Bouchard, F., Hufnagel, F., Koutnỳ, D., Abbas, A., Sit, A., Heshami, K., Fickler, R. & Karimi, E., Full characterization of a high-dimensional quantum communication channel, arXiv preprint arXiv:1806.08018 (2018).
arXiv:1806.08018
[58] Tomamichel, M. et al. Tight finite-key analysis for quantum cryptography, Nat. Commun. 3, 634 (2012).
https://doi.org/10.1038/ncomms1631
Cited by
[1] Dongkai Zhang, Xiaodong Qiu, Wuhong Zhang, and Lixiang Chen, "Verifying angular-position entanglement by Hardy's paradox with multisetting high-dimensional systems", Physical Review A 105 6, 062401 (2022).
[2] Aarón A. Aguilar-Cardoso, Cheng Li, Tobey J. B. Luck, Manuel F. Ferrer-Garcia, Jeremy Upham, Jeff S. Lundeen, and Robert W. Boyd, "All-optical turbulence mitigation for free-space quantum key distribution using stimulated parametric down-conversion", Optica 13 3, 386 (2026).
[3] Antonio Manzalini, "Topological Photonics for Optical Communications and Quantum Computing", Quantum Reports 2 4, 579 (2020).
[4] Natalia Korolkova, Luis Sánchez-Soto, and Gerd Leuchs, "An operational distinction between quantum entanglement and classical non-separability", Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 382 2287, 20230342 (2024).
[5] A. A. Aguilar-Cardoso, C. Li, T. J. B. Luck, M. F. Ferrer-Garcia, J. Upham, J. S. Lundeen, and R. W. Boyd, "Tailoring spatial modes produced by stimulated parametric down-conversion", Physical Review A 112 4, 043541 (2025).
[6] Wei Li, Le Wang, and Shengmei Zhao, "Phase Matching Quantum Key Distribution based on Single-Photon Entanglement", Scientific Reports 9 1, 15466 (2019).
[7] NOORULDEN BASIL, "WITHDRAWN: Ultrafast Quantum Communication with Time Bin Qubits Applied for Biomedical Engineering", (2023).
[8] Qian-Ke Wang, Fang-Xiang Wang, Jun Liu, Wei Chen, Zheng-Fu Han, Andrew Forbes, and Jian Wang, "High-Dimensional Quantum Cryptography with Hybrid Orbital-Angular-Momentum States through 25 km of Ring-Core Fiber: A Proof-of-Concept Demonstration", Physical Review Applied 15 6, 064034 (2021).
[9] Keshav Das Agarwal, Debkanta Ghosh, Pritam Halder, and Aditi Sen(De), "Revealing effects of the local dimension on a variable-range interacting model by connecting Lieb-Robinson bounds and multipartite entanglement", Physical Review B 113 5, 054302 (2026).
[10] Tareq Jaouni, Xiaoqin Gao, Sören Arlt, Mario Krenn, and Ebrahim Karimi, "Experimental solutions to the high-dimensional mean king’s problem", Optica Quantum 1 2, 49 (2023).
[11] Katarzyna Siudzińska, "Equivalence Relations Between Conical 2-Designs and Mutually Unbiased Generalized Equiangular Tight Frames", Mathematics 14 1, 128 (2025).
[12] Hasnaa Hajji and Morad El Baz, "Qutrit-based semi-quantum key distribution protocol", Quantum Information Processing 20 1, 4 (2021).
[13] Ayan Patra, Rivu Gupta, Tamoghna Das, and Aditi Sen(De), "Dimensional advantage in secure information trading via the noisy dense-coding protocol", Physical Review A 110 3, 032419 (2024).
[14] Jiapeng Zhao, Yiyu Zhou, Boris Braverman, Cong Liu, Kai Pang, Nicholas K. Steinhoff, Glenn A. Tyler, Alan E. Willner, and Robert W. Boyd, "Performance of real-time adaptive optics compensation in a turbulent channel with high-dimensional spatial-mode encoding", Optics Express 28 10, 15376 (2020).
[15] Alessio D'Errico and Ebrahim Karimi, Electromagnetic Vortices 423 (2021) ISBN:9781119662822.
[16] Zhiwei Tao, Yichong Ren, Azezigul Abdukirim, Shiwei Liu, and Ruizhong Rao, "Mitigating the effect of atmospheric turbulence on orbital angular momentum-based quantum key distribution using real-time adaptive optics with phase unwrapping", Optics Express 29 20, 31078 (2021).
[17] Joseph C. Chapman, Charles C.W. Lim, and Paul G. Kwiat, "Hyperentangled Time-Bin and Polarization Quantum Key Distribution", Physical Review Applied 18 4, 044027 (2022).
[18] Felix Hufnagel, Alicia Sit, Florence Grenapin, Frédéric Bouchard, Khabat Heshami, Duncan England, Yingwen Zhang, Benjamin J. Sussman, Robert W. Boyd, Gerd Leuchs, and Ebrahim Karimi, "Characterization of an underwater channel for quantum communications in the Ottawa River", Optics Express 27 19, 26346 (2019).
[19] Aayam Bista, Baibhav Sharma, and Enrique J. Galvez, "A demonstration of quantum key distribution with entangled photons for the undergraduate laboratory", American Journal of Physics 89 1, 111 (2021).
[20] Amir Hossein Fahim Raouf, Majid Safari, and Murat Uysal, "Performance analysis of decoy state quantum key distribution over underwater turbulence channels", Journal of the Optical Society of America B 39 6, 1470 (2022).
[21] Zehong Chang, Fumin Wang, Junliang Jia, Xiaoli Wang, Yi Lv, and Pei Zhang, "Security analysis for a mutually partially unbiased bases–based protocol", Journal of the Optical Society of America B 39 10, 2823 (2022).
[22] Frédéric Bouchard, Kent Bonsma-Fisher, Khabat Heshami, Philip J. Bustard, Duncan England, and Benjamin Sussman, "Measuring ultrafast time-bin qudits", Physical Review A 107 2, 022618 (2023).
[23] Kfir Sulimany, Guy Pelc, Rom Dudkiewicz, Simcha Korenblit, Hagai S. Eisenberg, Yaron Bromberg, and Michael Ben-Or, "High-dimensional coherent one-way quantum key distribution", npj Quantum Information 11 1, 16 (2025).
[24] Daniele Cozzolino, Beatrice Da Lio, Davide Bacco, and Leif Katsuo Oxenløwe, "High‐Dimensional Quantum Communication: Benefits, Progress, and Future Challenges", Advanced Quantum Technologies 2 12, 1900038 (2019).
[25] Manuel B. Santos, Paulo Mateus, and Chrysoula Vlachou, "Quantum Universally Composable Oblivious Linear Evaluation", Quantum 8, 1507 (2024).
[26] Robert Fickler, Frédéric Bouchard, Enno Giese, Vincenzo Grillo, Gerd Leuchs, and Ebrahim Karimi, "Full-field mode sorter using two optimized phase transformations for high-dimensional quantum cryptography", Journal of Optics 22 2, 024001 (2020).
[27] Sheng Wang, Yu Lin, Xun Ye, Tianbing Zhang, Bao Feng, and Xiang Bian Yu, "The A Satellite-to-Ground Quantum Key Distribution Protocol Based on Orbital Angular Momentum of Light", Journal of Physics: Conference Series 1757 1, 012173 (2021).
[28] Andrew Forbes, Mostafa Youssef, Sachleen Singh, Isaac Nape, and Bora Ung, "Quantum cryptography with structured photons", Applied Physics Letters 124 11, 110501 (2024).
[29] Armin Tavakoli, Máté Farkas, Denis Rosset, Jean-Daniel Bancal, and Jedrzej Kaniewski, "Mutually unbiased bases and symmetric informationally complete measurements in Bell experiments", Science Advances 7 7, eabc3847 (2021).
[30] Mikka Stasiuk, Felix Hufnagel, Xiaoqin Gao, Aaron Z. Goldberg, Frédéric Bouchard, Ebrahim Karimi, and Khabat Heshami, "High-dimensional Encoding in the Round-Robin Differential-Phase-Shift Protocol", Quantum 7, 1207 (2023).
[31] Huan Zhang, Zhenyu Cao, Yu Sun, and Hu Jin, "A survey of OAM-encoded high-dimensional quantum key distribution: Foundations, experiments, and recent trends", ICT Express (2026).
[32] Shuang-Yin Huang, Zhou-Xiang Wang, Min Wang, Qian-Qian Tian, Chenghou Tu, Yongnan Li, and Hui-Tian Wang, Conference on Lasers and Electro-Optics JW1A.94 (2021) ISBN:978-1-943580-91-0.
[33] Keshav Das Agarwal, Sudip Kumar Haldar, and Aditi Sen(De), "Creating a two-qudit maximally entangled quantum link through bulk", Physical Review A 113 5, 052424 (2026).
[34] Alicia Sit, Felix Hufnagel, and Ebrahim Karimi, Structured Light for Optical Communication 139 (2021) ISBN:9780128215104.
[35] Jaesung Heo, Taek Jeong, Nam Hun Park, and Yonggi Jo, "True image construction in quantum-secured single-pixel imaging under spoofing attack", APL Photonics 9 7, 076111 (2024).
[36] Calum Maitland and Fabio Biancalana, "Angular momentum supercontinuum from fibre rings", Journal of Optics 22 1, 015503 (2020).
[37] F. Zhu, M. Tyler, N. H. Valencia, M. Malik, and J. Leach, "Is high-dimensional photonic entanglement robust to noise?", AVS Quantum Science 3 1, 011401 (2021).
[38] Wen-Zhe Yan, Yunting Li, Zhibo Hou, Huangjun Zhu, Guo-Yong Xiang, Chuan-Feng Li, and Guang-Can Guo, "Experimental Demonstration of Inequivalent Mutually Unbiased Bases", Physical Review Letters 132 8, 080202 (2024).
[39] Víctor López Pastor, Jeff Lundeen, and Florian Marquardt, "Arbitrary optical wave evolution with Fourier transforms and phase masks", Optics Express 29 23, 38441 (2021).
[40] Wei Li and Shengmei Zhao, "Generation of two-photon orbital-angular-momentum entanglement with a high degree of entanglement", Applied Physics Letters 114 4, 041105 (2019).
[41] Gerd Leuchs, Christoph Marquardt, Luis L. Sánchez-Soto, and Dmitry V. Strekalov, Optical Fiber Telecommunications VII 495 (2020) ISBN:9780128165027.
[42] Yangsheng Yuan, Xinyue Xiao, Dong Liu, Peng Fu, Jun Qu, Greg Gbur, and Yangjian Cai, "Mitigating orbital angular momentum crosstalk in an optical communication uplink channel using cylindrical vector beams", Waves in Random and Complex Media 35 2, 3635 (2025).
[43] Comfort Sekga, Mhlambululi Mafu, and Makhamisa Senekane, "High-dimensional quantum key distribution implemented with biphotons", Scientific Reports 13 1, 1229 (2023).
[44] Joonwoo Bae, Anindita Bera, Dariusz Chruściński, Beatrix C Hiesmayr, and Daniel McNulty, "How many mutually unbiased bases are needed to detect bound entangled states?", Journal of Physics A: Mathematical and Theoretical 55 50, 505303 (2022).
[45] Simon Morelli, Marcus Huber, and Armin Tavakoli, "Resource-Efficient High-Dimensional Entanglement Detection via Symmetric Projections", Physical Review Letters 131 17, 170201 (2023).
[46] Markus Hiekkamäki, Shashi Prabhakar, and Robert Fickler, "Near-perfect measuring of full-field transverse-spatial modes of light", Optics Express 27 22, 31456 (2019).
[47] Isaac Nape, Bereneice Sephton, Pedro Ornelas, Chane Moodley, and Andrew Forbes, "Quantum structured light in high dimensions", APL Photonics 8 5, 051101 (2023).
[48] Suman Karan, Radhika Prasad, and Anand K. Jha, "Postselection-free controlled generation of a high-dimensional orbital-angular-momentum entangled state", Physical Review Applied 20 5, 054027 (2023).
[49] Yundu Zhao, Shan Huang, and Shengjun Wu, "Entropic uncertainty relations and entanglement detection from quantum designs", Journal of Physics A: Mathematical and Theoretical 57 39, 395305 (2024).
[50] Vikas S. Bhat, Rounak Chatterjee, Kiran Bajar, and Sushil Mujumdar, "Optimizing the qudit dimensions of position-momentum entangled photons for quantum key distribution", Physical Review Applied 24 2, 024042 (2025).
[51] Lukas Scarfe, Felix Hufnagel, Manuel F. Ferrer-Garcia, Alessio D’Errico, Khabat Heshami, and Ebrahim Karimi, "Fast adaptive optics for high-dimensional quantum communications in turbulent channels", Communications Physics 8 1, 79 (2025).
[52] Jaroslav Kysela, "Arbitrary unitaries in orbital angular momentum of single photons", EPJ Quantum Technology 9 1, 22 (2022).
[53] Rojan Abolhassani, Lukas Scarfe, Francesco Di Colandrea, Alessio D’Errico, Khabat Heshami, and Ebrahim Karimi, "Investigating the performance of adaptive optics on different bases of spatial modes in turbulent channels", Optics Express 34 3, 3732 (2026).
[54] Abhishek Muhuri, Ayan Patra, Rivu Gupta, Tamoghna Das, and Aditi Sen(De), "Security of two-way deterministic quantum key distribution with higher-dimensional systems", Physics Letters A 583, 131550 (2026).
[55] Daniel McNulty and Stefan Weigert, "Mutually Unbiased Bases in Composite Dimensions – A Review", Quantum 10, 2051 (2026).
[56] Gianluca Ruffato, Michele Massari, Pietro Capaldo, and Filippo Romanato, "Holographic Silicon Metasurfaces for Total Angular Momentum Demultiplexing Applications in Telecom", Applied Sciences 9 11, 2387 (2019).
[57] Ohad Lib, Kfir Sulimany, Mateus Araújo, Michael Ben-Or, and Yaron Bromberg, "High-dimensional quantum key distribution using a multi-plane light converter", Optica Quantum 3 2, 182 (2025).
[58] E. A. Vashukevich and T. Yu. Golubeva, "Parallel two-qubit entangling gates via a quantum nondemolition interaction controlled by rotation", Physical Review A 110 3, 032616 (2024).
[59] Saheli Mukherjee, Bivas Mallick, Arun Kumar Das, Amit Kundu, and Pratik Ghosal, "Measurement-device-independent Schmidt number certification of all entangled states", Physical Review A 112 6, 062434 (2025).
[60] Matias Eriksson, Benjamin A. Stickler, Lea Kopf, Markus Hiekkamäki, Regina Gumenyuk, Yuri Chamorovskiy, Sven Ramelow, and Robert Fickler, "Talbot self-imaging and two-photon interference in ring-core fibers", Physical Review A 104 6, 063512 (2021).
[61] Rong Wang, Zhen-Qiang Yin, Hang Liu, Shuang Wang, Wei Chen, Guang-Can Guo, and Zheng-Fu Han, "Tight finite-key analysis for generalized high-dimensional quantum key distribution", Physical Review Research 3 2, 023019 (2021).
[62] Daniel Martínez, Esteban S. Gómez, Jaime Cariñe, Luciano Pereira, Aldo Delgado, Stephen P. Walborn, Armin Tavakoli, and Gustavo Lima, "Certification of a non-projective qudit measurement using multiport beamsplitters", Nature Physics 19 2, 190 (2022).
[63] Yonggi Jo, Hee Su Park, Seung-Woo Lee, and Wonmin Son, "Efficient High-Dimensional Quantum Key Distribution with Hybrid Encoding", Entropy 21 1, 80 (2019).
[64] Hong Lai and Linchun Wan, "Cryptographic algorithm for multi-path distribution of entangled states of orbital angular momentum based on Fibonacci values", Laser Physics Letters 21 6, 065209 (2024).
[65] Maciej Ogrodnik, Adam Widomski, Dagmar Bruẞ, Giovanni Chesi, Federico Grasselli, Hermann Kampermann, Chiara Macchiavello, Nathan Walk, Nikolai Wyderka, and Michał Karpiński, "High-dimensional quantum key distribution with resource-efficient detection", Optica Quantum 3 4, 372 (2025).
[66] Frédéric Bouchard, Duncan England, Philip J. Bustard, Khabat Heshami, and Benjamin Sussman, "Quantum Communication with Ultrafast Time-Bin Qubits", PRX Quantum 3 1, 010332 (2022).
[67] Yichi Zhang, Haoqi Zhao, Tianwei Wu, Zihe Gao, Li Ge, and Liang Feng, "High-Dimensional Quantum Key Distribution by a Spin-Orbit Microlaser", Physical Review X 15 1, 011024 (2025).
[68] Florian Kanitschar, Alexandra Bergmayr-Mann, Matej Pivoluska, and Marcus Huber, "Harnessing high-dimensional temporal entanglement using limited interferometric setups", Physical Review Applied 22 5, 054054 (2024).
[69] Katarzyna Siudzińska, "Measures from conical 2-designs depend only on two constants", Journal of Physics A: Mathematical and Theoretical 58 37, 375302 (2025).
[70] "WITHDRAWN: Ultrafast Quantum Communication with Time Bin Qubits Applied for Biomedical Engineering", (2023).
[71] Frédéric Bouchard, Felix Hufnagel, Dominik Koutný, Aazad Abbas, Alicia Sit, Khabat Heshami, Robert Fickler, and Ebrahim Karimi, "Quantum process tomography of a high-dimensional quantum communication channel", Quantum 3, 138 (2019).
[72] Randy Kuang, "Quantum encryption in phase space with dynamic displacement operators and quantum permutation pad", Academia Quantum 2 1(2025).
[73] Ziqing Wang, Robert Malaney, and Benjamin Burnett, "Satellite-To-Earth Quantum Key Distribution via Orbital Angular Momentum", Physical Review Applied 14 6, 064031 (2020).
[74] Abel Demeke, "A Systematic Literature Review: Quantum Key Distribution Networks: Challenges and Future Research Issues in Security", Science Discovery Physics 1 1, 29 (2026).
[75] Stephen Z. D. Plachta, Markus Hiekkamäki, Abuzer Yakaryılmaz, and Robert Fickler, "Quantum advantage using high-dimensional twisted photons as quantum finite automata", Quantum 6, 752 (2022).
[76] Konstantin Y Bliokh, Ebrahim Karimi, Miles J Padgett, Miguel A Alonso, Mark R Dennis, Angela Dudley, Andrew Forbes, Sina Zahedpour, Scott W Hancock, Howard M Milchberg, Stefan Rotter, Franco Nori, Şahin K Özdemir, Nicholas Bender, Hui Cao, Paul B Corkum, Carlos Hernández-García, Haoran Ren, Yuri Kivshar, Mário G Silveirinha, Nader Engheta, Arno Rauschenbeutel, Philipp Schneeweiss, Jürgen Volz, Daniel Leykam, Daria A Smirnova, Kexiu Rong, Bo Wang, Erez Hasman, Michela F Picardi, Anatoly V Zayats, Francisco J Rodríguez-Fortuño, Chenwen Yang, Jie Ren, Alexander B Khanikaev, Andrea Alù, Etienne Brasselet, Michael Shats, Jo Verbeeck, Peter Schattschneider, Dusan Sarenac, David G Cory, Dmitry A Pushin, Michael Birk, Alexey Gorlach, Ido Kaminer, Filippo Cardano, Lorenzo Marrucci, Mario Krenn, and Florian Marquardt, "Roadmap on structured waves", Journal of Optics 25 10, 103001 (2023).
[77] Sebastian Ecker, Frédéric Bouchard, Lukas Bulla, Florian Brandt, Oskar Kohout, Fabian Steinlechner, Robert Fickler, Mehul Malik, Yelena Guryanova, Rupert Ursin, and Marcus Huber, "Overcoming Noise in Entanglement Distribution", Physical Review X 9 4, 041042 (2019).
[78] Katarzyna Siudzińska, "Informationally overcomplete measurements from generalized equiangular tight frames", Journal of Physics A: Mathematical and Theoretical 57 33, 335302 (2024).
[79] Katherine Stevens, Amy McWilliam, Rair Macêdo, and Sonja Franke-Arnold, "BB84 protocol using rotationally symmetric states and a Fresnel cone", Optics Express 34 14, 25261 (2026).
[80] Dongxu Chen, Liyun Zhang, and Junhua Zhang, "Quantum teleportation with mutually unbiased bases", Quantum Information Processing 19 4, 121 (2020).
[81] Eileen Otte, Isaac Nape, Carmelo Rosales-Guzmán, Cornelia Denz, Andrew Forbes, and Bienvenu Ndagano, "High-dimensional cryptography with spatial modes of light: tutorial", Journal of the Optical Society of America B 37 11, A309 (2020).
[82] Armin Tavakoli, Denis Rosset, and Marc-Olivier Renou, "Enabling Computation of Correlation Bounds for Finite-Dimensional Quantum Systems via Symmetrization", Physical Review Letters 122 7, 070501 (2019).
[83] Alessio D’Errico, Felix Hufnagel, Filippo Miatto, Mohammadreza Rezaee, and Ebrahim Karimi, "Full-mode characterization of correlated photon pairs generated in spontaneous downconversion", Optics Letters 46 10, 2388 (2021).
[84] Hooman Barati Sedeh and Natalia M. Litchinitser, "Bridging meta-optics to quantum-optics: manipulating nonclassical light with structured materials [Invited]", Photonics Research 14 3, B249 (2026).
[85] George Claudiu Crisan, Antoine Henry, Dario A. Fioretto, Juan R. Alvarez, Stéphane Monfray, Frédéric Boeuf, Laurent Vivien, Eric Cassan, Carlos Alonso-Ramos, and Nadia Belabas, "Multi-dimensional frequency-bin entanglement-based quantum key distribution network", npj Quantum Information 12 1, 99 (2026).
[86] D. A. Turaykhanov, D. O. Akat'ev, A. V. Vasiliev, F. M. Ablayev, and A. A. Kalachev, "Quantum hashing via single-photon states with orbital angular momentum", Physical Review A 104 5, 052606 (2021).
[87] Jaesung Heo, Junghyun Kim, Taek Jeong, Yong Sup Ihn, Duk Y. Kim, Zaeill Kim, and Yonggi Jo, "Quantum-secured single-pixel imaging with enhanced security", Optica 10 11, 1461 (2023).
[88] Felix Hufnagel, Alicia Sit, Frédéric Bouchard, Yingwen Zhang, Duncan England, Khabat Heshami, Benjamin J Sussman, and Ebrahim Karimi, "Investigation of underwater quantum channels in a 30 meter flume tank using structured photons", New Journal of Physics 22 9, 093074 (2020).
[89] Shravan Kumar Sehgal and Rashmi Gupta, 2021 International Conference on Industrial Electronics Research and Applications (ICIERA) 1 (2021) ISBN:978-1-6654-3542-0.
[90] Jaesung Heo, Taek Jeong, Nam Hun Park, Su-Yong Lee, Jihwan Kim, Duk Y. Kim, Zaeill Kim, and Yonggi Jo, "Imaging through strong external illumination via mode-error analysis", Optics Express 33 17, 36973 (2025).
[91] Armin Tavakoli, Ingemar Bengtsson, Nicolas Gisin, and Joseph M. Renes, "Compounds of symmetric informationally complete measurements and their application in quantum key distribution", Physical Review Research 2 4, 043122 (2020).
[92] Katarzyna Siudzińska, "All classes of informationally complete symmetric measurements in finite dimensions", Physical Review A 105 4, 042209 (2022).
[93] Xingyu Wang, Tianyi Wu, Chen Dong, Haonan Zhu, Zhuodan Zhu, and Shanghong Zhao, "Integrating deep learning to achieve phase compensation for free-space orbital-angular-momentum-encoded quantum key distribution under atmospheric turbulence", Photonics Research 9 2, B9 (2021).
[94] Ilaria Gianani, Marco Sbroscia, and Marco Barbieri, "Measuring the time–frequency properties of photon pairs: A short review", AVS Quantum Science 2 1, 011701 (2020).
[95] Fan Wu and Liang Tang, "Quantifying high-dimensional entanglement via classes of symmetric measurements", Physical Review A 113 2, 022440 (2026).
[96] Dotan Halevi, Boaz Lubotzky, Kfir Sulimany, Eric G. Bowes, Jennifer A. Hollingsworth, Yaron Bromberg, and Ronen Rapaport, "High-dimensional quantum key distribution using orbital angular momentum of single photons from a colloidal quantum dot at room temperature", Optica Quantum 2 5, 351 (2024).
[97] Hugo Defienne, Matthew Reichert, and Jason W. Fleischer, "Adaptive Quantum Optics with Spatially Entangled Photon Pairs", Physical Review Letters 121 23, 233601 (2018).
[98] Fang-Xiang Wang, Wei Chen, Zhen-Qiang Yin, Shuang Wang, Guang-Can Guo, and Zheng-Fu Han, "Characterizing High-Quality High-Dimensional Quantum Key Distribution by State Mapping Between Different Degrees of Freedom", Physical Review Applied 11 2, 024070 (2019).
[99] J. Miguel-Ramiro and W. Dür, "Efficient entanglement purification protocols for d -level systems", Physical Review A 98 4, 042309 (2018).
[100] Frédéric Bouchard, Alicia Sit, Khabat Heshami, Robert Fickler, and Ebrahim Karimi, "Round-robin differential-phase-shift quantum key distribution with twisted photons", Physical Review A 98 1, 010301 (2018).
[101] Dongkai Zhang, Xiaodong Qiu, Wuhong Zhang, and Lixiang Chen, "Violation of a Bell inequality in two-dimensional state spaces for radial quantum number", Physical Review A 98 4, 042134 (2018).
[102] Yuval Idan and Avihai Didi, "A Review of Quantum communication using high-dimensional Hilbert spaces", arXiv:2402.01319, (2024).
The above citations are from Crossref's cited-by service (last updated successfully 2026-07-15 17:03:41) and SAO/NASA ADS (last updated successfully 2026-07-15 17:03:42). The list may be incomplete as not all publishers provide suitable and complete citation data.
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.