Quantum process tomography of a high-dimensional quantum communication channel
1Department of Physics, University of Ottawa, 25 Templeton Street, Ottawa, ON, K1N 6N5, Canada.
2Department of Optics, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic.
3National Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario K1A 0R6, Canada
4Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria.
5Current address: Photonics Laboratory, Physics Unit, Tampere University, Tampere, FI-33720, Finland.
6Department of Physics, Institute for Advanced Studies in Basic Sciences, 45137-66731 Zanjan, Iran.
| Published: | 2019-05-06, volume 3, page 138 |
| Eprint: | arXiv:1806.08018v3 |
| Doi: | https://doi.org/10.22331/q-2019-05-06-138 |
| Citation: | Quantum 3, 138 (2019). |
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
The characterization of quantum processes, e.g. communication channels, is an essential ingredient for establishing quantum information systems. For quantum key distribution protocols, the amount of overall noise in the channel determines the rate at which secret bits are distributed between authorized partners. In particular, tomographic protocols allow for the full reconstruction, and thus characterization, of the channel. Here, we perform quantum process tomography of high-dimensional quantum communication channels with dimensions ranging from 2 to 5. We can thus explicitly demonstrate the effect of an eavesdropper performing an optimal cloning attack or an intercept-resend attack during a quantum cryptographic protocol. Moreover, our study shows that quantum process tomography enables a more detailed understanding of the channel conditions compared to a coarse-grained measure, such as quantum bit error rates. This full characterization technique allows us to optimize the performance of quantum key distribution under asymmetric experimental conditions, which is particularly useful when considering high-dimensional encoding schemes.
► BibTeX data
► References
[1] Dowling, J. P. & Milburn, G. J. Quantum technology: the second quantum revolution, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 361, 1655–1674 (2003).
https://doi.org/10.1098/rsta.2003.1227
[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] Nielsen, M. A. & Chuang, I. Quantum computation and quantum information. Cambridge: Cambridge University Press (2002).
https://doi.org/10.1017/CBO9780511976667
[4] Degen, C., Reinhard, F. & Cappellaro, P. Quantum sensing. Rev. Mod. Phys. 89, 035002 (2017).
https://doi.org/10.1103/RevModPhys.89.035002
[5] 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
[6] Poyatos, J., Cirac, J. I. & Zoller, P. Complete characterization of a quantum process: the two-bit quantum gate. Phys. Rev. Lett. 78, 390 (1997).
https://doi.org/10.1103/PhysRevLett.78.390
[7] Nielsen, M. A., Knill, E. & Laflamme, R. Complete quantum teleportation using nuclear magnetic resonance. Nature 396, 52 (1998).
https://doi.org/10.1038/23891
[8] Childs, A. M., Chuang, I. L. & Leung, D. W. Realization of quantum process tomography in NMR. Phys. Rev. A 64, 012314 (2001).
https://doi.org/10.1103/PhysRevA.64.012314
[9] O'Brien, J. L. et al. Quantum process tomography of a controlled-not gate. Phys. Rev. Lett. 93, 080502 (2004).
https://doi.org/10.1103/PhysRevLett.93.080502
[10] Myrskog, S. H., Fox, J. K., Mitchell, M. W. & Steinberg, A. M. Quantum process tomography on vibrational states of atoms in an optical lattice. Phys. Rev. A 72, 013615 (2005).
https://doi.org/10.1103/PhysRevA.72.013615
[11] Riebe, M., Kim, K., Schindler, P., Monz, T., Schmidt, P. O., Korber, T. K., Hansel, W., Haffner, H., Roos, C. F. & Blatt, R. Process tomography of ion trap quantum gates. Phys. Rev. Lett. 97, 220407 (2006).
https://doi.org/10.1103/PhysRevLett.97.220407
[12] Howard, M. et al. Quantum process tomography and linblad estimation of a solid-state qubit. New Journal of Physics 8, 33 (2006).
https://doi.org/10.1088/1367-2630/8/3/033
[13] Lobino, M. et al. Complete characterization of quantum-optical processes. Science 322, 563–566 (2008).
https://doi.org/10.1126/science.1162086
[14] Kim, D. et al. Quantum control and process tomography of a semiconductor quantum dot hybrid qubit. Nature 511, 70 (2014).
https://doi.org/10.1038/nature13407
[15] Jacob, K. V., Mirasola, A. E., Adhikari, S. & Dowling, J. P. Direct characterization of linear and quadratically nonlinear optical systems. Phys. Rev. A 98, 052327 (2018).
https://doi.org/10.1103/PhysRevA.98.052327
[16] Gisin, N., Ribordy, G., Tittel, W. & Zbinden, H. Quantum cryptography. Rev. Mod. Phys. 74, 145 (2002).
https://doi.org/10.1103/RevModPhys.74.145
[17] Bienvenu, N., Perez-Garcia, B., Roux, F. S., McLaren, M., Rosales-Guzman, C., Zhang, Y., Mouane, O., Hernandez-Aranda, R. I., Konrad, T. & Forbes, A. Characterizing quantum channels with non-separable states of classical light. Nat. Phys. 13, 397 (2017).
https://doi.org/10.1038/nphys4003
[18] Muller, A., Zbinden, H. & Gisin, N. Underwater quantum coding. Nature 378, 449 (1995).
https://doi.org/10.1038/378449a0
[19] Buttler, W. T., Hughes, R. J., Kwiat, P. G., Lamoreaux, S. K., Luther, G. G., Morgan, G. L., Nordholt, J. E., Peterson, C. G. & Simmons, C. M. Practical free-space quantum key distribution over 1 km. Phys. Rev. Lett. 81, 3283 (1998).
https://doi.org/10.1103/PhysRevLett.81.3283
[20] Yin, J., Cao, Y., Li, Y. H., Ren, J. G., Liao, S. K., Zhang, L., Cai, W. Q., Liu, W. Y., Li, B., Dai, H., Li, M., Huang, Y. M., Deng, L., Li, L., Zhang, Q., Liu, N. L., Chen, Y. A., Lu, C. Y., Shu, R., Peng, C. Z., Wang, J. Y. & Pan, J. W. Satellite-to-ground entanglement-based quantum key distribution. Phys. Rev. Lett. 119, 200501 (2017).
https://doi.org/10.1103/PhysRevLett.119.200501
[21] 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
[22] Coles, P. J., Metodiev, E. M. & Lütkenhaus, N. Numerical approach for unstructured quantum key distribution. Nature Commun. 7, 11712 (2016).
https://doi.org/10.1038/ncomms11712
[23] Bechmann-Pasquinucci, H. & Tittel, W. Quantum cryptography using larger alphabets. Phys. Rev. A 61, 062308 (2000).
https://doi.org/10.1103/PhysRevA.61.062308
[24] 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
[25] Molina-Terriza, G., Torres, J. P. & Torner, L. Twisted photons. Nature Physics 3, 305 (2007).
https://doi.org/10.1038/nphys607
[26] Erhard, M., Fickler, R., Krenn, M. & Zeilinger, A. Twisted photons: New quantum perspectives in high dimensions. Light: Sci. & Appl. 7, 17146 (2018).
https://doi.org/10.1038/lsa.2017.146
[27] 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
[28] Mafu, M., Dudley, A., Goyal, S., Giovannini, D., McLaren, M., Padgett, M. J., Konrad, T., Petruccione, F., Lutkenhaus, N. & Forbes, A. 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
[29] 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
[30] Bouchard, F. et al. Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons. Quantum 2, 111 (2018).
https://doi.org/10.22331/q-2018-12-04-111
[31] Vallone, G., DAmbrosio, V., Sponselli, A., Slussarenko, S., Marrucci, L., Sciarrino, F. & Villoresi, P. Free-space quantum key distribution by rotation-invariant twisted photons. Phys. Rev. Lett. 113, 060503 (2014).
https://doi.org/10.1103/PhysRevLett.113.060503
[32] 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
[33] 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
[34] Bouchard, F. et al. Quantum cryptography with twisted photons through an outdoor underwater channel. Opt. Exp. 26, 22563-22573 (2018).
https://doi.org/10.1364/OE.26.022563
[35] 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
[36] Wootters, W. K. & Fields, B. D. Optimal state-determination by mutually unbiased measurements. Annals of Physics 191, 363–381 (1989).
https://doi.org/10.1016/0003-4916(89)90322-9
[37] Knee, G. C., Bolduc, E., Leach, J. & Gauger, E. M. Quantum process tomography via completely positive and trace-preserving projection. Phys. Rev. A 98, 062336 (2018).
https://doi.org/10.1103/PhysRevA.98.062336
[38] Ježek, M., Fiurášek, J. & Hradil, Z. Quantum inference of states and processes. Phys. Rev. A 68, 012305 (2003).
https://doi.org/10.1103/PhysRevA.68.012305
[39] 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
[40] 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
[41] Fernández-Pérez, A., Klimov, A. B. & Saavedra, C. Quantum process reconstruction based on mutually unbiased basis. Phys. Rev. A 83, 052332 (2011).
https://doi.org/10.1103/PhysRevA.83.052332
[42] 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
[43] Bouchard, F., Valencia, N. H., Brandt, F., Fickler, R., Huber, M., & Malik, M., Measuring azimuthal and radial modes of photons. Optics Express 26, 31925 (2018).
https://doi.org/10.1364/OE.26.031925
[44] Gilchrist, A., Langford, N. K. & Nielsen, M. A. Distance measures to compare real and ideal quantum processes. Phys. Rev. A 71, 062310 (2005).
https://doi.org/10.1103/PhysRevA.71.062310
[45] Irvine, W. T. M., Linares, A. L., deDood, M. J. A. & Bouwmeester, D. Optimal quantum cloning on a beam splitter. Phys. Rev. Lett. 92, 047902 (2004).
https://doi.org/10.1103/PhysRevLett.92.047902
[46] Ricci, M., Sciarrino, F., Sias, C. & DeMartini, F. Teleportation scheme implementing the universal optimal quantum cloning machine and the universal not gate. Phys. Rev. Lett. 92, 047901 (2004).
https://doi.org/10.1103/PhysRevLett.92.047901
[47] Nagali, E. et al. Optimal quantum cloning of orbital angular momentum photon qubits through Hong–Ou–Mandel coalescence. Nature Photonics 3, 720 (2009).
https://doi.org/10.1038/nphoton.2009.214
[48] Nagali, E., Giovannini, D., Marrucci, L., Slussarenko, S., Santamato, E. & Sciarrino, F. Experimental optimal cloning of four-dimensional quantum states of photons. Phys. Rev. Lett. 105, 073602 (2010).
https://doi.org/10.1103/PhysRevLett.105.073602
[49] 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
[50] Bruß, D. & Macchiavello, C. Optimal state estimation for d-dimensional quantum systems. Phys. Lett. A 253, 249-251 (1999).
https://doi.org/10.1016/S0375-9601(99)00099-7
[51] Thekkadath, G. S., Saaltink, R. Y., Giner, L. & Lundeen, J. S. Determining complementary properties with quantum clones. Phys. Rev. Lett. 119, 050405 (2017).
https://doi.org/10.1103/PhysRevLett.119.050405
[52] Hong, C.-K., Ou, Z.-Y. & Mandel, L. Measurement of subpicosecond time intervals between two photons by interference. Phys. Rev. Lett. 59, 2044 (1987).
https://doi.org/10.1103/PhysRevLett.59.2044
[53] 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
[54] Devetak, I. & Winter, A. Distillation of secret key and entanglement from quantum states. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 461, 207–235 (The Royal Society, 2005).
https://doi.org/10.1098/rspa.2004.1372
[55] 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
[56] Berkhout, C. G. et al. Efficient sorting of orbital angular momentum states of light. Phys. Rev. Lett. 105, 153601 (2010).
https://doi.org/10.1103/PhysRevLett.105.153601
[57] Fontaine, N. K. et al. Laguerre-Gaussian mode sorter. arXiv preprint arXiv:1803.04126v2 (2018).
arXiv:1803.04126v2
Cited by
[1] Muhammad AbuGhanem, "Full Quantum Process Tomography of a Universal Entangling Gate on an IBM’s Quantum Computer", Arabian Journal for Science and Engineering 50 23, 19923 (2025).
[2] Francesco Di Colandrea, Nazanin Dehghan, Alessio D’Errico, and Ebrahim Karimi, "Fourier Quantum Process Tomography", npj Quantum Information 10 1, 49 (2024).
[3] Maolin Luo, Xiaoqian Zhang, and Xiaoqi Zhou, "Proof-of-principle experimental demonstration of quantum gate verification", Physical Review A 105 1, 012614 (2022).
[4] Mohamed Barhoumi, Riccardo Bassoli, and Frank H.P. Fitzek, "Quantum qubit-optical cavity node: Effects of the temperature and coupling strengths on the cavity Fock-state distribution occupation probability and entropy of the subsystems using Dicke model", Results in Physics 72, 108214 (2025).
[5] Masahito Hayashi, Yu-Ao Chen, Chenghong Zhu, and Xin Wang, "Optimal sample complexity for testing unitary properties", Physical Review A 113 4, L040402 (2026).
[6] 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).
[7] Luigi Di Lauro, Stefania Sciara, Bennet Fischer, Junliang Dong, Imtiaz Alamgir, Benjamin Wetzel, Goëry Genty, Mitchell Nichols, Armaghan Eshaghi, David J. Moss, and Roberto Morandotti, "Optimization methods for integrated and programmable photonics in next-generation classical and quantum smart communication and signal processing systems", Advances in Optics and Photonics 17 3, 526 (2025).
[8] Ali Abu-Nada, Subhashish Banerjee, and Vivek Balasaheb Sabale, "Exploring the non-Markovian dynamics in depolarizing maps", Physical Review A 110 5, 052209 (2024).
[9] V. Cimini, I. Gianani, M. F. Sacchi, C. Macchiavello, and M. Barbieri, "Experimental witnessing of the quantum channel capacity in the presence of correlated noise", Physical Review A 102 5, 052404 (2020).
[10] Istiak Mahmud and Ahmed Abdelhadi, "Artificial Intelligence in Quantum Communications: A Comprehensive Survey", IEEE Access 13, 121174 (2025).
[11] Tareq Jaouni, Francesco Di Colandrea, Lorenzo Amato, Filippo Cardano, and Ebrahim Karimi, "Process tomography of structured optical gates with convolutional neural networks", Machine Learning: Science and Technology 5 4, 045071 (2024).
[12] Alicia Sit, Felix Hufnagel, and Ebrahim Karimi, Structured Light for Optical Communication 139 (2021) ISBN:9780128215104.
[13] Alessio D'Errico and Ebrahim Karimi, Electromagnetic Vortices 423 (2021) ISBN:9781119662822.
[14] Ryan Levy, Di Luo, and Bryan K. Clark, "Classical shadows for quantum process tomography on near-term quantum computers", Physical Review Research 6 1, 013029 (2024).
[15] Robert Fickler and Shashi Prabhakar, Structured Light for Optical Communication 205 (2021) ISBN:9780128215104.
[16] Sophie Engineer, Suraj Goel, Sophie Egelhaaf, Will McCutcheon, Vatshal Srivastav, Saroch Leedumrongwatthanakun, Sabine Wollmann, Benjamin D. M. Jones, Thomas Cope, Nicolas Brunner, Roope Uola, and Mehul Malik, "Certifying high-dimensional quantum channels", Physical Review Research 7 3, 033233 (2025).
[17] Francesco Di Colandrea, Lorenzo Amato, Roberto Schiattarella, Alexandre Dauphin, and Filippo Cardano, "Retrieving space-dependent polarization transformations via near-optimal quantum process tomography", Optics Express 31 20, 31698 (2023).
[18] Nazanin Dehghan, Alessio D’Errico, Francesco Di Colandrea, and Ebrahim Karimi, "Biphoton state reconstruction via phase retrieval methods", Optica 11 8, 1115 (2024).
[19] Frédéric Bouchard, Khabat Heshami, Duncan England, Robert Fickler, Robert W. Boyd, Berthold-Georg Englert, Luis L. Sánchez-Soto, and Ebrahim Karimi, "Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons", Quantum 2, 111 (2018).
[20] Salini Rajeev and Mayukh Lahiri, "Characterizing a high-dimensional unitary transformation without measuring the qudit it transforms", Optics Letters 51 4, 853 (2026).
[21] Q. Pears Stefano, I. Perito, and L. Rebón, "Selective and Efficient Quantum Process Tomography for Non-Trace-Preserving Maps: Implementation with a Superconducting Quantum Processor", Physical Review Applied 19 4, 044065 (2023).
[22] Yidan Sun, Yingjun Luo, Peiwen Peng, Yi Liu, and Dongmei Deng, "Dual coaxial electric field components in tightly focused circularly polarized Pearcey beams", Optics Communications 572, 130973 (2024).
[23] 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).
[24] Florian Brandt, Markus Hiekkamäki, Frédéric Bouchard, Marcus Huber, and Robert Fickler, "High-dimensional quantum gates using full-field spatial modes of photons", Optica 7 2, 98 (2020).
[25] Frédéric Bouchard, Natalia Herrera Valencia, Florian Brandt, Robert Fickler, Marcus Huber, and Mehul Malik, "Measuring azimuthal and radial modes of photons", Optics Express 26 24, 31925 (2018).
[26] 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).
[27] Ying Zuo, Chenfeng Cao, Ningping Cao, Xuanying Lai, Bei Zeng, and Shengwang Du, "Optical neural network quantum state tomography", Advanced Photonics 4, 026004 (2022).
[28] Bienvenu Ndagano and Andrew Forbes, "Characterization and mitigation of information loss in a six-state quantum-key-distribution protocol with spatial modes of light through turbulence", Physical Review A 98 6, 062330 (2018).
[29] Suraj Goel, Bohnishikha Ghosh, and Mehul Malik, "Quantum information processing with spatially structured light", Advanced Photonics 8, 014005 (2026).
[30] Huynh Le Dan Linh, Vu Tuan Hai, and Le Bin Ho, "Advancing quantum process tomography through quantum compilation", arXiv:2504.14958, (2025).
The above citations are from Crossref's cited-by service (last updated successfully 2026-08-13 19:14:11) and SAO/NASA ADS (last updated successfully 2026-08-13 19:14:12). 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.