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). |
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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.
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► 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
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[3] Hugo Defienne, Matthew Reichert, and Jason W. Fleischer, "Adaptive Quantum Optics with Spatially Entangled Photon Pairs", Physical Review Letters 121 23, 233601 (2018).
[4] 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).
[5] 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).
[6] 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 General 55 50, 505303 (2022).
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[8] 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).
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[12] J. Miguel-Ramiro and W. Dür, "Efficient entanglement purification protocols for d -level systems", Physical Review A 98 4, 042309 (2018).
[13] Katarzyna Siudzińska, "All classes of informationally complete symmetric measurements in finite dimensions", Physical Review A 105 4, 042209 (2022).
[14] 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).
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[17] 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).
[18] 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).
[19] 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).
[20] 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).
[21] 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).
[22] Hasnaa Hajji and Morad El Baz, "Qutrit-based semi-quantum key distribution protocol", Quantum Information Processing 20 1, 4 (2021).
[23] Andrew Forbes, Mostafa Youssef, Sachleen Singh, Isaac Nape, and Bora Ung, "Quantum cryptography with structured photons", Applied Physics Letters 124 11, 110501 (2024).
[24] Wei Li, Le Wang, and Shengmei Zhao, "Phase Matching Quantum Key Distribution based on Single-Photon Entanglement", Scientific Reports 9, 15466 (2019).
[25] Simon Morelli, Marcus Huber, and Armin Tavakoli, "Resource-Efficient High-Dimensional Entanglement Detection via Symmetric Projections", Physical Review Letters 131 17, 170201 (2023).
[26] 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).
[27] 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).
[28] 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).
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