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J. Theor. Phys.</a> <a href="/?q=in%3A478327" title="Articles in this Issue">61, No. 2, Paper No. 26, 13 p. (2022)</a>. </div> <div class="abstract">Summary: With the development of quantum computer, the research of quantum key distribution (QKD) protocols and high-dimensional quantum key distribution (HD-QKD) protocols has emerged as an important subject. Most of the existing QKD protocols are based on complex physical devices or have high technical requirements for users. Hadamard gate can realize the interconversion of mutually unbiased bases (MUB) and have advantages in the QKD protocol construction. In this paper, we propose a ternary quantum key distribution protocol using three-dimensional Hadamard gate. This protocol can enrich the research of high-dimensional key distribution protocols. The protocol can directly send and receive ternary quantum with security and efficiency. By logic tree analysis, the protocol is demonstrated to show higher security than the two-dimensional protocol in a perfect channel. In principle, this protocol could be extended to higher dimensions and would benefit the research of high-dimensional key distribution protocols.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7489104">Cited in <strong>1</strong> Document</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A81P94" title="MSC2020">81P94</a> </td> <td class="space"> Quantum cryptography (quantum-theoretic aspects) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A94A60" title="MSC2020">94A60</a> </td> <td class="space"> Cryptography </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A81P10" title="MSC2020">81P10</a> </td> <td class="space"> Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A68P25" title="MSC2020">68P25</a> </td> <td class="space"> Data encryption (aspects in computer science) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A81P68" title="MSC2020">81P68</a> </td> <td class="space"> Quantum computation </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A81P70" title="MSC2020">81P70</a> </td> <td class="space"> Quantum coding (general) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05B20" title="MSC2020">05B20</a> </td> <td class="space"> Combinatorial aspects of matrices (incidence, Hadamard, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A15B34" title="MSC2020">15B34</a> </td> <td class="space"> Boolean and Hadamard matrices </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A81P55" title="MSC2020">81P55</a> </td> <td class="space"> Special bases (entangled, mutual unbiased, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C05" title="MSC2020">05C05</a> </td> <td class="space"> Trees </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AHD-QKD">HD-QKD</a>; <a href="/?q=ut%3Aternary+quantum">ternary quantum</a>; <a href="/?q=ut%3AHadamard+gate">Hadamard gate</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">&times;</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;"> <!-- alert for connection errors etc --> </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1486.81074" data-ciurl="/ci/07489104" data-biburl="/bibtex/07489104.bib" data-amsurl="/amsrefs/07489104.bib" data-xmlurl="/xml/07489104.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07489104.pdf" title="Zbl 1486.81074 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s10773-022-05041-w" aria-label="DOI for “Ternary quantum key distribution protocol based on Hadamard gate”" title="10.1007/s10773-022-05041-w">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Zbinden, H.; Bechmann-Pasquinucci, H.; Gisin, N.; Ribordy, G., Quantum cryptography, Appl. Phys. B, 67, 6, 743-748 (1998)&nbsp;&middot; <a href="https://doi.org/10.1007/s003400050574" class="nowrap">doi:10.1007/s003400050574</a></td> </tr><tr> <td>[2]</td> <td class="space">Sasaki, M., Fujiwara, M., Ishizuka, H., Klaus, W., Wakui, K., Takeoka, M., Miki, S., Yamashita, T., Wang, Z., Tanaka, A.: Field test of quantum key distribution in the tokyo qkd network. Opt. Express 19 (2011)</td> </tr><tr> <td>[3]</td> <td class="space">Pirandola, S., Andersen, U.L., Banchi, L., Berta, M., Bunandar, D., Colbeck, R., Englund, D., Gehring, T., Lupo, C., Ottaviani, C.: Advances in quantum cryptography (2019)</td> </tr><tr> <td>[4]</td> <td class="space">Bennett, CH; Brassard, G., Quantum cryptography: Public key distribution and coin tossing, Theor. Comput. Sci., 560, 7-11 (2014) &middot; <a href="/1306.81030" class="nowrap">Zbl 1306.81030</a>&nbsp;&middot; <a href="https://doi.org/10.1016/j.tcs.2014.05.025" class="nowrap">doi:10.1016/j.tcs.2014.05.025</a></td> </tr><tr> <td>[5]</td> <td class="space">Watanabe, S., Matsumoto, R., Uyematsu, T.: Noise tolerance of the bb84 protocol with random privacy amplification. In: International Symposium On Information Theory, 2005. ISIT 2005. Proceedings (2005) &middot; <a href="/1110.81061" class="nowrap">Zbl 1110.81061</a></td> </tr><tr> <td>[6]</td> <td class="space">Nambu, Y.; Hatanaka, T.; Nakamura, K., Bb84 quantum key distribution system based on silica-based planar lightwave circuits, Jpn. J. Appl. Phys., 43, No. 8B, 1109-1110 (2014)&nbsp;&middot; <a href="https://doi.org/10.1143/JJAP.43.L1109" class="nowrap">doi:10.1143/JJAP.43.L1109</a></td> </tr><tr> <td>[7]</td> <td class="space">Inamori, H., Security of practical bb84 quantum key distribution, Algorithmica, 34, 4, 366-371 (2002) &middot; <a href="/1012.68064" class="nowrap">Zbl 1012.68064</a>&nbsp;&middot; <a href="https://doi.org/10.1007/s00453-002-0984-3" class="nowrap">doi:10.1007/s00453-002-0984-3</a></td> </tr><tr> <td>[8]</td> <td class="space">Shor, PW; Preskill, J., Simple proof of security of the bb84 quantum key distribution protocol, Phys. Rev. Lett., 85, 2, 441-444 (2000)&nbsp;&middot; <a href="https://doi.org/10.1103/PhysRevLett.85.441" class="nowrap">doi:10.1103/PhysRevLett.85.441</a></td> </tr><tr> <td>[9]</td> <td class="space">Chong, SK; Hwang, T., Quantum key agreement protocol based on bb84, Opt. Commun., 283, 6, 1192-1195 (2010)&nbsp;&middot; <a href="https://doi.org/10.1016/j.optcom.2009.11.007" class="nowrap">doi:10.1016/j.optcom.2009.11.007</a></td> </tr><tr> <td>[10]</td> <td class="space">Chen, X. , Wang, F.Q., Lu, Y.Q., Zhao, F., Liu, S.H.: A differential phase shift key distribution qkd system combing with efficient bb84 scheme. Acta Photonica Sinica (2008)</td> </tr><tr> <td>[11]</td> <td class="space">Li, X.K., Song, X.Q., Guo, Q.W., Zhou, X.Y., Wang, Q.: Practical decoy-state bb84 quantum key distribution with quantum memory. Chin. Phys. B 30(6) (2021)</td> </tr><tr> <td>[12]</td> <td class="space">Liu, B.X., Jiang, D.H., Liang, X.Q., Zhang, Y.H.: A novel quantum voting scheme based on bb84-state. Int. J. Theor. Phys. (3) (2021) &middot; <a href="/1522.81072" class="nowrap">Zbl 1522.81072</a></td> </tr><tr> <td>[13]</td> <td class="space">Rezazadeh, F., Mani, A., Karimipour, V.: Quantum key distribution with no shared reference frame. Quantum Inf. Process. 19(2) (2019)</td> </tr><tr> <td>[14]</td> <td class="space">Hendrych, M.; Gallego, R.; MiUda, M.; Brunner, N.; Torres, JP, Experimental estimation of the dimension of classical and quantum systems, Nat. Phys., 8, 8, 588-591 (2012)&nbsp;&middot; <a href="https://doi.org/10.1038/nphys2334" class="nowrap">doi:10.1038/nphys2334</a></td> </tr><tr> <td>[15]</td> <td class="space">Deutsch, D., Quantum theory, the church-turing principle and the universal quantum computer, Proc. R. Soc. Lond., 400, 1818, 97-117 (1985) &middot; <a href="/0900.81019" class="nowrap">Zbl 0900.81019</a></td> </tr><tr> <td>[16]</td> <td class="space">Biamonte, J.D., Perkowski, M.A.: Testing a quantum computer. Physics (2009)</td> </tr><tr> <td>[17]</td> <td class="space">Monfared, AT; Haghparast, M., Design of new quantum/reversible ternary subtractor circuits, Journal of Circuits, Systems and Computers, 25, 2, 1650014 (2016)&nbsp;&middot; <a href="https://doi.org/10.1142/S0218126616500146" class="nowrap">doi:10.1142/S0218126616500146</a></td> </tr><tr> <td>[18]</td> <td class="space">Cerf, NJ; Bourennane, M.; Karlsson, A.; Gisin, N., Security of quantum key distribution using d-level systems, Phys. Rev. Lett., 88, 12, 127902 (2002)&nbsp;&middot; <a href="https://doi.org/10.1103/PhysRevLett.88.127902" class="nowrap">doi:10.1103/PhysRevLett.88.127902</a></td> </tr><tr> <td>[19]</td> <td class="space">Wang, YQ; She, K.; Huang, RF; Zhong, O., Optimal symmetric ternary quantum encryption schemes, Int. J. Theor. Phys., 55, 11, 1-14 (2016) &middot; <a href="/1358.81094" class="nowrap">Zbl 1358.81094</a>&nbsp;&middot; <a href="https://doi.org/10.1007/s10773-016-3094-8" class="nowrap">doi:10.1007/s10773-016-3094-8</a></td> </tr><tr> <td>[20]</td> <td class="space">Ding, Y., Bacco, D., Dalgaard, K., Cai, X., Zhou, X., Rottwitt, K., Oxenlwe, L.K.: High-dimensional quantum key distribution based on multicore fiber using silicon photonic integrated circuits. npj Quantum Information 3(1) (2017)</td> </tr><tr> <td>[21]</td> <td class="space">Duo, H., Zhihui, L., Chengji, L., Feifei, G.: The phase matching quantum key distribution protocol with 3-state systems (2020)</td> </tr><tr> <td>[22]</td> <td class="space">Etcheverry, S.; Cañas, G.; Gómez, E.; Nogueira, W.; Saavedra, C.; Xavier, GB; Lima, G., Quantum key distribution session with 16-dimensional photonic states, Scientific Reports, 3, 2316 (2013)&nbsp;&middot; <a href="https://doi.org/10.1038/srep02316" class="nowrap">doi:10.1038/srep02316</a></td> </tr><tr> <td>[23]</td> <td class="space">Iqbal, H., Krawec, W.O.: Analysis of a High-Dimensional Extended B92 Protocol (2021) &middot; <a href="/1509.81380" class="nowrap">Zbl 1509.81380</a></td> </tr><tr> <td>[24]</td> <td class="space">Bocharov, A., Cui, S.X., Roetteler, M., Svore, K.M.: Improved quantum ternary arithmetics. Computer Science (2015)</td> </tr><tr> <td>[25]</td> <td class="space">Brandt, F., Hiekkamki, M., Bouchard, F., Huber, M., Fickler, R.: High-dimensional quantum gates using full-field spatial modes of photons. Optica 7(2) (2019)</td> </tr><tr> <td>[26]</td> <td class="space">Wang, Y.; She, K.; Luo, Q.; Fan, Y.; Chao, Z., Symmetric weak ternary quantum homomorphic encryption schemes, Modern Physics Letters B, 30, 7, 1650076 (2016)&nbsp;&middot; <a href="https://doi.org/10.1142/S0217984916500767" class="nowrap">doi:10.1142/S0217984916500767</a></td> </tr><tr> <td>[27]</td> <td class="space">Di, Y.M., Wei, H.R.: Elementary gates for ternary quantum logic circuit (2011)</td> </tr><tr> <td>[28]</td> <td class="space">Quantum key distribution based on interferometry and interaction-free measurement. Int. J. Theor. Phys. 55(1), 98-106 (2016) &middot; <a href="/1335.81058" class="nowrap">Zbl 1335.81058</a></td> </tr><tr> <td>[29]</td> <td class="space">Knirsch, F.; Eibl, G.; Engel, D., Error-resilient masking approaches for privacy preserving data aggregation, IEEE Transactions on Smart Grid, PP, 99, 1-1 (2016)</td> </tr><tr> <td>[30]</td> <td class="space">Nimbe, P., Weyori, B.A., Yeng, P.K.: A framework for quantum-classical cryptographic translation. Int. J. Theor. Phys. 60(1) (2021) &middot; <a href="/1523.81056" class="nowrap">Zbl 1523.81056</a></td> </tr><tr> <td>[31]</td> <td class="space">Zhang, C.M., Zhu, J.R., Wang, Q.: Reference-frame-independent measurement-device-independent quantum key distribution with modified coherent states. IEEE Photonics Journal (2018)</td> </tr><tr> <td>[32]</td> <td class="space">Wang, L, Zhou, Y.-Y., Xue-Jun, Zhou, Chen, X., Zhang, Z.: Correction to: New scheme for measurement-device-independent quantum key distribution. Quantum Inf. Process. (2019) &middot; <a href="/1417.81118" class="nowrap">Zbl 1417.81118</a></td> </tr><tr> <td>[33]</td> <td class="space">Feng, J.; Zhou, C.; Wang, B.; Zheng, J.; Peng, L., Three-port beam splitter of a binary fused-silica grating, Appl. Opt., 47, 35, 6638-6643 (2009)&nbsp;&middot; <a href="https://doi.org/10.1364/AO.47.006638" class="nowrap">doi:10.1364/AO.47.006638</a></td> </tr><tr> <td>[34]</td> <td class="space">Su, H.Y.: Simple analysis of security of the bb84 quantum key distribution protocol. Quantum Inf. Process. 19(6) (2020) &middot; <a href="/1508.81744" class="nowrap">Zbl 1508.81744</a></td> </tr><tr> <td>[35]</td> <td class="space">Wang, YQ; She, K.; Huang, RF; Zhong, O., Optimal symmetric ternary quantum encryption schemes, Int. J. Theor. Phys., 55, 11, 1-14 (2016) &middot; <a href="/1358.81094" class="nowrap">Zbl 1358.81094</a>&nbsp;&middot; <a href="https://doi.org/10.1007/s10773-016-3094-8" class="nowrap">doi:10.1007/s10773-016-3094-8</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH&nbsp;Open. 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