{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,23]],"date-time":"2026-02-23T23:58:27Z","timestamp":1771891107747,"version":"3.50.1"},"reference-count":52,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T00:00:00Z","timestamp":1769040000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T00:00:00Z","timestamp":1769040000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12064012"],"award-info":[{"award-number":["12064012"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["11464015"],"award-info":[{"award-number":["11464015"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Quantum Inf Process"],"DOI":"10.1007\/s11128-026-05061-z","type":"journal-article","created":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T07:02:38Z","timestamp":1769065358000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Measurement-device-independent high-dimensional two-way quantum key distribution"],"prefix":"10.1007","volume":"25","author":[{"given":"Guo-Dong","family":"Kang","sequence":"first","affiliation":[]},{"given":"Xiao-Lu","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Ting","family":"Zhang","sequence":"additional","affiliation":[]},{"given":"Jie","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Mao-Fa","family":"Fang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,1,22]]},"reference":[{"issue":"5886","key":"5061_CR1","doi-asserted-by":"publisher","first-page":"802","DOI":"10.1038\/299802A0","volume":"299","author":"WK Wootters","year":"1982","unstructured":"Wootters, W.K., Zurek, W.H.: A single quantum cannot be cloned. Nature 299(5886), 802\u2013803 (1982). https:\/\/doi.org\/10.1038\/299802A0","journal-title":"Nature"},{"key":"5061_CR2","doi-asserted-by":"publisher","unstructured":"Bennett, C.H., Brassard, G.: Quantum cryptography: public key distribution and coin tossing. Proc. Int. Conf. Comput., Syst. Signal Process., 175\u2013179 (1984). https:\/\/doi.org\/10.1016\/j.tcs.2014.05.025","DOI":"10.1016\/j.tcs.2014.05.025"},{"issue":"6","key":"5061_CR3","doi-asserted-by":"publisher","first-page":"661","DOI":"10.1103\/PhysRevLett.67.661","volume":"67","author":"AK Ekert","year":"1991","unstructured":"Ekert, A.K.: Quantum cryptography based on Bell\u2019s theorem. Phys. Rev. Lett. 67(6), 661\u2013663 (1991). https:\/\/doi.org\/10.1103\/PhysRevLett.67.661","journal-title":"Phys. Rev. Lett."},{"key":"5061_CR4","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/BF00191318","volume":"5","author":"CH Bennett","year":"1992","unstructured":"Bennett, C.H., Bessette, F., Brassard, G., Salvail, L., Smolin, J.: Experimental quantum cryptography. J. Cryptol. 5, 3 (1992). https:\/\/doi.org\/10.1007\/BF00191318","journal-title":"J. Cryptol."},{"issue":"5410","key":"5061_CR5","doi-asserted-by":"publisher","first-page":"2050","DOI":"10.1126\/science.283.5410.2050","volume":"283","author":"H-K Lo","year":"1999","unstructured":"Lo, H.-K., Chau, H.F.: Unconditional security of quantum key distribution over arbitrarily long distances. Science 283(5410), 2050\u20132056 (1999). https:\/\/doi.org\/10.1126\/science.283.5410.2050","journal-title":"Science"},{"key":"5061_CR6","doi-asserted-by":"publisher","first-page":"441","DOI":"10.1103\/PhysRevLett.85.441","volume":"85","author":"PW Shor","year":"2000","unstructured":"Shor, P.W., Preskill, J.: Simple proof of security of the BB84 quantum key distribution protocol. Phys. Rev. Lett. 85, 441 (2000). https:\/\/doi.org\/10.1103\/PhysRevLett.85.441","journal-title":"Phys. Rev. Lett."},{"key":"5061_CR7","doi-asserted-by":"publisher","unstructured":"Liao, S.-K., Cai, W.-Q., Liu, W.-Y., Zhang, L., Li, Y., Ren, J.-G., Yin, J., Shen, Q., Cao, Y., Li, Z.-P., Li, F.-Z., Chen, X.-W., Sun, L.-H., Jia, J.-J., Wu, J.-C., Jiang, X.-J., Wang, J.-F., Huang, Y.-M., Wang, Q., Zhou, Y.-L., Deng, L., Xi, T., Ma, L., Hu, T., Zhang, Q., Chen, Y.-A., Liu, N.-L., Wang, X.-B., Zhu, Z.-C., Lu, C.-Y., Shu, R., Peng, C.-Z., Wang, J.-Y., Pan, J.-W.: Satellite-to-ground quantum key distribution. Nature 549(7670), 43\u201347 (2017). https:\/\/doi.org\/10.1038\/nature23655","DOI":"10.1038\/nature23655"},{"issue":"19","key":"5061_CR8","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1103\/PhysRevLett.121.190502","volume":"121","author":"A Boaron","year":"2018","unstructured":"Boaron, A., Boso, G., Rusca, D., Vulliez, C., Autebert, C., Caloz, M., Perrenoud, M., Gras, G., Bussi\u00e8res, F., Li, M.-J., Nolan, D., Martin, A., Zbinden, H.: Secure quantum key distribution over 421km of optical fiber. Phys. Rev. Lett. 121(19), 9\u201313 (2018). https:\/\/doi.org\/10.1103\/PhysRevLett.121.190502","journal-title":"Phys. Rev. Lett."},{"issue":"4","key":"5061_CR9","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.78.042333","volume":"78","author":"Y Zhao","year":"2008","unstructured":"Zhao, Y., Fred Fung, C.-H., Qi, B., Chen, C., Lo, H.-K.: Quantum hacking: experimental demonstration of time-shift attack against practical quantum-key-distribution systems. Phys. Rev. A 78(4), 042333 (2008). https:\/\/doi.org\/10.1103\/PhysRevA.78.042333","journal-title":"Phys. Rev. A"},{"issue":"3","key":"5061_CR10","doi-asserted-by":"publisher","first-page":"723","DOI":"10.1103\/PhysRevA.75.032314","volume":"75","author":"C-H Fred Fung","year":"2006","unstructured":"Fred Fung, C.-H., Qi, B., Tamaki, K., Lo, H.-K.: Phase-remapping attack in practical quantum key distribution systems. Phys. Rev. A 75(3), 723\u2013727 (2006). https:\/\/doi.org\/10.1103\/PhysRevA.75.032314","journal-title":"Phys. Rev. A"},{"issue":"10","key":"5061_CR11","doi-asserted-by":"publisher","first-page":"686","DOI":"10.1038\/nphoton.2010.214","volume":"4","author":"L Lydersen","year":"2010","unstructured":"Lydersen, L., Wiechers, C., Wittmann, C., Elser, D., Skaar, J., Makarov, V.: Hacking commercial quantum cryptography systems by tailored bright illumination. Nat. Photonics 4(10), 686\u2013689 (2010). https:\/\/doi.org\/10.1038\/nphoton.2010.214","journal-title":"Nat. Photonics"},{"issue":"13","key":"5061_CR12","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.108.130503","volume":"108","author":"H-K Lo","year":"2012","unstructured":"Lo, H.-K., Curty, M., Qi, B.: Measurement-device-independent quantum key distribution. Phys. Rev. Lett. 108(13), 130503 (2012). https:\/\/doi.org\/10.1103\/PhysRevLett.108.130503","journal-title":"Phys. Rev. Lett."},{"issue":"19","key":"5061_CR13","doi-asserted-by":"publisher","first-page":"190501","DOI":"10.1103\/PhysRevLett.117.19050","volume":"117","author":"H-L Yin","year":"2016","unstructured":"Yin, H.-L., Chen, T.-Y., Yu, Z.-W., Liu, H., You, L.-X., Zhou, Y.-H., Chen, S.-J., Mao, Y.Q., Huang, M.-Q., Zhang, W.-J., Chen, H., Li, M.-J., Nolan, D., Zhou, F., Jiang, X., Wang, Z., Zhang, Q., Wang, X.-B., Pan, J.-W.: Measurement-device-independent quantum key distribution over a 404 km optical fiber. Phys. Rev. Lett. 117(19), 190501\u2013190515 (2016). https:\/\/doi.org\/10.1103\/PhysRevLett.117.19050","journal-title":"Phys. Rev. Lett."},{"issue":"7705","key":"5061_CR14","doi-asserted-by":"publisher","first-page":"400","DOI":"10.1038\/s41586-018-0066-6","volume":"557","author":"M Lucamarini","year":"2018","unstructured":"Lucamarini, M., Yuan, Z.L., Dynes, J.F., Shields, A.J.: Overcoming the rate-distance limit of quantum key distribution without quantum repeaters. Nature 557(7705), 400\u2013403 (2018). https:\/\/doi.org\/10.1038\/s41586-018-0066-6","journal-title":"Nature"},{"issue":"3","key":"5061_CR15","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevX.8.031043","volume":"8","author":"X Ma","year":"2018","unstructured":"Ma, X., Zeng, P., Zhou, H.Y.: Phase-matching quantum key distribution. Phys. Rev. X 8(3), 031043 (2018). https:\/\/doi.org\/10.1103\/PhysRevX.8.031043","journal-title":"Phys. Rev. X"},{"key":"5061_CR16","doi-asserted-by":"publisher","first-page":"422","DOI":"10.1038\/s41566-020-0599-8","volume":"14","author":"X-T Fang","year":"2020","unstructured":"Fang, X.-T., Zeng, P., Liu, H., Zou, M., Wu, W.-J., Tang, Y.-L., Sheng, Y.-J., Xiang, Y., Zhang, W.-J., Li, H., Wang, Z., You, L.-X., Li, M.-J., Chen, H., Chen, Y.-A., Zhang, Q., Peng, C.-Z., Ma, X.-F., Chen, T.-Y., Pan, J.-W.: Implementation of quantum key distribution surpassing the linear rate-transmittance bound. Nat. Photonics 14, 422\u2013425 (2020). https:\/\/doi.org\/10.1038\/s41566-020-0599-8","journal-title":"Nat. Photonics"},{"key":"5061_CR17","doi-asserted-by":"publisher","unstructured":"Chen, J.-P., Zhang, C., Liu, Y., Jiang, C., Zhang, W.-J., Han, Z.-Y., Ma, S.-Z., Hu, X.-L., Li, Y.-H., Liu, H.: Twin-field quantum key distribution over a 511 km optical fiber linking two distant metropolitan areas. Nat. Photonics 15(8), 570\u2013575 (2021). https:\/\/doi.org\/10.21203\/rs.3.rs-182908","DOI":"10.21203\/rs.3.rs-182908"},{"key":"5061_CR18","doi-asserted-by":"publisher","first-page":"154","DOI":"10.1038\/s41566-021-00928-2","volume":"16","author":"S Wang","year":"2022","unstructured":"Wang, S., Yin, Z.-Q., He, D.-Y., Wang, R.-Q., Ye, P., Zhou, Y., Fan, Y., Guan, J., Wang, F.-X., Chen, W.-Z., Yong, G., Morozov, P.V., Divochiy, A.V., Zhou, Z., Guo, G.-C., Han, Z.-F.: Twin-field quantum key distribution over 830 km fiber. Nat. Photonics 16, 154\u2013161 (2022). https:\/\/doi.org\/10.1038\/s41566-021-00928-2","journal-title":"Nat. Photonics"},{"issue":"18","key":"5061_CR19","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.89.187902","volume":"89","author":"K Bostrom","year":"2002","unstructured":"Bostrom, K., Felbinger, T.: Deterministic secure direct communication using entanglement. Phys. Rev. Lett. 89(18), 187902 (2002). https:\/\/doi.org\/10.1103\/PhysRevLett.89.187902","journal-title":"Phys. Rev. Lett."},{"issue":"14","key":"5061_CR20","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.94.140501","volume":"94","author":"M Lucamarini","year":"2005","unstructured":"Lucamarini, M., Mancini, S.: Secure deterministic communication without entanglement. Phys. Rev. Lett. 94(14), 140501 (2005). https:\/\/doi.org\/10.1103\/PhysRevLett.94.140501","journal-title":"Phys. Rev. Lett."},{"issue":"4","key":"5061_CR21","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.84.042344","volume":"84","author":"H Lu","year":"2011","unstructured":"Lu, H., Fred Fung, C.-H., Ma, X.-F., Cai, Q.-Y.: Unconditional security proof of a deterministic quantum key distribution with a two-way quantum channel. Phys. Rev. A 84(4), 042344 (2011). https:\/\/doi.org\/10.1103\/PhysRevA.84.042344","journal-title":"Phys. Rev. A"},{"issue":"3","key":"5061_CR22","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.85.032308","volume":"85","author":"C-H Fred Fung","year":"2012","unstructured":"Fred Fung, C.-H., Ma, X.-F., Chau, H.-F., Cai, Q.-Y.: Quantum key distribution with delayed privacy amplification and its application to the security proof of a two-way deterministic protocol. Phys. Rev. A 85(3), 032308 (2012). https:\/\/doi.org\/10.1103\/PhysRevA.85.032308","journal-title":"Phys. Rev. A"},{"issue":"1","key":"5061_CR23","doi-asserted-by":"publisher","first-page":"1950010","DOI":"10.1142\/S0219749919500102","volume":"17","author":"G-D Kang","year":"2019","unstructured":"Kang, G.-D., Zhou, Q.-P., Fang, M.-F.: Two-way quantum key distribution in a uniformly distributed quantum space via a special mapping and its analytical security proofs. Int. J. Quant. Inf. 17(1), 1950010 (2019). https:\/\/doi.org\/10.1142\/S0219749919500102","journal-title":"Int. J. Quant. Inf."},{"issue":"169","key":"5061_CR24","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1007\/s11080-007-9045-3","volume":"14","author":"M Lucamarini","year":"2007","unstructured":"Lucamarini, M., C\u00e9re, A., Di Giuseppe, G., Mancini, S., Vitali, D., Tombesi, P.: Two-way protocol for quantum cryptography with imperfect devices. Open. Syst. Inf. Dyn. 14(169), 169\u2013178 (2007). https:\/\/doi.org\/10.1007\/s11080-007-9045-3","journal-title":"Open. Syst. Inf. Dyn."},{"issue":"4","key":"5061_CR25","doi-asserted-by":"publisher","first-page":"0443021","DOI":"10.1103\/PhysRevA.88.044302","volume":"88","author":"H Lu","year":"2013","unstructured":"Lu, H., Fred Fung, C.-H., Cai, Q.-Y.: Two-way deterministic quantum key distribution against detector-side-channel attacks. Phys. Rev. A 88(4), 0443021\u2013044302 (2013). https:\/\/doi.org\/10.1103\/PhysRevA.88.044302","journal-title":"Phys. Rev. A"},{"issue":"6","key":"5061_CR26","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11128-024-04419-5","volume":"23","author":"G-D Kang","year":"2024","unstructured":"Kang, G.-D., Liu, J., Zhang, T., Zhou, Q.-P., Fang, M.-F.: Fully measurement-device-independent two-way quantum key distribution with finite single-photon sources. Quantum Inf. Process. 23(6), 1\u201318 (2024). https:\/\/doi.org\/10.1007\/s11128-024-04419-5","journal-title":"Quantum Inf. Process."},{"key":"5061_CR27","doi-asserted-by":"publisher","unstructured":"Ding, Y.-H., 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 Inf. 3(1), 25 (2017). https:\/\/doi.org\/10.1038\/s41534-017-0026-2","DOI":"10.1038\/s41534-017-0026-2"},{"issue":"11","key":"5061_CR28","doi-asserted-by":"publisher","first-page":"4709","DOI":"10.1007\/s10773-016-3094-8","volume":"55","author":"Y-Q Wang","year":"2016","unstructured":"Wang, Y.-Q., She, K., Huang, R.-F., Zhong, O.: Optimal symmetric ternary quantum encryption schemes. Int. J. Theor. Phys. 55(11), 4709\u20134722 (2016). https:\/\/doi.org\/10.1007\/s10773-016-3094-8","journal-title":"Int. J. Theor. Phys."},{"issue":"4","key":"5061_CR29","doi-asserted-by":"publisher","first-page":"1012","DOI":"10.1364\/AOP.361502","volume":"12","author":"S Pirandola","year":"2020","unstructured":"Pirandola, S., Andersen, U.L., Banchi, L., Berta, M., Bunandar, D., Colbeck, R., Englund, D., Gehring, T., Lupo, C., Ottaviani, C., Pereira, J., Razavi, M., Shaari, J.S., Tomamichel, M., Usenko, V.C., Vallone, G., Villoresi, P., Wallden, P.: Advances in quantum cryptography. Adv. Opt. Photonics 12(4), 1012\u20131236 (2020). https:\/\/doi.org\/10.1364\/AOP.361502","journal-title":"Adv. Opt. Photonics"},{"issue":"3","key":"5061_CR30","doi-asserted-by":"publisher","first-page":"1301","DOI":"10.1103\/RevModPhys.81.1301","volume":"81","author":"V Scarani","year":"2009","unstructured":"Scarani, V., Bechmann-Pasquinucci, H., Cerf, N.J., Du\u0161ek, M., L\u00fctkenhaus, N., Peev, M.: The security of practical quantum key distribution. Rev. Mod. Phys. 81(3), 1301\u20131350 (2009). https:\/\/doi.org\/10.1103\/RevModPhys.81.1301","journal-title":"Rev. Mod. Phys."},{"key":"5061_CR31","doi-asserted-by":"crossref","unstructured":"Gan, Y.-H., Wang, Y., Bao, W.-S., He, R.-S., Zhou, C., Jiang, M.-S.: Finite-key analysis for a practical high-dimensional quantum key distribution system based on time phase states. Chin. Phys. Lett. 36(4), 040301(2019). https:\/\/iopscience.iop.org\/article\/10.1088\/0256-307X\/36\/4\/040301","DOI":"10.1088\/0256-307X\/36\/4\/040301"},{"issue":"10","key":"5061_CR32","first-page":"2511","volume":"5","author":"WO Krawec","year":"2022","unstructured":"Krawec, W.O.: Security of a High Dimensional Two?Way Quantum Key Distribution Protocol. Adv. Quant. Tech. 5(10), 2511\u20139044 (2022). arxiv:2203.02989","journal-title":"Adv. Quant. Tech."},{"issue":"3","key":"5061_CR33","doi-asserted-by":"publisher","first-page":"1","DOI":"10.48550\/arXiv.1805.07228","volume":"63","author":"Z-R Zhou","year":"2020","unstructured":"Zhou, Z.-R., Sheng, Y.-B., Niu, P.-H., Yin, L.-G., Long, G.-L.: Measurement-device-independent quantum secure direct communication. Sci. China Phys. Mech. 63(3), 1\u20136 (2020). https:\/\/doi.org\/10.48550\/arXiv.1805.07228","journal-title":"Sci. China Phys. Mech."},{"key":"5061_CR34","doi-asserted-by":"publisher","unstructured":"Liu, L., Niu, J.-L., Fan, C.-R., Feng, X.-T., Wang, C.: High-dimensional measurement-device-independent quantum secure direct communication. Quant. Inf. Process. 19(11): 404-415. https:\/\/doi.org\/10.1007\/s11128-020-02908-x.","DOI":"10.1007\/s11128-020-02908-x."},{"key":"5061_CR35","doi-asserted-by":"publisher","unstructured":"Sun, Z.-Z., Cheng, Y.-B., Ruan, D., Pan, D.: Single-photon measurement-device-independent quantum secure direct communication. Opt. Commun.569(000):8(2024).https:\/\/doi.org\/10.1016\/j.optcom.2024.130745","DOI":"10.1016\/j.optcom.2024.130745"},{"key":"5061_CR36","doi-asserted-by":"publisher","unstructured":"Hu, X.-M., Xing, W.-B., Liu, B.-H., Huang, Y.-F., Huber, M.: Efficient Generation of High-Dimensional Entanglement through Multi-path Down-Conversion.Phys. Rev. Lett. 125, 090503 (2020). https:\/\/doi.org\/10.1103\/PhysRevLett.125.090503","DOI":"10.1103\/PhysRevLett.125.090503"},{"key":"5061_CR37","doi-asserted-by":"crossref","unstructured":"Zhang, C., Miao, J.-L., Hu, X.-M., Pauwels, J., Guo, Y., Li, C.-F.,et al.: Quantum stochastic communication via high-dimensional entanglement.Phys. Rev. Lett. 135, 120802, (2025). https:\/\/journals.aps.org\/prl\/abstract\/10.1103\/rq78-1qbh","DOI":"10.1103\/rq78-1qbh"},{"key":"5061_CR38","doi-asserted-by":"crossref","unstructured":"Kong, L.-J., Zhang, F., Zhang, J., Zhang, X.: High-dimensional entanglement-enabled holography. Phys. Rev. Lett. 130, 053602 (2023). arxiv:2302.01485","DOI":"10.1103\/PhysRevLett.130.053602"},{"issue":"19","key":"5061_CR39","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.120.193904","volume":"120","author":"Y Wen","year":"2018","unstructured":"Wen, Y., Chremmos, I., Chen, Y., Zhu, J., Zhang, Y., Yu, S.: Spiral transformation for high-resolution and efficient sorting of optical vortex modes. Phys. Rev. Lett. 120(19), 193904 (2018)","journal-title":"Phys. Rev. Lett."},{"key":"5061_CR40","doi-asserted-by":"publisher","unstructured":"El Gamal, A., Kim, Y.H.: Network information theory. Cambridge Univ. Press (2011). https:\/\/doi.org\/10.1017\/CBO9781139030687","DOI":"10.1017\/CBO9781139030687"},{"issue":"20","key":"5061_CR41","doi-asserted-by":"publisher","first-page":"4162","DOI":"10.1016\/j.optcom.2009.07.011","volume":"282","author":"H-W Li","year":"2009","unstructured":"Li, H.-W., Zhao, Y.-B., Yin, Z.-Q., Shuang, W., Han, Z.-F., Bao, W.-S., Guo, G.-C.: Security of decoy states QKD with finite resources against collective attacks. Opt. Commun. 282(20), 4162\u20134166 (2009). https:\/\/doi.org\/10.1016\/j.optcom.2009.07.011","journal-title":"Opt. Commun."},{"issue":"2","key":"5061_CR42","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.102.020504","volume":"102","author":"M Christandl","year":"2009","unstructured":"Christandl, M., K\u00f6nig, R., Renner, R.: Postselection technique for quantum channels with applications to quantum cryptography. Phys. Rev. Lett. 102(2), 020504 (2009). https:\/\/doi.org\/10.1103\/PhysRevLett.102.020504","journal-title":"Phys. Rev. Lett."},{"key":"5061_CR43","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.96.200501","volume":"96","author":"A Cer\u00e9","year":"2006","unstructured":"Cer\u00e9, A., Lucamarini, M., Di Giuseppe, G., Tombesi, P.: Experimental test of two-way quantum key distribution in the presence of controlled noise. Phys. Rev. Lett. 96, 200501 (2006). https:\/\/doi.org\/10.1103\/PhysRevLett.96.200501","journal-title":"Phys. Rev. Lett."},{"issue":"6498","key":"5061_CR44","doi-asserted-by":"publisher","first-page":"1487","DOI":"10.1126\/science.aba9779","volume":"368","author":"L Li","year":"2020","unstructured":"Li, L., Liu, Z., Ren, X., Wang, S., Su, V.-C., Chen, M.-K., Chu, C.-H., Kuo, H.-Y., Liu, B.-Y., Zang, W.-B., Guo, G.-C., Zhang, L.-J., Wang, Z.-L., Zhu, S.-N., Tsai, D.-P.: Metalens-array-based high-dimensional and multiphoton quantum source. Science 368(6498), 1487\u20131490 (2020). https:\/\/doi.org\/10.1126\/science.aba9779","journal-title":"Science"},{"issue":"1","key":"5061_CR45","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1103\/PhysRevA.72.012326","volume":"72","author":"X Ma","year":"2005","unstructured":"Ma, X., Qi, B., Zhao, Y., Lo, H.-K.: Practical decoy state for quantum key distribution. Phys. Rev. A 72(1), 1\u2013127 (2005). https:\/\/doi.org\/10.1103\/PhysRevA.72.012326","journal-title":"Phys. Rev. A"},{"issue":"19","key":"5061_CR46","doi-asserted-by":"publisher","first-page":"3762","DOI":"10.1063\/1.1738173","volume":"84","author":"C Gobby","year":"2004","unstructured":"Gobby, C., Yuan, Z.L., Shields, A.J.: Quantum key distribution over 122 km of standard telecom fiber. Appl. Phys. Lett. 84(19), 3762\u20133764 (2004). https:\/\/doi.org\/10.1063\/1.1738173","journal-title":"Appl. Phys. Lett."},{"key":"5061_CR47","doi-asserted-by":"publisher","DOI":"10.1088\/1367-2630\/18\/7\/073030","volume":"18","author":"K Br\u00e1dler","year":"2016","unstructured":"Br\u00e1dler, K., Mirhosseini, M., Fickler, R., Broadbent, A., Boyd, R.: Finite-key security analysis for multilevel quantum key distribution. New J. Phys. 18, 073030 (2016). https:\/\/doi.org\/10.1088\/1367-2630\/18\/7\/073030","journal-title":"New J. Phys."},{"key":"5061_CR48","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.130.250801","volume":"130","author":"Z Lai","year":"2023","unstructured":"Lai, Z., Lin, J., Xie, Y.-M., Lu, Y.-S., Jing, Y.-M., Yin, H.-L., Yuan, Z.-L.: Experimental quantum communication overcomes the rate-loss limit without global phase tracking. Phys. Rev. Lett. 130, 250801 (2023). https:\/\/doi.org\/10.1103\/PhysRevLett.130.250801","journal-title":"Phys. Rev. Lett."},{"issue":"7","key":"5061_CR49","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.115.070502.Epub","volume":"115","author":"Z-Q Zhou","year":"2015","unstructured":"Zhou, Z.-Q., Hua, Y.-L., Liu, X., Chen, G., Xu, J.-S., Han, Y.-J., Li, C.-F., Guo, G.-C.: Quantum Storage of Three-Dimensional Orbital-Angular-Momentum Entanglement in a Crystal. Phys. Rev. Lett. 115(7), 070502 (2015). https:\/\/doi.org\/10.1103\/PhysRevLett.115.070502.Epub","journal-title":"Phys. Rev. Lett."},{"key":"5061_CR50","doi-asserted-by":"crossref","unstructured":"Niek J. B., Serge F.: Sampling in a quantum population, and applications. In Annual Cryptology Conference, 724-741 (2010). arxiv:0907.4246","DOI":"10.1007\/978-3-642-14623-7_39"},{"key":"5061_CR51","doi-asserted-by":"publisher","unstructured":"Yao, K., Krawec, W. O., Zhu, J.-D.: Quantum sampling for finite key rates in high dimensional quantum cryptography. IEEE Trans. Inf. Theory. 68(5), 3144\u20133163 (2022). https:\/\/doi.org\/10.48550\/arXiv.2012.04151","DOI":"10.48550\/arXiv.2012.04151"},{"issue":"1","key":"5061_CR52","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1142\/S0219749908003256","volume":"6","author":"R Renner","year":"2008","unstructured":"Renner, R.: Security of quantum key distribution. Int. J. Quant. Inf. 6(1), 1\u2013127 (2008). https:\/\/doi.org\/10.1142\/S0219749908003256","journal-title":"Int. J. Quant. Inf."}],"container-title":["Quantum Information Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-026-05061-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11128-026-05061-z","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-026-05061-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,23]],"date-time":"2026-02-23T23:02:23Z","timestamp":1771887743000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11128-026-05061-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,22]]},"references-count":52,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2026,2]]}},"alternative-id":["5061"],"URL":"https:\/\/doi.org\/10.1007\/s11128-026-05061-z","relation":{},"ISSN":["1573-1332"],"issn-type":[{"value":"1573-1332","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,22]]},"assertion":[{"value":"31 July 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 January 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 January 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that there are no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"36"}}