{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:18:45Z","timestamp":1760235525469,"version":"build-2065373602"},"reference-count":51,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2021,8,25]],"date-time":"2021-08-25T00:00:00Z","timestamp":1629849600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2020YFA0309702"],"award-info":[{"award-number":["2020YFA0309702"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61605248, 61675235 and 61505261"],"award-info":[{"award-number":["61605248, 61675235 and 61505261"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2021M691536"],"award-info":[{"award-number":["2021M691536"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006407","name":"Natural Science Foundation of Henan Province","doi-asserted-by":"publisher","award":["202300410534 and 202300410532"],"award-info":[{"award-number":["202300410534 and 202300410532"]}],"id":[{"id":"10.13039\/501100006407","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Anhui Initiative in Quantum Information Technologies","award":["none"],"award-info":[{"award-number":["none"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Twin-field quantum key distribution (TF-QKD) has attracted considerable attention and developed rapidly due to its ability to surpass the fundamental rate-distance limit of QKD. However, the device imperfections may compromise its practical implementations. The goal of this paper is to make it robust against the state preparation flaws (SPFs) and side channels at the light source. We adopt the sending or not-sending (SNS) TF-QKD protocol to accommodate the SPFs and multiple optical modes in the emitted states. We analyze that the flaws of the phase modulation can be overcome by regarding the deviation of the phase as phase noise and eliminating it with the post-selection of phase. To overcome the side channels, we extend the generalized loss-tolerant (GLT) method to the four-intensity decoy-state SNS protocol. Remarkably, by decomposing of the two-mode single-photon states, the phase error rate can be estimated with only four parameters. The practical security of the SNS protocol with flawed and leaky source can be guaranteed. Our results might constitute a crucial step towards guaranteeing the practical implementation of the SNS protocol.<\/jats:p>","DOI":"10.3390\/e23091103","type":"journal-article","created":{"date-parts":[[2021,8,26]],"date-time":"2021-08-26T00:19:26Z","timestamp":1629937166000},"page":"1103","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Sending or Not-Sending Twin-Field Quantum Key Distribution with Flawed and Leaky Sources"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8212-0556","authenticated-orcid":false,"given":"Yi-Fei","family":"Lu","sequence":"first","affiliation":[{"name":"Henan Key Laboratory of Quantum Information and Cryptography, SSF IEU, Zhengzhou 450001, China"},{"name":"Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9774-1064","authenticated-orcid":false,"given":"Yang","family":"Wang","sequence":"additional","affiliation":[{"name":"Henan Key Laboratory of Quantum Information and Cryptography, SSF IEU, Zhengzhou 450001, China"},{"name":"Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"National Laboratory of Solid State Microstructures, School of Physics and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mu-Sheng","family":"Jiang","sequence":"additional","affiliation":[{"name":"Henan Key Laboratory of Quantum Information and Cryptography, SSF IEU, Zhengzhou 450001, China"},{"name":"Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2789-5938","authenticated-orcid":false,"given":"Xiao-Xu","family":"Zhang","sequence":"additional","affiliation":[{"name":"Henan Key Laboratory of Quantum Information and Cryptography, SSF IEU, Zhengzhou 450001, China"},{"name":"Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fan","family":"Liu","sequence":"additional","affiliation":[{"name":"Henan Key Laboratory of Quantum Information and Cryptography, SSF IEU, Zhengzhou 450001, China"},{"name":"Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hong-Wei","family":"Li","sequence":"additional","affiliation":[{"name":"Henan Key Laboratory of Quantum Information and Cryptography, SSF IEU, Zhengzhou 450001, China"},{"name":"Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chun","family":"Zhou","sequence":"additional","affiliation":[{"name":"Henan Key Laboratory of Quantum Information and Cryptography, SSF IEU, Zhengzhou 450001, China"},{"name":"Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shi-Biao","family":"Tang","sequence":"additional","affiliation":[{"name":"QuantumCTek Co., Ltd., Hefei 230088, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jia-Yong","family":"Wang","sequence":"additional","affiliation":[{"name":"CAS Quantum Network Co., Ltd., Shanghai 201315, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wan-Su","family":"Bao","sequence":"additional","affiliation":[{"name":"Henan Key Laboratory of Quantum Information and Cryptography, SSF IEU, Zhengzhou 450001, China"},{"name":"Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/j.tcs.2014.05.025","article-title":"Quantum cryptography: Public key distribution and coin tossing","volume":"560","author":"Bennett","year":"2014","journal-title":"Theor. Comput. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1012","DOI":"10.1364\/AOP.361502","article-title":"Advances in quantum cryptography","volume":"12","author":"Pirandola","year":"2020","journal-title":"Adv. Opt. Photon."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"025002","DOI":"10.1103\/RevModPhys.92.025002","article-title":"Secure quantum key distribution with realistic devices","volume":"92","author":"Xu","year":"2020","journal-title":"Rev. Mod. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"057901","DOI":"10.1103\/PhysRevLett.91.057901","article-title":"Quantum key distribution with high loss: Toward global secure communication","volume":"91","author":"Hwang","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"230503","DOI":"10.1103\/PhysRevLett.94.230503","article-title":"Beating the photon-number-splitting attack in practical quantum cryptography","volume":"94","author":"Wang","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"230504","DOI":"10.1103\/PhysRevLett.94.230504","article-title":"Decoy state quantum key distribution","volume":"94","author":"Lo","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"130503","DOI":"10.1103\/PhysRevLett.108.130503","article-title":"Measurement-device-independent quantum key distribution","volume":"108","author":"Lo","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"5235","DOI":"10.1038\/ncomms6235","article-title":"Fundamental rate-loss tradeoff for optical quantum key distribution","volume":"5","author":"Takeoka","year":"2014","journal-title":"Nat. Commun."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"15043","DOI":"10.1038\/ncomms15043","article-title":"Fundamental limits of repeaterless quantum communications","volume":"8","author":"Pirandola","year":"2017","journal-title":"Nat. Commun."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1038\/s41586-018-0066-6","article-title":"Overcoming the rate-distance limit of quantum key distribution without quantum repeaters","volume":"557","author":"Lucamarini","year":"2018","journal-title":"Nature"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"062323","DOI":"10.1103\/PhysRevA.98.062323","article-title":"Twin-field quantum key distribution with large misalignment error","volume":"98","author":"Wang","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_12","first-page":"031043","article-title":"Phase-Matching Quantum Key Distribution","volume":"8","author":"Ma","year":"2018","journal-title":"Phys. Rev. X"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1038\/s41534-019-0175-6","article-title":"Simple security proof of twin-field type quantum key distribution protocol","volume":"5","author":"Curty","year":"2019","journal-title":"NPJ Quant. Inf."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"034053","DOI":"10.1103\/PhysRevApplied.11.034053","article-title":"Twin-Field Quantum Key Distribution without Phase Postselection","volume":"11","author":"Cui","year":"2019","journal-title":"Phys. Rev. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"042332","DOI":"10.1103\/PhysRevA.98.042332","article-title":"Simple security analysis of phase-matching measurement-device-independent quantum key distribution","volume":"98","author":"Lin","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_16","unstructured":"Tamaki, K., Lo, H., Wang, W., and Lucamarini, M. (2018). Information theoretic security of quantum key distribution overcoming the repeaterless secret key capacity bound. arXiv."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3080","DOI":"10.1038\/s41598-019-39225-y","article-title":"Sending-or-not-sending twin-field quantum key distribution in practice","volume":"9","author":"Yu","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"024061","DOI":"10.1103\/PhysRevApplied.12.024061","article-title":"Unconditional Security of Sending or Not Sending Twin-Field Quantum Key Distribution with Finite Pulses","volume":"12","author":"Jiang","year":"2019","journal-title":"Phys. Rev. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"054034","DOI":"10.1103\/PhysRevApplied.12.054034","article-title":"Practical Long-Distance Side-Channel-Free Quantum Key Distribution","volume":"12","author":"Wang","year":"2019","journal-title":"Phys. Rev. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"062337","DOI":"10.1103\/PhysRevA.100.062337","article-title":"Sending-or-not-sending twin-field protocol for quantum key distribution with asymmetric source parameters","volume":"100","author":"Hu","year":"2019","journal-title":"Phys. Rev. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"062316","DOI":"10.1103\/PhysRevA.99.062316","article-title":"Asymmetric sending or not sending twin-field quantum key distribution in practice","volume":"99","author":"Zhou","year":"2019","journal-title":"Phys. Rev. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"042330","DOI":"10.1103\/PhysRevA.101.042330","article-title":"Sending-or-not-sending twin-field quantum key distribution: Breaking the direct transmission key rate","volume":"101","author":"Xu","year":"2020","journal-title":"Phys. Rev. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"053048","DOI":"10.1088\/1367-2630\/ab81b7","article-title":"Zigzag approach to higher key rate of sending-or-not-sending twin field quantum key distribution with finite-key effects","volume":"22","author":"Jiang","year":"2020","journal-title":"New J. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1007\/s11128-021-03070-8","article-title":"Finite-key analysis of sending-or-not-sending twin-field quantum key distribution with intensity fluctuations","volume":"20","author":"Lu","year":"2021","journal-title":"Quantum Inf. Process."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"100505","DOI":"10.1103\/PhysRevLett.123.100505","article-title":"Experimental Twin-Field Quantum Key Distribution through Sending or Not Sending","volume":"123","author":"Liu","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"070501","DOI":"10.1103\/PhysRevLett.124.070501","article-title":"Sending-or-Not-Sending with Independent Lasers: Secure Twin-Field Quantum Key Distribution over 509 km","volume":"124","author":"Chen","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"250502","DOI":"10.1103\/PhysRevLett.126.250502","article-title":"Field Test of Twin-Field Quantum Key Distribution through Sending-or-Not-Sending over 428 km","volume":"126","author":"Liu","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4487","DOI":"10.1126\/sciadv.aaz4487","article-title":"Quantum key distribution with correlated sources","volume":"6","author":"Pereira","year":"2020","journal-title":"Sci. Adv."},{"key":"ref_29","first-page":"325","article-title":"Security of quantum key distribution with imperfect devices","volume":"4","author":"Gottesman","year":"2004","journal-title":"Quantum Inf. Comput."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"052314","DOI":"10.1103\/PhysRevA.90.052314","article-title":"Loss-tolerant quantum cryptography with imperfect sources","volume":"90","author":"Tamaki","year":"2014","journal-title":"Phys. Rev. A"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"093011","DOI":"10.1088\/1367-2630\/17\/9\/093011","article-title":"Finite-key security analysis of quantum key distribution with imperfect light sources","volume":"17","author":"Mizutani","year":"2015","journal-title":"New J. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"065008","DOI":"10.1088\/1367-2630\/18\/6\/065008","article-title":"Decoy-state quantum key distribution with a leaky source","volume":"18","author":"Tamaki","year":"2016","journal-title":"New J. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1038\/s41534-017-0057-8","article-title":"Quantum key distribution with an efficient countermeasure against correlated intensity fluctuations in optical pulses","volume":"4","author":"Yoshino","year":"2018","journal-title":"NPJ Quant. Inf."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1038\/s41534-018-0122-y","article-title":"Quantum key distribution with setting-choice-independently correlated light sources","volume":"5","author":"Mizutani","year":"2019","journal-title":"NPJ Quant. Inf."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1038\/s41534-019-0194-3","article-title":"Quantum key distribution with simply characterized light sources","volume":"5","author":"Mizutani","year":"2019","journal-title":"NPJ Quant. Inf."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1038\/s41534-019-0133-3","article-title":"Characterising the correlations of prepare-and-measure quantum networks","volume":"5","author":"Wang","year":"2019","journal-title":"NPJ Quant. Inf."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"034072","DOI":"10.1103\/PhysRevApplied.15.034072","article-title":"Practical Quantum Key Distribution That is Secure Against Side Channels","volume":"15","author":"Navarrete","year":"2021","journal-title":"Phys. Rev. Appl."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"023011","DOI":"10.1088\/1367-2630\/abdf9b","article-title":"Security of quantum key distribution with source and detection imperfections","volume":"23","author":"Sun","year":"2021","journal-title":"New J. Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"032305","DOI":"10.1103\/PhysRevA.92.032305","article-title":"Experimental quantum key distribution with source flaws","volume":"92","author":"Xu","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"042308","DOI":"10.1103\/PhysRevA.93.042308","article-title":"Experimental measurement-device-independent quantum key distribution with imperfect sources","volume":"93","author":"Tang","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"190502","DOI":"10.1103\/PhysRevLett.121.190502","article-title":"Secure Quantum Key Distribution over 421 km of Optical Fiber","volume":"121","author":"Boaron","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1038\/s41534-019-0180-9","article-title":"Quantum key distribution with flawed and leaky sources","volume":"5","author":"Pereira","year":"2019","journal-title":"NPJ Quant. Inf."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"634","DOI":"10.1038\/ncomms1631","article-title":"Tight finite-key analysis for quantum cryptography","volume":"3","author":"Tomamichel","year":"2012","journal-title":"Nat. Commun."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"032304","DOI":"10.1103\/PhysRevA.85.032304","article-title":"Partially random phase attack to the practical two-way quantum-key-distribution system","volume":"85","author":"Sun","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"022308","DOI":"10.1103\/PhysRevA.88.022308","article-title":"Source attack of decoy-state quantum key distribution using phase information","volume":"88","author":"Tang","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"2023","DOI":"10.1080\/09500340108240904","article-title":"Large pulse attack as a method of conventional optical eavesdropping in quantum cryptography","volume":"48","author":"Vakhitov","year":"2001","journal-title":"J. Mod. Opt."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"123030","DOI":"10.1088\/1367-2630\/16\/12\/123030","article-title":"Trojan-horse attacks threaten the security of practical quantum cryptography","volume":"16","author":"Jain","year":"2014","journal-title":"New J. Phys."},{"key":"ref_48","first-page":"031030","article-title":"Practical Security Bounds Against the Trojan-Horse Attack in Quantum Key Distribution","volume":"5","author":"Lucamarini","year":"2015","journal-title":"Phys. Rev. X"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"034017","DOI":"10.1103\/PhysRevApplied.13.034017","article-title":"Laser-Damage Attack Against Optical Attenuators in Quantum Key Distribution","volume":"13","author":"Huang","year":"2020","journal-title":"Phys. Rev. Appl."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"042325","DOI":"10.1103\/PhysRevA.93.042325","article-title":"Security of quantum key distribution with light sources that are not independently and identically distributed","volume":"93","author":"Nagamatsu","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"357","DOI":"10.2748\/tmj\/1178243286","article-title":"Weighted sums of certain dependent random variables","volume":"19","author":"Azuma","year":"1967","journal-title":"Tohoku Math. J."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/9\/1103\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:51:24Z","timestamp":1760165484000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/9\/1103"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,25]]},"references-count":51,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2021,9]]}},"alternative-id":["e23091103"],"URL":"https:\/\/doi.org\/10.3390\/e23091103","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2021,8,25]]}}}