{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T17:26:52Z","timestamp":1775496412429,"version":"3.50.1"},"reference-count":41,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,12]],"date-time":"2023-01-12T00:00:00Z","timestamp":1673481600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Key Lab of Guangzhou for Quantum Precision Measurement","award":["202201000010"],"award-info":[{"award-number":["202201000010"]}]},{"name":"Key Lab of Guangzhou for Quantum Precision Measurement","award":["2019A1515011703"],"award-info":[{"award-number":["2019A1515011703"]}]},{"name":"Key Lab of Guangzhou for Quantum Precision Measurement","award":["2020B0303300001"],"award-info":[{"award-number":["2020B0303300001"]}]},{"name":"Key Lab of Guangzhou for Quantum Precision Measurement","award":["2020B1515310016"],"award-info":[{"award-number":["2020B1515310016"]}]},{"name":"Key Lab of Guangzhou for Quantum Precision Measurement","award":["12075159"],"award-info":[{"award-number":["12075159"]}]},{"name":"Key Lab of Guangzhou for Quantum Precision Measurement","award":["12171044"],"award-info":[{"award-number":["12171044"]}]},{"name":"Key Lab of Guangzhou for Quantum Precision Measurement","award":["Z190005"],"award-info":[{"award-number":["Z190005"]}]},{"name":"the Guangdong Basic and Applied Basic Research Foundation","award":["202201000010"],"award-info":[{"award-number":["202201000010"]}]},{"name":"the Guangdong Basic and Applied Basic Research Foundation","award":["2019A1515011703"],"award-info":[{"award-number":["2019A1515011703"]}]},{"name":"the Guangdong Basic and Applied Basic Research Foundation","award":["2020B0303300001"],"award-info":[{"award-number":["2020B0303300001"]}]},{"name":"the Guangdong Basic and Applied Basic Research Foundation","award":["2020B1515310016"],"award-info":[{"award-number":["2020B1515310016"]}]},{"name":"the Guangdong Basic and Applied Basic Research Foundation","award":["12075159"],"award-info":[{"award-number":["12075159"]}]},{"name":"the Guangdong Basic and Applied Basic Research Foundation","award":["12171044"],"award-info":[{"award-number":["12171044"]}]},{"name":"the Guangdong Basic and Applied Basic Research Foundation","award":["Z190005"],"award-info":[{"award-number":["Z190005"]}]},{"name":"the Key Research and Development Project of Guangdong Province","award":["202201000010"],"award-info":[{"award-number":["202201000010"]}]},{"name":"the Key Research and Development Project of Guangdong Province","award":["2019A1515011703"],"award-info":[{"award-number":["2019A1515011703"]}]},{"name":"the Key Research and Development Project of Guangdong Province","award":["2020B0303300001"],"award-info":[{"award-number":["2020B0303300001"]}]},{"name":"the Key Research and Development Project of Guangdong Province","award":["2020B1515310016"],"award-info":[{"award-number":["2020B1515310016"]}]},{"name":"the Key Research and Development Project of Guangdong Province","award":["12075159"],"award-info":[{"award-number":["12075159"]}]},{"name":"the Key Research and Development Project of Guangdong Province","award":["12171044"],"award-info":[{"award-number":["12171044"]}]},{"name":"the Key Research and Development Project of Guangdong Province","award":["Z190005"],"award-info":[{"award-number":["Z190005"]}]},{"name":"the Guangdong Basic and Applied Basic Research Foundation","award":["202201000010"],"award-info":[{"award-number":["202201000010"]}]},{"name":"the Guangdong Basic and Applied Basic Research Foundation","award":["2019A1515011703"],"award-info":[{"award-number":["2019A1515011703"]}]},{"name":"the Guangdong Basic and Applied Basic Research Foundation","award":["2020B0303300001"],"award-info":[{"award-number":["2020B0303300001"]}]},{"name":"the Guangdong Basic and Applied Basic Research Foundation","award":["2020B1515310016"],"award-info":[{"award-number":["2020B1515310016"]}]},{"name":"the Guangdong Basic and Applied Basic Research Foundation","award":["12075159"],"award-info":[{"award-number":["12075159"]}]},{"name":"the Guangdong Basic and Applied Basic Research Foundation","award":["12171044"],"award-info":[{"award-number":["12171044"]}]},{"name":"the Guangdong Basic and Applied Basic Research Foundation","award":["Z190005"],"award-info":[{"award-number":["Z190005"]}]},{"name":"the National Natural Science Foundation of China","award":["202201000010"],"award-info":[{"award-number":["202201000010"]}]},{"name":"the National Natural Science Foundation of China","award":["2019A1515011703"],"award-info":[{"award-number":["2019A1515011703"]}]},{"name":"the National Natural Science Foundation of China","award":["2020B0303300001"],"award-info":[{"award-number":["2020B0303300001"]}]},{"name":"the National Natural Science Foundation of China","award":["2020B1515310016"],"award-info":[{"award-number":["2020B1515310016"]}]},{"name":"the National Natural Science Foundation of China","award":["12075159"],"award-info":[{"award-number":["12075159"]}]},{"name":"the National Natural Science Foundation of China","award":["12171044"],"award-info":[{"award-number":["12171044"]}]},{"name":"the National Natural Science Foundation of China","award":["Z190005"],"award-info":[{"award-number":["Z190005"]}]},{"DOI":"10.13039\/501100004826","name":"Beijing Natural Science Foundation","doi-asserted-by":"publisher","award":["202201000010"],"award-info":[{"award-number":["202201000010"]}],"id":[{"id":"10.13039\/501100004826","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004826","name":"Beijing Natural Science Foundation","doi-asserted-by":"publisher","award":["2019A1515011703"],"award-info":[{"award-number":["2019A1515011703"]}],"id":[{"id":"10.13039\/501100004826","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004826","name":"Beijing Natural Science Foundation","doi-asserted-by":"publisher","award":["2020B0303300001"],"award-info":[{"award-number":["2020B0303300001"]}],"id":[{"id":"10.13039\/501100004826","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004826","name":"Beijing Natural Science Foundation","doi-asserted-by":"publisher","award":["2020B1515310016"],"award-info":[{"award-number":["2020B1515310016"]}],"id":[{"id":"10.13039\/501100004826","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004826","name":"Beijing Natural Science Foundation","doi-asserted-by":"publisher","award":["12075159"],"award-info":[{"award-number":["12075159"]}],"id":[{"id":"10.13039\/501100004826","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004826","name":"Beijing Natural Science Foundation","doi-asserted-by":"publisher","award":["12171044"],"award-info":[{"award-number":["12171044"]}],"id":[{"id":"10.13039\/501100004826","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004826","name":"Beijing Natural Science Foundation","doi-asserted-by":"publisher","award":["Z190005"],"award-info":[{"award-number":["Z190005"]}],"id":[{"id":"10.13039\/501100004826","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the Academician Innovation Platform of Hainan Province","award":["202201000010"],"award-info":[{"award-number":["202201000010"]}]},{"name":"the Academician Innovation Platform of Hainan Province","award":["2019A1515011703"],"award-info":[{"award-number":["2019A1515011703"]}]},{"name":"the Academician Innovation Platform of Hainan Province","award":["2020B0303300001"],"award-info":[{"award-number":["2020B0303300001"]}]},{"name":"the Academician Innovation Platform of Hainan Province","award":["2020B1515310016"],"award-info":[{"award-number":["2020B1515310016"]}]},{"name":"the Academician Innovation Platform of Hainan Province","award":["12075159"],"award-info":[{"award-number":["12075159"]}]},{"name":"the Academician Innovation Platform of Hainan Province","award":["12171044"],"award-info":[{"award-number":["12171044"]}]},{"name":"the Academician Innovation Platform of Hainan Province","award":["Z190005"],"award-info":[{"award-number":["Z190005"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Noise exists inherently in realistic quantum systems and affects the evolution of quantum systems. We investigate the dynamics of quantum networks in noisy environments by using the fidelity of the quantum evolved states and the classical percolation theory. We propose an analytical framework that allows us to characterize the stability of quantum networks in terms of quantum noises and network topologies. The calculation results of the framework determine the maximal time that quantum networks with different network topologies can maintain the ability to communicate under noise. We demonstrate the results of the framework through examples of specific graphs under amplitude damping and phase damping noises. We further consider the capacity of the quantum network in a noisy environment according to the proposed framework. The analytical framework helps us better understand the evolution time of a quantum network and provides a reference for designing large quantum networks.<\/jats:p>","DOI":"10.3390\/e25010157","type":"journal-article","created":{"date-parts":[[2023,1,13]],"date-time":"2023-01-13T02:05:52Z","timestamp":1673575552000},"page":"157","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Dynamics of Quantum Networks in Noisy Environments"],"prefix":"10.3390","volume":"25","author":[{"given":"Chang-Yue","family":"Zhang","sequence":"first","affiliation":[{"name":"Department of Mathematics, South China University of Technology, Guangzhou 510641, China"}]},{"given":"Zhu-Jun","family":"Zheng","sequence":"additional","affiliation":[{"name":"Department of Mathematics, South China University of Technology, Guangzhou 510641, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2412-8626","authenticated-orcid":false,"given":"Shao-Ming","family":"Fei","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Capital Normal University, Beijing 100048, China"},{"name":"Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8652-6086","authenticated-orcid":false,"given":"Mang","family":"Feng","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics Innovation Academy of Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430071, China"},{"name":"Research Center for Quantum Precision Measurement, Institute of Industry Technology, Guangzhou and Chinese Academy of Sciences, Guangzhou 511458, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,12]]},"reference":[{"key":"ref_1","unstructured":"Bennett, C.H., and Brassard, G. (1984, January 10\u201312). Quantum Cryptography: Public Key Distribution and Coin Tossing. Proceedings of the IEEE International Conference on Computers. Systems and Signal Processing, Bangalore, India."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1900005","DOI":"10.1002\/qute.201900005","article-title":"Implementation security certification of decoy-BB84 quantum key distribution systems","volume":"2","author":"Tomita","year":"2019","journal-title":"Adv. Quantum Technol."},{"key":"ref_3","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_4","first-page":"031043","article-title":"Phase-matching quantum key distribution","volume":"8","author":"Ma","year":"2018","journal-title":"Phys. Rev. X"},{"key":"ref_5","unstructured":"Elliott1, C., Colvin, A., Pearson, D., Pikalo, O., Schlafer, J., and Yeh, H. (2005). Current status of the DARPA quantum network. Proc. SPIE Int. Soc. Opt. Eng., 5815, 138\u2013149."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"10387","DOI":"10.1364\/OE.19.010387","article-title":"Field test of quantum key distribution in the Tokyo QKD Network","volume":"19","author":"Sasaki","year":"2011","journal-title":"Opt. Express"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"030501","DOI":"10.1103\/PhysRevLett.120.030501","article-title":"Satellite-relayed intercontinental quantum network","volume":"120","author":"Liao","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","unstructured":"Nielsen, M.A., and Chuang, I.L. (2000). Quantum Computation and Quantum Information, Cambridge University Press."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"12909","DOI":"10.1038\/s41598-020-68498-x","article-title":"Dynamics of entangled networks of the quantum Internet","volume":"10","author":"Gyongyosi","year":"2020","journal-title":"Sci. Rep."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"012407","DOI":"10.1103\/PhysRevA.103.012407","article-title":"Quantum networking with short-range entanglement assistance","volume":"103","author":"Santra","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1895","DOI":"10.1103\/PhysRevLett.70.1895","article-title":"Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels","volume":"70","author":"Bennett","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1038\/nphys549","article-title":"Entanglement percolation in quantum network","volume":"3","author":"Cirac","year":"2007","journal-title":"Nat. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Grimmett, G.R. (1999). Percolation, Springer.","DOI":"10.1007\/978-3-662-03981-6"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"240503","DOI":"10.1103\/PhysRevLett.103.240503","article-title":"Entanglement percolation in quantum complex networks","volume":"103","author":"Cuquet","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"032327","DOI":"10.1103\/PhysRevA.81.032327","article-title":"Multipartite entanglement percolation","volume":"81","author":"Perseguers","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"032319","DOI":"10.1103\/PhysRevA.83.032319","article-title":"Limited-path-length entanglement percolation in quantum complex networks","volume":"83","author":"Cuquet","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"052315","DOI":"10.1103\/PhysRevA.101.052315","article-title":"Structural vulnerability of quantum networks","volume":"101","author":"Wu","year":"2020","journal-title":"Phys. Rev. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"722","DOI":"10.1103\/PhysRevLett.76.722","article-title":"Purification of noisy entanglement and faithful teleportation via noisy channel","volume":"76","author":"Bennett","year":"1996","journal-title":"Phys. Rev. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"012338","DOI":"10.1103\/PhysRevA.92.012338","article-title":"Fighting noise with noise in realistic quantum teleportation","volume":"92","author":"Fortes","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"042332","DOI":"10.1103\/PhysRevA.90.042332","article-title":"Noisy quantum teleportation: An experimental study on the influence of local environments","volume":"90","author":"Knoll","year":"2014","journal-title":"Phys. Rev. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"012312","DOI":"10.1103\/PhysRevA.78.012312","article-title":"Greenberger-Horne-Zeilinger versus W states: Quantum teleportation through noisy channels","volume":"78","author":"Jung","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"022316","DOI":"10.1103\/PhysRevA.66.022316","article-title":"Fidelity of quantum teleportation through noisy channels","volume":"66","author":"Oh","year":"2002","journal-title":"Phys. Rev. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"022321","DOI":"10.1103\/PhysRevA.78.022321","article-title":"Efficient quantum key distribution over a collective noise channel","volume":"78","author":"Li","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1007\/s11128-021-03080-6","article-title":"Entanglement-based quantum key distribution with untrusted third party","volume":"20","author":"Zhang","year":"2021","journal-title":"Quantum Inf. Process."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"19201","DOI":"10.1038\/srep19201","article-title":"Long-distance continuous-variable quantum key distribution by controlling excess noise","volume":"6","author":"Huang","year":"2016","journal-title":"Sci. Rep."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"306","DOI":"10.1007\/s11128-017-1759-8","article-title":"Effect of quantum noise on deterministic remote state preparation of an arbitrary two-particle state via various quantum entangled channels","volume":"16","author":"Qu","year":"2017","journal-title":"Quantum Inf. Process."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1007\/s11128-018-1917-7","article-title":"Cyclic joint remote state preparation in noisy environment","volume":"17","author":"Zhang","year":"2018","journal-title":"Quantum Inf. Process."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Passian, A., and Imam, N. (2019). Nanosystems, Edge Computing, and the Next Generation Computing Systems. Sensors, 19.","DOI":"10.3390\/s19184048"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"4287","DOI":"10.1103\/PhysRevLett.71.4287","article-title":"\u2018Event-ready-detectors\u2019 Bell experiment via entanglement swapping","volume":"71","author":"Zukowski","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1098","DOI":"10.1038\/nature07241","article-title":"Experimental demonstration of a BDCZ quantum repeater node","volume":"454","author":"Yuan","year":"2008","journal-title":"Nature"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1038\/nature12016","article-title":"Heralded entanglement between solid-state qubits separated by three metres","volume":"497","author":"Bernien","year":"2013","journal-title":"Nature"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1038\/nature06118","article-title":"Entanglement of single-atom quantum bits at a distance","volume":"449","author":"Moehring","year":"2007","journal-title":"Nature"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Stauffer, D., and Aharony, A. (2018). Introduction To Percolation Theory, Taylor & Francis.","DOI":"10.1201\/9781315274386"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1182","DOI":"10.1214\/aoap\/1019487612","article-title":"Mixed Percolation as a bridge between site and bond percoaltion","volume":"10","author":"Chayes","year":"2000","journal-title":"Ann. Appl. Probab."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1016\/0021-9797(90)90267-R","article-title":"The Mixed Bond-Site Percolation Problem and Its Application to Capillary Phenomena in Porous Media","volume":"134","author":"Yanuka","year":"1990","journal-title":"J. Colloid Interface Sci."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"014130","DOI":"10.1103\/PhysRevE.104.014130","article-title":"Site-bond percolation in two-dimensional kagome lattices: Analytical approach and numerical simulations","volume":"104","author":"Torres","year":"2021","journal-title":"Phys. Rev. E"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1007\/BF01608499","article-title":"On the generators of quantum dynamical semigroups","volume":"48","author":"Lindblad","year":"1976","journal-title":"Commun. Math. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1016\/0034-4877(76)90060-4","article-title":"The \u2018transition probabilit\u2019 in the state space of a \u2217-algebra","volume":"9","author":"Uhlmann","year":"1976","journal-title":"Rep. Math. Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1007\/s12043-020-02027-3","article-title":"Using a Lindbladian approach to model decoherence in two coupled nuclear spins via correlated phase damping and amplitude damping noise channels","volume":"94","author":"Dorai","year":"2020","journal-title":"Pramana J. Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"388","DOI":"10.1109\/18.825799","article-title":"The capacity of wireless networks","volume":"46","author":"Gupta","year":"2000","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"012316","DOI":"10.1103\/PhysRevA.98.012316","article-title":"Capacity of a continuously distributed quantum network","volume":"98","author":"Zhou","year":"2018","journal-title":"Phys. Rev. A"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/157\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:04:38Z","timestamp":1760119478000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/157"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,12]]},"references-count":41,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["e25010157"],"URL":"https:\/\/doi.org\/10.3390\/e25010157","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,12]]}}}