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increasingly important to understand their large-scale features from the viewpoint of statistical physics. This review paper discusses a fundamental question: how can entanglement be effectively and indirectly (e.g., through intermediate nodes) distributed between distant nodes in an imperfect quantum network, where the connections are only partially entangled and subject to quantum noise? We survey recent studies addressing this issue by drawing exact or approximate mappings to percolation theory, a branch of statistical physics centered on network connectivity. Notably, we show that the classical percolation frameworks do not uniquely define the network\u2019s indirect connectivity. This realization leads to the emergence of an alternative theory called \u201cconcurrence percolation\u201d, which uncovers a previously unrecognized quantum advantage that emerges at large scales, suggesting that quantum networks are more resilient than initially assumed within classical percolation contexts, offering refreshing insights into future quantum network design.<\/jats:p>","DOI":"10.3390\/e25111564","type":"journal-article","created":{"date-parts":[[2023,11,20]],"date-time":"2023-11-20T10:32:32Z","timestamp":1700476352000},"page":"1564","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Percolation Theories for Quantum Networks"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5184-7648","authenticated-orcid":false,"given":"Xiangyi","family":"Meng","sequence":"first","affiliation":[{"name":"Network Science Institute, Northeastern University, Boston, MA 02115, USA"},{"name":"Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-5018-1934","authenticated-orcid":false,"given":"Xinqi","family":"Hu","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3872-3971","authenticated-orcid":false,"given":"Yu","family":"Tian","sequence":"additional","affiliation":[{"name":"Nordita, KTH Royal Institute of Technology and Stockholm University, SE-106 91 Stockholm, Sweden"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gaogao","family":"Dong","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0583-4595","authenticated-orcid":false,"given":"Renaud","family":"Lambiotte","sequence":"additional","affiliation":[{"name":"Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK"},{"name":"Turing Institute, London NW1 2DB, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3952-208X","authenticated-orcid":false,"given":"Jianxi","family":"Gao","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Rensselaer Polytechnic Institute, Troy, NY 12180, USA"},{"name":"Network Science and Technology Center, Rensselaer Polytechnic Institute, Troy, NY 12180, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shlomo","family":"Havlin","sequence":"additional","affiliation":[{"name":"Department of Physics, Bar-Ilan University, Ramat Gan 52900, Israel"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,20]]},"reference":[{"key":"ref_1","unstructured":"Nielsen, M.A., and Chuang, I.L. 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