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The task becomes even more complex when the particles have many accessible levels, i.e., they are of high dimension, which leads to a potentially high-dimensional multipartite entangled state. These are important resources for an ever-increasing number of tasks, especially when a network of parties needs to share highly entangled states, e.g., for communicating more efficiently and securely. For these applications, as well as for purely theoretical arguments, it is important to be able to certify both the high-dimensional and the genuine multipartite nature of entangled states, possibly based on simple measurements. Here we derive a novel method that achieves this and improves over typical entanglement witnesses like the fidelity with respect to states of a Greenberger-Horne-Zeilinger (GHZ) form, without needing more complex measurements. We test our condition on paradigmatic classes of high-dimensional multipartite entangled states like imperfect GHZ states with random noise, as well as on purely randomly chosen ones and find that, in comparison with other available criteria our method provides a significant advantage and is often also simpler to evaluate.<\/jats:p>","DOI":"10.22331\/q-2026-02-03-1995","type":"journal-article","created":{"date-parts":[[2026,2,3]],"date-time":"2026-02-03T13:04:51Z","timestamp":1770123891000},"page":"1995","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":["Characterizing high-dimensional multipartite entanglement beyond Greenberger-Horne-Zeilinger fidelities"],"prefix":"10.22331","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7130-1888","authenticated-orcid":false,"given":"Shuheng","family":"Liu","sequence":"first","affiliation":[{"name":"State Key Laboratory for Mesoscopic Physics, School of Physics, Frontiers Science Center for Nano-optoelectronics, Peking University, Beijing 100871, China"},{"name":"Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2408-4320","authenticated-orcid":false,"given":"Qiongyi","family":"He","sequence":"additional","affiliation":[{"name":"State Key Laboratory for Mesoscopic Physics, School of Physics, Frontiers Science Center for Nano-optoelectronics, Peking University, Beijing 100871, China"},{"name":"Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China"},{"name":"Hefei National Laboratory, Hefei 230088, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1985-4623","authenticated-orcid":false,"given":"Marcus","family":"Huber","sequence":"additional","affiliation":[{"name":"Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria"},{"name":"Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, 1090 Vienna, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5563-3222","authenticated-orcid":false,"given":"Giuseppe","family":"Vitagliano","sequence":"additional","affiliation":[{"name":"Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2026,2,3]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Ryszard Horodecki, Pawe\u0142 Horodecki, Micha\u0142 Horodecki, and Karol Horodecki. ``Quantum entanglement''. 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