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Its certification is often difficult, and most widely used methods for experiments are based on fidelity measurements with respect to highly entangled states. Here, instead, we consider covariances of collective observables, as in the well-known Covariance Matrix Criterion (CMC) \\cite{guhnecova} and present a generalization of the CMC for determining the Schmidt number of a bipartite system. This is potentially particularly advantageous in many-body systems, such as cold atoms, where the set of practical measurements is very limited and only variances of collective operators can typically be estimated. To show the practical relevance of our results, we derive simpler Schmidt-number criteria that require similar information as the fidelity-based witnesses, yet can detect a wider set of states. We also consider paradigmatic criteria based on spin covariances, which would be very helpful for experimental detection of high-dimensional entanglement in cold atom systems. We conclude by discussing the applicability of our results to a multiparticle ensemble and some open questions for future work.<\/jats:p>","DOI":"10.22331\/q-2024-01-30-1236","type":"journal-article","created":{"date-parts":[[2024,1,30]],"date-time":"2024-01-30T11:12:58Z","timestamp":1706613178000},"page":"1236","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":17,"title":["Bounding entanglement dimensionality from the covariance matrix"],"prefix":"10.22331","volume":"8","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, & Collaborative Innovation Center of Quantum Matter, Peking University, Beijing 100871, China"},{"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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3653-0030","authenticated-orcid":false,"given":"Matteo","family":"Fadel","sequence":"additional","affiliation":[{"name":"Department of Physics, ETH Z\u00fcrich, 8093 Z\u00fcrich, Switzerland"}]},{"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, & Collaborative Innovation Center of Quantum Matter, 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"}]},{"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"}]},{"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"},{"name":"Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, 1090 Vienna, Austria"}]}],"member":"9598","published-online":{"date-parts":[[2024,1,30]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"O. 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