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This exponential complexity na\u00efvely renders a complete state characterization, for instance via the complete set of bipartite Renyi entropies for all disjoint regions, a challenging task. Recently, a compact way of storing subregions&amp;apos; purities by encoding them as amplitudes of a fictitious quantum wave function, known as entanglement feature, was proposed. Notably, the entanglement feature can be a simple object even for highly entangled quantum states. However the complexity and practical usage of the entanglement feature for general quantum states has not been explored. In this work, we demonstrate that the entanglement feature can be efficiently learned using only a polynomial amount of samples in the number of degrees of freedom through the so-called tensor cross interpolation (TCI) algorithm, assuming it is expressible as a finite bond dimension MPS. We benchmark this learning process on Haar and random MPS states, confirming analytic expectations. Applying the TCI algorithm to quantum eigenstates of various one dimensional quantum systems, we identify cases where eigenstates have entanglement feature learnable with TCI. We conclude with possible applications of the learned entanglement feature, such as quantifying the distance between different entanglement patterns and finding the optimal one-dimensional ordering of physical indices in a given state, highlighting the potential utility of the proposed purity interpolation method.<\/jats:p>","DOI":"10.22331\/q-2026-05-22-2114","type":"journal-article","created":{"date-parts":[[2026,5,22]],"date-time":"2026-05-22T11:45:48Z","timestamp":1779450348000},"page":"2114","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":["Tensor Cross Interpolation of Purities in Quantum Many-Body Systems"],"prefix":"10.22331","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8612-8202","authenticated-orcid":false,"given":"Dmytro","family":"Kolisnyk","sequence":"first","affiliation":[{"name":"Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5383-2869","authenticated-orcid":false,"given":"Raimel A.","family":"Medina","sequence":"additional","affiliation":[{"name":"Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4636-4139","authenticated-orcid":false,"given":"Romain","family":"Vasseur","sequence":"additional","affiliation":[{"name":"Department of Theoretical Physics, University of Geneva, 24 quai Ernest-Ansermet, 1211 Geneva, Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2399-5827","authenticated-orcid":false,"given":"Maksym","family":"Serbyn","sequence":"additional","affiliation":[{"name":"Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2026,5,22]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"A. Browaeys and T. Lahaye, Many-body physics with individually controlled rydberg atoms, Nat. Phys. 16, 132 (2020).","DOI":"10.1038\/s41567-019-0733-z"},{"key":"1","doi-asserted-by":"publisher","unstructured":"R. Blatt and C. F. Roos, Quantum simulations with trapped ions, Nat. Phys. 8, 277 (2012).","DOI":"10.1038\/nphys2252"},{"key":"2","doi-asserted-by":"publisher","unstructured":"C. Gross and I. Bloch, Quantum simulations with ultracold atoms in optical lattices, Science 357, 995 (2017).","DOI":"10.1126\/science.aal3837"},{"key":"3","doi-asserted-by":"publisher","unstructured":"C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V. Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano, P. Richerme, C. Senko, and N. Y. Yao, Programmable quantum simulations of spin systems with trapped ions, Rev. Mod. Phys. 93, 025001 (2021).","DOI":"10.1103\/RevModPhys.93.025001"},{"key":"4","doi-asserted-by":"publisher","unstructured":"I. Bloch, J. Dalibard, and W. 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