{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T15:16:41Z","timestamp":1783696601538,"version":"3.55.0"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T00:00:00Z","timestamp":1775692800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T00:00:00Z","timestamp":1775692800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"the National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12371315"],"award-info":[{"award-number":["12371315"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"the Natural Science Foundation of Hunan Province of China","award":["2021JJ40662"],"award-info":[{"award-number":["2021JJ40662"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Quantum Inf Process"],"DOI":"10.1007\/s11128-026-05160-x","type":"journal-article","created":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T13:56:16Z","timestamp":1775742976000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A discussion on the well-definetness of a geometric measure for quantum mixed states"],"prefix":"10.1007","volume":"25","author":[{"given":"Mengyu","family":"He","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Guyan","family":"Ni","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ying","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mengshi","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,4,9]]},"reference":[{"key":"5160_CR1","doi-asserted-by":"publisher","first-page":"675","DOI":"10.1111\/j.1749-6632.1995.tb39008.x","volume":"755","author":"A Shimony","year":"1995","unstructured":"Shimony, A.: Degree of entanglement$$^a$$. Ann. N. Y. Acad. Sci. 755, 675 (1995)","journal-title":"Ann. N. Y. Acad. Sci."},{"key":"5160_CR2","doi-asserted-by":"publisher","first-page":"6787","DOI":"10.1088\/0305-4470\/34\/35\/305","volume":"34","author":"H Barnum","year":"2001","unstructured":"Barnum, H., Linden, N.: Monotones and invariants for multi-particle quantumstates. J. Phys. A: Math. Gen. 34, 6787 (2001)","journal-title":"J. Phys. A: Math. Gen."},{"key":"5160_CR3","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.68.042307","volume":"68","author":"T-C Wei","year":"2003","unstructured":"Wei, T.-C., Goldbart, P.M.: Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys. Rev. A 68, 042307 (2003)","journal-title":"Phys. Rev. A"},{"key":"5160_CR4","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.108.052217","volume":"108","author":"B Yang","year":"2023","unstructured":"Yang, B., Li, Y., Ni, G.: Separability discrimination and decomposition of quantum mixed states based on the Broyden-Fletcher-Goldfarb-Shanno algorithm. Phys. Rev. A 108, 052217 (2023)","journal-title":"Phys. Rev. A"},{"key":"5160_CR5","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-49725-8","volume-title":"A Group Theoretic Approach to Quantum Information","author":"M Hayashi","year":"2017","unstructured":"Hayashi, M.: A Group Theoretic Approach to Quantum Information. Springer International Publishing, Cham (2017)"},{"key":"5160_CR6","doi-asserted-by":"crossref","unstructured":"Mu$$\\tilde{n}$$oz-Moller, A.D., Pereira, L., Zambrano, L., Cort$$\\acute{e}$$s-Vega, J., Delgado, A.: Variational determination of multiqubit geometrical entanglement in noisy intermediate-scale quantum computers. Phys. Rev. Appl. 18, 024048 (2022)","DOI":"10.1103\/PhysRevApplied.18.024048"},{"key":"5160_CR7","doi-asserted-by":"crossref","unstructured":"Zambrano, L., Mu$$\\tilde{n}$$oz-Moller, A.D., Mu$$\\tilde{n}$$oz, M., Pereira, L., Delgado, A.: Avoiding barren plateaus in the variational determination of geometric entanglement. Quantum Sci. Tech. 9, 025016 (2024)","DOI":"10.1088\/2058-9565\/ad2a16"},{"key":"5160_CR8","doi-asserted-by":"publisher","first-page":"208","DOI":"10.21468\/SciPostPhys.15.5.208","volume":"15","author":"J Denis","year":"2023","unstructured":"Denis, J., Damanet, F., Martin, J.: Estimation of the geometric measure of entanglement with Wehrl moments through artificial neural networks. SciPost Phys. 15, 208 (2023)","journal-title":"SciPost Phys."},{"key":"5160_CR9","volume":"15","author":"Z Li","year":"2025","unstructured":"Li, Z., Lee, D., Yoshida, B.: How much entanglement is needed for topological codes and mixed states with anomalous symmetry? Phys. Rev. X 15, 021090 (2025)","journal-title":"Phys. Rev. X"},{"key":"5160_CR10","doi-asserted-by":"publisher","DOI":"10.1103\/4ct9-2xhs","volume":"112","author":"D Dilley","year":"2025","unstructured":"Dilley, D., Chang, J., Larson, J., Chitambar, E.: Exploring non-multiplicativity in the geometric measure of entanglement. Phys. Rev. A 112, 032425 (2025)","journal-title":"Phys. Rev. A"},{"key":"5160_CR11","volume":"15","author":"LA Lessa","year":"2025","unstructured":"Lessa, L.A., Cheng, M., Wang, C.: Mixed-state quantum anomaly and multipartite entanglement. Phys. Rev. X 15, 011069 (2025)","journal-title":"Phys. Rev. X"},{"key":"5160_CR12","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.92.042329","volume":"92","author":"S Szalay","year":"2015","unstructured":"Szalay, S.: Multipartite entanglement measures. Phys. Rev. A 92, 042329 (2015)","journal-title":"Phys. Rev. A"},{"key":"5160_CR13","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.134.210602","volume":"134","author":"S Bravyi","year":"2025","unstructured":"Bravyi, S., Lee, D., Li, Z., Yoshida, B.: How much entanglement is needed for quantum error correction? Phys. Rev. Lett. 134, 210602 (2025)","journal-title":"Phys. Rev. Lett."},{"key":"5160_CR14","doi-asserted-by":"publisher","DOI":"10.1063\/1.3451264","volume":"51","author":"J Hilling","year":"2010","unstructured":"Hilling, J., Sudbery, A.: The geometric measure of multipartite entanglement and the singular values of a hypermatrix. J. Math. Phys. 51, 072102 (2010)","journal-title":"J. Math. Phys."},{"key":"5160_CR15","doi-asserted-by":"publisher","DOI":"10.1063\/1.3271041","volume":"50","author":"M Hayashi","year":"2009","unstructured":"Hayashi, M., Markham, D., Murao, M., Owari, M., Virmani, S.: The geometric measure of entanglement for a symmetric pure state with non-negative amplitudes. J. Math. Phys. 50, 122104 (2009)","journal-title":"J. Math. Phys."},{"key":"5160_CR16","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.93.012304","volume":"93","author":"S Hu","year":"2016","unstructured":"Hu, S., Qi, L., Zhang, G.: Computing the geometric measure of entanglement of multipartite pure states by means of non-negative tensors. Phys. Rev. A 93, 012304 (2016)","journal-title":"Phys. Rev. A"},{"key":"5160_CR17","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1137\/120892891","volume":"35","author":"G Ni","year":"2014","unstructured":"Ni, G., Qi, L., Bai, M.: Geometric measure of entanglement and U-eigenvalues of tensors. SIAM J. Matrix Anal. Appl. 35, 73 (2014)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"5160_CR18","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1007\/s11128-017-1633-8","volume":"16","author":"P Teng","year":"2017","unstructured":"Teng, P.: Accurate calculation of the geometric measure of entanglement for multipartite quantum states. Quantum Inf. Process. 16, 181 (2017)","journal-title":"Quantum Inf. Process."},{"key":"5160_CR19","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1016\/j.neucom.2018.05.094","volume":"313","author":"M Che","year":"2018","unstructured":"Che, M., Qi, L., Wei, Y., Zhang, G.: Geometric measures of entanglement in multipartite pure states via complex-valued neural networks. Neurocomputing 313, 25 (2018)","journal-title":"Neurocomputing"},{"issue":"22","key":"5160_CR20","doi-asserted-by":"publisher","first-page":"1465","DOI":"10.1016\/j.physleta.2018.04.007","volume":"382","author":"L Qi","year":"2018","unstructured":"Qi, L., Zhang, G., Ni, G.: How entangled can a multi-party system possibly be? Phys. Lett. A 382(22), 1465\u20131471 (2018)","journal-title":"Phys. Lett. A"},{"key":"5160_CR21","doi-asserted-by":"crossref","unstructured":"Bai, M., Yan, S., Zeng, Q.: Extremal geometric measure of entanglement and Riemannian optimization methods. J. Oper. Res. Soc. China, 1-23 (2023)","DOI":"10.1007\/s40305-023-00504-1"},{"key":"5160_CR22","unstructured":"Semenov, A., Murphy, N., Patscheider, S., Bernardi, A., Blokhina, E.: A quantum implementation of high-order power method for estimating geometric entanglement of pure states, arxiv:2405.19134 (2024)"},{"key":"5160_CR23","doi-asserted-by":"publisher","DOI":"10.1007\/s11432-022-3681-2","volume":"66","author":"Y Guo","year":"2023","unstructured":"Guo, Y., Lin, L., Cao, H., et al.: Experimental entanglement quantification for unknown quantum states in a semi-device-independent manner. Sci. China Inf. Sci. 66, 180506 (2023)","journal-title":"Sci. China Inf. Sci."},{"key":"5160_CR24","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.110.012452","volume":"110","author":"X Zhu","year":"2024","unstructured":"Zhu, X., Zhang, C., Zeng, B.: Quantifying subspace entanglement with geometric measures. Phys. Rev. A 110, 012452 (2024)","journal-title":"Phys. Rev. A"},{"key":"5160_CR25","doi-asserted-by":"publisher","first-page":"306","DOI":"10.1007\/s11128-024-04501-y","volume":"23","author":"H Zhao","year":"2024","unstructured":"Zhao, H., Ma, P., Fei, S.: Geometric genuine N-partite entanglement measure for arbitrary dimensions. Quantum Inf. Process. 23, 306 (2024)","journal-title":"Quantum Inf. Process."},{"key":"5160_CR26","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.102.012402","volume":"102","author":"Y Li","year":"2020","unstructured":"Li, Y., Ni, G.: Separability discrimination and decomposition of m-partite quantum mixed states. Phys. Rev. A 102, 012402 (2020)","journal-title":"Phys. Rev. A"},{"key":"5160_CR27","doi-asserted-by":"crossref","unstructured":"H$$\\ddot{u}$$bener, R., Kleinmann, M., Wei, T.-C., Guill$$\\acute{e}$$n, C.G., Gonzalez-Guillen, C., G$$\\ddot{u}$$hne, O.: Geometric measure of entanglement for symmetric states. Phys. Rev. A 80, 032324 (2009)","DOI":"10.1103\/PhysRevA.80.032324"},{"key":"5160_CR28","volume-title":"Complex Tensor Optimization and Its Applications in Quantum Information","author":"G Ni","year":"2022","unstructured":"Ni, G., Li, Y., Zhang, M.: Complex Tensor Optimization and Its Applications in Quantum Information. Science Press, Beijing (2022)"},{"key":"5160_CR29","doi-asserted-by":"publisher","first-page":"1115","DOI":"10.1137\/19M1306889","volume":"41","author":"J Nie","year":"2020","unstructured":"Nie, J., Yang, Z.: Hermitian tensor decompositions. SIAM J. on Matrix Anal. Appl. 41, 1115 (2020)","journal-title":"SIAM J. on Matrix Anal. Appl."},{"key":"5160_CR30","doi-asserted-by":"publisher","first-page":"6581","DOI":"10.1080\/03081087.2021.1965078","volume":"70","author":"M Dressler","year":"2022","unstructured":"Dressler, M., Nie, J., Yang, Z.: Separability of Hermitian tensors and PSD decompositions. Linear Multilinear Algebra 70, 6581 (2022)","journal-title":"Linear Multilinear Algebra"},{"key":"5160_CR31","doi-asserted-by":"publisher","first-page":"961","DOI":"10.1007\/s11464-021-0927-4","volume":"17","author":"G Ni","year":"2022","unstructured":"Ni, G., Yang, B.: Symmetric Hermitian decomposability criterion, decomposition, and its applications. Front. Math. China 17, 961 (2022)","journal-title":"Front. Math. China"},{"key":"5160_CR32","doi-asserted-by":"publisher","first-page":"1084","DOI":"10.1137\/06066518X","volume":"30","author":"V De Silva","year":"2008","unstructured":"De Silva, V., Lim, L.H.: Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 30, 1084 (2008)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"3","key":"5160_CR33","doi-asserted-by":"publisher","first-page":"1619","DOI":"10.1103\/PhysRevA.57.1619","volume":"57","author":"V Vedral","year":"1998","unstructured":"Vedral, V., Plenio, M.B.: Entanglement measures and purification procedures. Phys. Rev. A 57(3), 1619 (1998)","journal-title":"Phys. Rev. A"},{"key":"5160_CR34","doi-asserted-by":"publisher","first-page":"4206","DOI":"10.1103\/PhysRevA.59.4206","volume":"59","author":"M Horodecki","year":"1999","unstructured":"Horodecki, M., Horodecki, P.: Reduction criterion of separability and limits for a class of distillation protocols. Phys. Rev. A 59, 4206 (1999)","journal-title":"Phys. Rev. A"}],"container-title":["Quantum Information Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-026-05160-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11128-026-05160-x","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-026-05160-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T14:56:10Z","timestamp":1783695370000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11128-026-05160-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,9]]},"references-count":34,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2026,5]]}},"alternative-id":["5160"],"URL":"https:\/\/doi.org\/10.1007\/s11128-026-05160-x","relation":{},"ISSN":["1573-1332"],"issn-type":[{"value":"1573-1332","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,9]]},"assertion":[{"value":"16 November 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 March 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 April 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"139"}}