{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,2]],"date-time":"2026-04-02T00:54:04Z","timestamp":1775091244972,"version":"3.50.1"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"10","license":[{"start":{"date-parts":[[2021,10,1]],"date-time":"2021-10-01T00:00:00Z","timestamp":1633046400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,10,1]],"date-time":"2021-10-01T00:00:00Z","timestamp":1633046400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Quantum Inf Process"],"published-print":{"date-parts":[[2021,10]]},"DOI":"10.1007\/s11128-021-03278-8","type":"journal-article","created":{"date-parts":[[2021,10,22]],"date-time":"2021-10-22T10:06:34Z","timestamp":1634897194000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["$$H_2$$-reducible matrices in six-dimensional mutually unbiased bases"],"prefix":"10.1007","volume":"20","author":[{"given":"Xiaoyu","family":"Chen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9363-9932","authenticated-orcid":false,"given":"Mengfan","family":"Liang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mengyao","family":"Hu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lin","family":"Chen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,10,22]]},"reference":[{"issue":"4","key":"3278_CR1","doi-asserted-by":"publisher","first-page":"570","DOI":"10.1073\/pnas.46.4.570","volume":"46","author":"J Schwinger","year":"1960","unstructured":"Schwinger, J.: Unitary operator bases. Proc. Nat. Acad. Sci. U.S.A. 46(4), 570\u2013579 (1960)","journal-title":"Proc. Nat. Acad. Sci. U.S.A."},{"key":"3278_CR2","doi-asserted-by":"crossref","unstructured":"Amburg, I., Sharma, R., Sussman, D.M., Wootters, W.K.: States that \u201clook the same\u201d with respect to every basis in a mutually unbiased set. J. Math. Phys. 55(12), 122206 (2014)","DOI":"10.1063\/1.4904317"},{"key":"3278_CR3","doi-asserted-by":"crossref","unstructured":"Amburg, I., Sharma, R., Sussman, D.M., Wootters, W.K.: Erratum: \u201cstates that \u2018look the same\u2019 with respect to every basis in a mutually unbiased set\u201d. J. Math. Phys. 56(3), 039901 (2015)","DOI":"10.1063\/1.4916675"},{"key":"3278_CR4","doi-asserted-by":"crossref","unstructured":"Brierley, S.: Mutually unbiased bases in low dimensions. University of York, Department of Mathematics. Ph.D. thesis (2009)","DOI":"10.1103\/PhysRevA.79.052316"},{"key":"3278_CR5","doi-asserted-by":"publisher","first-page":"042312","DOI":"10.1103\/PhysRevA.78.042312","volume":"78","author":"S Brierley","year":"2008","unstructured":"Brierley, S., Weigert, S.: Maximal sets of mutually unbiased quantum states in dimension 6. Phys. Rev. A 78, 042312 (2008)","journal-title":"Phys. Rev. A"},{"key":"3278_CR6","doi-asserted-by":"publisher","first-page":"052316","DOI":"10.1103\/PhysRevA.79.052316","volume":"79","author":"S Brierley","year":"2009","unstructured":"Brierley, S., Weigert, S.: Constructing mutually unbiased bases in dimension six. Phys. Rev. A 79, 052316 (2009)","journal-title":"Phys. Rev. A"},{"key":"3278_CR7","first-page":"021008","volume":"254","author":"S Brierley","year":"2010","unstructured":"Brierley, S., Weigert, S.: Mutually unbiased bases and semi-definite programming. J. Phys. Conf. Ser. 254, 22 (2010)","journal-title":"J. Phys.: Conf. Ser."},{"key":"3278_CR8","doi-asserted-by":"publisher","first-page":"296","DOI":"10.1016\/j.laa.2014.10.017","volume":"466","author":"AS Maxwell","year":"2015","unstructured":"Maxwell, A.S.: Brierley, Stephen: On properties of Karlsson Hadamards and sets of mutually unbiased bases in dimension six. Linear Algebra Appl. 466, 296\u2013306 (2015)","journal-title":"Linear Algebra Appl."},{"issue":"3","key":"3278_CR9","doi-asserted-by":"publisher","first-page":"616","DOI":"10.1112\/jlms\/jdr052","volume":"85","author":"F Sz\u00f6ll\u00f6si","year":"2012","unstructured":"Sz\u00f6ll\u00f6si, F.: Complex hadamard matrices of order 6: a four-parameter family. Journal of the London Mathematical Society 85(3), 616\u2013632 (2012)","journal-title":"J. Lond. Math. Soc."},{"key":"3278_CR10","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1016\/j.laa.2019.01.018","volume":"569","author":"O Turek","year":"2016","unstructured":"Turek, O, Goyeneche D. A generalization of circulant Hadamard and conference matrices. Linear Algebra Appl. 569:241\u2013265 (2016)","journal-title":"Linear Algebra Appl."},{"key":"3278_CR11","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1016\/j.laa.2019.02.016","volume":"571","author":"R Nicoara","year":"2019","unstructured":"Nicoara, R.: Worley, Chase: A finiteness result for circulant core complex Hadamard matrices. Linear Algebra Appl. 571, 143\u2013153 (2019)","journal-title":"Linear Algebra Appl."},{"issue":"5","key":"3278_CR12","doi-asserted-by":"publisher","first-page":"1219","DOI":"10.1016\/j.ejc.2007.06.009","volume":"29","author":"F Sz\u00f6ll\u0151si","year":"2008","unstructured":"Sz\u00f6ll\u0151si, F.: Parametrizing complex hadamard matrices. Eur. J. Combin. 29(5), 1219\u20131234 (2008)","journal-title":"Eur. J. Combin."},{"issue":"10","key":"3278_CR13","doi-asserted-by":"publisher","first-page":"105301","DOI":"10.1088\/1751-8113\/46\/10\/105301","volume":"46","author":"D Goyeneche","year":"2013","unstructured":"Goyeneche, D.: Mutually unbiased triplets from non-affine families of complex hadamard matrices in dimension 6. J. Phys. A: Math. Theor. 46(10), 105301 (2013)","journal-title":"J. Phys. A: Math. Theor."},{"key":"3278_CR14","unstructured":"Boykin, P.O., Sitharam, M., Tiep, P.H., Wocjan, P.: Mutually unbiased bases and orthogonal decompositions of Lie algebras (2005). arXiv:quant-ph\/0506089"},{"issue":"24","key":"3278_CR15","first-page":"245305","volume":"42","author":"P Jaming","year":"2009","unstructured":"Jaming, P., Matolcsi, M., M\u00f3ra, P., Sz\u00f6ll\u00f6si, F., Weiner, M.: A generalized Pauli problem and an infinite family of MUB-triplets in dimension 6. J. Phys. Math. Theor. 42(24), 245305 (2009)","journal-title":"J. Phys.: Math. Theor."},{"issue":"4","key":"3278_CR16","doi-asserted-by":"publisher","first-page":"535","DOI":"10.1142\/S0219749910006502","volume":"8","author":"T Durt","year":"2010","unstructured":"Durt, T., Englert, B.-G., Bengtsson, I., Zyczkowski, K.: On mutually unbiased bases. Int. J. Quantum Information 8(4), 535\u2013640 (2010)","journal-title":"Int. J. Quant. Inform."},{"issue":"5","key":"3278_CR17","doi-asserted-by":"publisher","first-page":"053047","DOI":"10.1088\/1367-2630\/13\/5\/053047","volume":"13","author":"M Wiesniak","year":"2011","unstructured":"Wiesniak, M., Paterek, T., Zeilinger, A.: Entanglement in mutually unbiased bases. New J. Phys. 13(5), 053047 (2011)","journal-title":"New J. Phys."},{"issue":"05","key":"3278_CR18","doi-asserted-by":"publisher","first-page":"1250056","DOI":"10.1142\/S0219749912500566","volume":"10","author":"D Mcnulty","year":"2012","unstructured":"Mcnulty, D., Weigert, S.: On the impossibility to extend triples of mutually unbiased product bases in dimension six. Int. J. Quant. Inform. 10(05), 125 (2012)","journal-title":"Int. J. Quant. Inform."},{"issue":"13","key":"3278_CR19","doi-asserted-by":"publisher","first-page":"135307","DOI":"10.1088\/1751-8113\/45\/13\/135307","volume":"45","author":"D McNulty","year":"2012","unstructured":"McNulty, D.: Weigert, Stefan: All mutually unbiased product bases in dimension 6. J. Phys. A: Math. Theor. 45(13), 135307 (2012)","journal-title":"J. Phys. A: Math. Theor."},{"key":"3278_CR20","doi-asserted-by":"publisher","first-page":"062303","DOI":"10.1103\/PhysRevA.83.062303","volume":"83","author":"P Raynal","year":"2011","unstructured":"Raynal, P, L\u00fc, X, Englert, B-G.: Mutually unbiased bases in six dimensions: The four most distant bases. Phys. Rev. A 83, 33 (2011)","journal-title":"Phys. Rev. A"},{"issue":"10","key":"3278_CR21","doi-asserted-by":"publisher","first-page":"102001","DOI":"10.1088\/1751-8113\/45\/10\/102001","volume":"45","author":"D McNulty","year":"2012","unstructured":"McNulty, D.: Weigert, Stefan: The limited role of mutually unbiased product bases in dimension 6. J. Phys. A: Math. Theor. 45(10), 102001 (2012)","journal-title":"J. Phys. A: Math. Theor."},{"issue":"3","key":"3278_CR22","doi-asserted-by":"publisher","first-page":"032202","DOI":"10.1063\/1.4943301","volume":"57","author":"D McNulty","year":"2016","unstructured":"McNulty D, Pammer B, Weigert S: Mutually unbiased product bases for multiple qudits. J. Math. Phys. 57(3): 32 (2016)","journal-title":"J. Math. Phys."},{"issue":"47","key":"3278_CR23","first-page":"475304","volume":"50","author":"L Chen","year":"2017","unstructured":"Chen, L, Li, Y.: Product states and Schmidt rank of mutually unbiased bases in dimension six. J. Phys. Math. General 50(47), 475304 (2017)","journal-title":"J. Phys. Math. General"},{"issue":"8","key":"3278_CR24","doi-asserted-by":"publisher","first-page":"198","DOI":"10.1007\/s11128-018-1964-0","volume":"17","author":"L Chen","year":"2018","unstructured":"Chen, L, Li, Y.: Mutually unbiased bases in dimension six containing a product-vector basis. Quant. Inform. Process. 17(8), 198 (2018)","journal-title":"Quant. Inform. Process."},{"key":"3278_CR25","doi-asserted-by":"publisher","first-page":"050402","DOI":"10.1103\/PhysRevLett.122.050402","volume":"122","author":"S Designolle","year":"2018","unstructured":"Designolle, S, Skrzypczyk, P, Fr\u00f6wis, F, Brunner, N.: Quantifying measurement incompatibility of mutually unbiased bases. Phys. Rev. Lett. 122: 504 (2018)","journal-title":"Phys. Rev. Lett."},{"issue":"1","key":"3278_CR26","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1016\/j.laa.2010.08.020","volume":"434","author":"BR Karlsson","year":"2011","unstructured":"Karlsson, B.R.: Three-parameter complex hadamard matrices of order 6. Linear Algebra Appl. 434(1), 247\u2013258 (2011)","journal-title":"Linear Algebra Appl."},{"issue":"1","key":"3278_CR27","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1016\/j.laa.2010.08.021","volume":"434","author":"BR Karlsson","year":"2011","unstructured":"Karlsson, B.R.: $${H}_2$$-reducible complex hadamard matrices of order 6. Linear Algebra Appl. 434(1), 239\u2013246 (2011)","journal-title":"Linear Algebra Appl."},{"issue":"11","key":"3278_CR28","doi-asserted-by":"publisher","first-page":"352","DOI":"10.1007\/s11128-019-2467-3","volume":"18","author":"M Liang","year":"2019","unstructured":"Liang, M, Mengyao, H, Chen, L, Chen, X: The $${H}_2$$-reducible matrix in four six-dimensional mutually unbiased bases. Quantum Inform. Process. 18(11), 352 (2019)","journal-title":"Quantum Inform. Process."},{"key":"3278_CR29","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1016\/j.laa.2016.10.028","volume":"514","author":"S De Baerdemacker","year":"2017","unstructured":"De Baerdemacker, S, De Vos, A, Chen, L, Li, Y: The Birkhoff theorem for unitary matrices of arbitrary dimensions. Linear Algebra Appl 514, 151\u2013164 (2017)","journal-title":"Linear Algebra Appl"},{"key":"3278_CR30","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.laa.2017.12.015","volume":"543","author":"L Chen","year":"2018","unstructured":"Chen, L.: Friedland, Shmuel: The tensor rank of tensor product of two three-qubit W states is eight. Linear Algebra Appl. 543, 1\u201316 (2018)","journal-title":"Linear Algebra Appl."},{"key":"3278_CR31","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1080\/03081087.2019.1588849","volume":"69","author":"K Wang","year":"2019","unstructured":"Wang, K, Chen, L, Shen, Y, Sun, Y, Zhao, L-J.: Constructing $$2\\times 2\\times 4$$ and $$4\\times 4$$ unextendible product bases and positive-partial-transpose entangled states. Linear Multilinear Algebra 69:131\u201346 (2019)","journal-title":"Linear Multilinear Algebra"},{"key":"3278_CR32","unstructured":"Beauchamp, K., Nicoara, R.: Orthogonal maximal abelian $$*$$-subalgebras of the $$6\\times 6$$ matrices (2006). arXiv:math\/0609076v1"}],"container-title":["Quantum Information Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-021-03278-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11128-021-03278-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-021-03278-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,10,30]],"date-time":"2021-10-30T08:35:23Z","timestamp":1635582923000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11128-021-03278-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10]]},"references-count":32,"journal-issue":{"issue":"10","published-print":{"date-parts":[[2021,10]]}},"alternative-id":["3278"],"URL":"https:\/\/doi.org\/10.1007\/s11128-021-03278-8","relation":{},"ISSN":["1570-0755","1573-1332"],"issn-type":[{"value":"1570-0755","type":"print"},{"value":"1573-1332","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,10]]},"assertion":[{"value":"2 June 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 September 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 October 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"353"}}