{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:54:57Z","timestamp":1760151297701,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,3,6]],"date-time":"2022-03-06T00:00:00Z","timestamp":1646524800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we present a new method for the construction of maximally entangled states in Cd\u2297Cd\u2032 when d\u2032\u22652d. A systematic way of constructing a set of maximally entangled bases (MEBs) in Cd\u2297Cd\u2032 was established. Both cases when d\u2032 is divisible by d and not divisible by d are discussed. We give two examples of maximally entangled bases in C2\u2297C4, which are mutually unbiased bases. Finally, we found a new example of an unextendible maximally entangled basis (UMEB) in C2\u2297C5.<\/jats:p>","DOI":"10.3390\/e24030373","type":"journal-article","created":{"date-parts":[[2022,3,6]],"date-time":"2022-03-06T20:35:50Z","timestamp":1646598950000},"page":"373","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Construction of a Family of Maximally Entangled Bases in \u2102d \u2297 \u2102d\u2032"],"prefix":"10.3390","volume":"24","author":[{"given":"Chenghong","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematics, South China University of Technology, Guangzhou 510641, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6664-0455","authenticated-orcid":false,"given":"Kun","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, Zhoukou Normal University, Zhoukou 466001, China"}]},{"given":"Zhu-Jun","family":"Zheng","sequence":"additional","affiliation":[{"name":"School of Mathematics, South China University of Technology, Guangzhou 510641, China"},{"name":"Laboratory of Quantum Science and Engineering, South China University of Technology, Guangzhou 510641, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"865","DOI":"10.1103\/RevModPhys.81.865","article-title":"Quantum entanglement","volume":"81","author":"Horodecki","year":"2009","journal-title":"Rev. Mod. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"e45","DOI":"10.1002\/que2.45","article-title":"Mermin\u2019s inequalities of multiple qubits with orthogonal measurements on IBM Q 53-qubit system","volume":"2","author":"Huang","year":"2020","journal-title":"Quantum Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"032320","DOI":"10.1103\/PhysRevA.72.032320","article-title":"Security bound of two-basis quantum-key-distribution protocols using qudits","volume":"72","author":"Nikolopoulos","year":"2005","journal-title":"Phys. Rev. A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"032305","DOI":"10.1103\/PhysRevA.88.032305","article-title":"Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases","volume":"88","author":"Mafu","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1016\/0003-4916(89)90322-9","article-title":"Optimal state-determination by mutually unbiased measurements","volume":"191","author":"Wootters","year":"1989","journal-title":"Ann. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"052332","DOI":"10.1103\/PhysRevA.83.052332","article-title":"Quantum process reconstruction based on mutually unbiased basis","volume":"83","author":"Klimov","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Nielsen, M.A., and Chuang, I.L. (2002). Quantum Computation and Quantum Information, Cambridge University Press.","DOI":"10.1119\/1.1463744"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2009.02.004","article-title":"Entanglement detection","volume":"474","author":"Guhne","year":"2009","journal-title":"Phys. Rep."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"012301","DOI":"10.1103\/PhysRevA.66.012301","article-title":"Optimal teleportation based on Bell measurements","volume":"66","author":"Albeverio","year":"2002","journal-title":"Phys. Rev. A"},{"key":"ref_10","first-page":"398","article-title":"Teleportation via maximally and non-maximally entangled mixed states","volume":"10","author":"Adhikari","year":"2010","journal-title":"Quantum Inf. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"570","DOI":"10.1073\/pnas.46.4.570","article-title":"Unitary operator bases","volume":"46","author":"Schwinger","year":"1960","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3241","DOI":"10.1088\/0305-4470\/14\/12\/019","article-title":"Geometrical description of quantal state determination","volume":"14","author":"Ivanovic","year":"1981","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"512","DOI":"10.1007\/s00453-002-0980-7","article-title":"A new proof for the existence of mutually unbiased bases","volume":"34","author":"Bandyopadhyay","year":"2002","journal-title":"Algorithmica"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"5267","DOI":"10.1088\/0305-4470\/38\/23\/013","article-title":"About mutually unbiased bases in even and odd prime power dimensions","volume":"38","author":"Durt","year":"2005","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_15","first-page":"950","article-title":"Deterministic quantum distribution of a d-ary key","volume":"9","author":"Eusebi","year":"2009","journal-title":"Quantum Inf. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Mcconnell, G., Spencer, H., and Tahir, A. (2021). Evidence for and against Zauner\u2019s MUB conjecture in \u21026. arXiv.","DOI":"10.26421\/QIC21.9-10-1"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"042306","DOI":"10.1103\/PhysRevA.84.042306","article-title":"Unextendible maximally entangled bases","volume":"84","author":"Bravyi","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"034301","DOI":"10.1103\/PhysRevA.88.034301","article-title":"Unextendible maximally entangled bases and mutually unbiased bases","volume":"72","author":"Chen","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"927","DOI":"10.1007\/s10773-014-2288-1","article-title":"Unextendible maximally entangled bases and mutually unbiased bases in \u2102d\u2297\u2102d\u2032","volume":"54","author":"Nan","year":"2015","journal-title":"Int. J. Theor. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2291","DOI":"10.1007\/s11128-015-0980-6","article-title":"Mutually unbiased maximally entangled bases in \u2102d\u2297\u2102kd","volume":"14","author":"Tao","year":"2015","journal-title":"Quantum Inf. Process."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1007\/s11128-018-2094-4","article-title":"Unextendible maximally entangled bases in \u2102pd\u2297\u2102qd","volume":"17","author":"Zhang","year":"2018","journal-title":"Quantum Inf. Process."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1007\/s11128-018-1824-y","article-title":"Mutually unbiased special entanled bases with Schmidt number 2 in \u21023\u2297\u21024k","volume":"17","author":"Han","year":"2018","journal-title":"Quantum Inf. Process."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2150019","DOI":"10.1142\/S0217984921500196","article-title":"Mutually unbiased unextendible maximally entangled bases in \u2102d\u2297\u2102q(d+1)","volume":"34","author":"Tang","year":"2020","journal-title":"Mod. Phys. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/s11128-020-02670-0","article-title":"Construction of mutually unbiased maximally entangled bases in \u21022s\u2297\u21022s by using Galois rings","volume":"19","author":"Xu","year":"2020","journal-title":"Quantum Inf. Process."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"483001","DOI":"10.1088\/1751-8113\/47\/48\/483001","article-title":"Entanglement witness: Construction, analysis and classification","volume":"47","author":"Sarbicki","year":"2014","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3553","DOI":"10.1007\/s11128-015-1058-1","article-title":"Multipartite unextendible entangled basis","volume":"14","author":"Guo","year":"2015","journal-title":"Quantum Inf. Process."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"245301","DOI":"10.1088\/1751-8113\/48\/24\/245301","article-title":"Entangled bases with fixed Schmidt number","volume":"48","author":"Guo","year":"2015","journal-title":"J. Phys. A Math. Theor."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/3\/373\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:32:51Z","timestamp":1760135571000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/3\/373"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,6]]},"references-count":27,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,3]]}},"alternative-id":["e24030373"],"URL":"https:\/\/doi.org\/10.3390\/e24030373","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,3,6]]}}}