{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T04:10:35Z","timestamp":1771042235912,"version":"3.50.1"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"7","license":[{"start":{"date-parts":[[2019,5,15]],"date-time":"2019-05-15T00:00:00Z","timestamp":1557878400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,5,15]],"date-time":"2019-05-15T00:00:00Z","timestamp":1557878400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["11871089"],"award-info":[{"award-number":["11871089"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Quantum Inf Process"],"published-print":{"date-parts":[[2019,7]]},"DOI":"10.1007\/s11128-019-2324-4","type":"journal-article","created":{"date-parts":[[2019,5,15]],"date-time":"2019-05-15T23:34:21Z","timestamp":1557963261000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["$$4\\times 4$$ unextendible product basis and genuinely entangled space"],"prefix":"10.1007","volume":"18","author":[{"given":"Kai","family":"Wang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8367-3651","authenticated-orcid":false,"given":"Lin","family":"Chen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lijun","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yumin","family":"Guo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,5,15]]},"reference":[{"issue":"1","key":"2324_CR1","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1006\/jcta.2000.3122","volume":"95","author":"N Alon","year":"2001","unstructured":"Alon, N., Lov\u00e1sz, L.: Unextendible product bases. J. Comb. Theory Ser. A 95(1), 169\u2013179 (2001)","journal-title":"J. Comb. Theory Ser. A"},{"key":"2324_CR2","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1007\/s00220-003-0877-6","volume":"238","author":"DP DiVincenzo","year":"2003","unstructured":"DiVincenzo, D.P., Mor, T., Shor, P.W., Smolin, J.A., Terhal, B.M.: Unextendible product bases, uncompletable product bases and bound entanglement. Commun. Math. Phys. 238, 379\u2013410 (2003)","journal-title":"Commun. Math. Phys."},{"issue":"1","key":"2324_CR3","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1007\/s00220-014-2186-7","volume":"333","author":"J Chen","year":"2013","unstructured":"Chen, J., Johnston, N.: The minimum size of unextendible product bases in the bipartite case (and some multipartite cases). Commun. Math. Phys. 333(1), 351\u2013365 (2013)","journal-title":"Commun. Math. Phys."},{"issue":"6","key":"2324_CR4","doi-asserted-by":"publisher","first-page":"060302","DOI":"10.1103\/PhysRevA.85.060302","volume":"85","author":"J Tura","year":"2012","unstructured":"Tura, J., Augusiak, R., Hyllus, P., Ku\u015b, M., Samsonowicz, J., Lewenstein, M.: Four-qubit entangled symmetric states with positive partial transpositions. Phys. Rev. A 85(6), 060302 (2012)","journal-title":"Phys. Rev. A"},{"issue":"8","key":"2324_CR5","doi-asserted-by":"publisher","first-page":"082207","DOI":"10.1063\/1.4893358","volume":"55","author":"J Chen","year":"2014","unstructured":"Chen, J., Chen, L., Zeng, B.: Unextendible product basis for fermionic systems. J. Math. Phys. 55(8), 082207 (2014)","journal-title":"J. Math. Phys."},{"issue":"4","key":"2324_CR6","doi-asserted-by":"publisher","first-page":"4233","DOI":"10.1103\/PhysRevA.85.042113","volume":"85","author":"R Augusiak","year":"2012","unstructured":"Augusiak, R., Fritz, T., Kotowski, M., Kotowski, M., Lewenstein, M., Ac\u00edn, A.: Tight bell inequalities with no quantum violation from qubit unextendible product bases. Phys. Rev. A 85(4), 4233\u20134237 (2012)","journal-title":"Phys. Rev. A"},{"key":"2324_CR7","doi-asserted-by":"publisher","first-page":"012313","DOI":"10.1103\/PhysRevA.98.012313","volume":"98","author":"M Demianowicz","year":"2018","unstructured":"Demianowicz, M., Augusiak, R.: From unextendible product bases to genuinely entangled subspaces. Phys. Rev. A 98, 012313 (2018)","journal-title":"Phys. Rev. A"},{"issue":"7315","key":"2324_CR8","doi-asserted-by":"publisher","first-page":"574","DOI":"10.1038\/nature09416","volume":"467","author":"L Dicarlo","year":"2010","unstructured":"Dicarlo, L., Reed, M.D., Sun, L., Johnson, B.R., Chow, J.M., Gambetta, J.M., Frunzio, L., Girvin, S.M., Devoret, M.H., Schoelkopf, R.J.: Preparation and measurement of three-qubit entanglement in a superconducting circuit. Nature 467(7315), 574\u20138 (2010)","journal-title":"Nature"},{"issue":"7385","key":"2324_CR9","doi-asserted-by":"publisher","first-page":"382","DOI":"10.1038\/nature10786","volume":"482","author":"MD Reed","year":"2012","unstructured":"Reed, M.D., Dicarlo, L., Nigg, S.E., Sun, L., Frunzio, L., Girvin, S.M., Schoelkopf, R.J.: Realization of three-qubit quantum error correction with superconducting circuits. Nature 482(7385), 382\u20135 (2012)","journal-title":"Nature"},{"issue":"2","key":"2324_CR10","doi-asserted-by":"publisher","first-page":"24","DOI":"10.1007\/s11128-017-1791-8","volume":"17","author":"L Chen","year":"2018","unstructured":"Chen, L., Djokovic, D.Z.: Nonexistence of n -qubit unextendible product bases of size $$2^n-5$$. Quantum Inf. Process. 17(2), 24 (2018)","journal-title":"Quantum Inf. Process."},{"issue":"47","key":"2324_CR11","doi-asserted-by":"publisher","first-page":"424034","DOI":"10.1088\/1751-8113\/47\/42\/424034","volume":"47","author":"N Johnston","year":"2014","unstructured":"Johnston, N.: The structure of qubit unextendible product bases. J. Phys. A Math. Theor. 47(47), 424034 (2014)","journal-title":"J. Phys. A Math. Theor."},{"issue":"39","key":"2324_CR12","doi-asserted-by":"publisher","first-page":"395301","DOI":"10.1088\/1751-8121\/aa8546","volume":"50","author":"L Chen","year":"2017","unstructured":"Chen, L., Djokovic, D.Z.: Orthogonal product bases of four qubits. J. Phys. A:Math. Theor. 50(39), 395301 (2017)","journal-title":"J. Phys. A:Math. Theor."},{"key":"2324_CR13","unstructured":"Johnston, N.: The minimum size of qubit unextendible product bases (2013). \n                    arXiv:1302.1604"},{"key":"2324_CR14","doi-asserted-by":"publisher","first-page":"5385","DOI":"10.1103\/PhysRevLett.82.5385","volume":"82","author":"CH Bennett","year":"1999","unstructured":"Bennett, C.H., Divincenzo, D.P., Mor, T., Shor, P.W., Smolin, J.A., Terhal, B.M.: Unextendible product bases and bound entanglement. Phys. Rev. Lett. 82, 5385\u20135388 (1999)","journal-title":"Phys. Rev. Lett."},{"key":"2324_CR15","doi-asserted-by":"publisher","first-page":"910","DOI":"10.1103\/PhysRevLett.86.910","volume":"86","author":"HJ Briegel","year":"2001","unstructured":"Briegel, H.J., Raussendorf, R.: Persistent entanglement in arrays of interacting particles. Phys. Rev. Lett. 86, 910\u2013913 (2001)","journal-title":"Phys. Rev. Lett."},{"key":"2324_CR16","doi-asserted-by":"publisher","first-page":"250405","DOI":"10.1103\/PhysRevLett.99.250405","volume":"99","author":"G Toth","year":"2007","unstructured":"Toth, G., Knapp, C., Guhne, O., Briegel, H.J.: Optimal spin squeezing inequalities detect bound entanglement in spin models. Phys. Rev. Lett. 99, 250405 (2007)","journal-title":"Phys. Rev. Lett."},{"key":"2324_CR17","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, S.: 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":"2324_CR18","doi-asserted-by":"crossref","unstructured":"Gurvits, L.: Classical deterministic complexity of Edmonds\u2019 problem and quantum entanglement (2003)","DOI":"10.1145\/780542.780545"},{"issue":"17","key":"2324_CR19","doi-asserted-by":"publisher","first-page":"170504","DOI":"10.1103\/PhysRevLett.114.170504","volume":"114","author":"F Monteiro","year":"2015","unstructured":"Monteiro, F., Caprara, V.V., Guerreiro, T., Martin, A., Bancal, J.D., Zbinden, H., Thew, R.T., Sangouard, N.: Revealing genuine optical-path entanglement. Phys. Rev. Lett. 114(17), 170504 (2015)","journal-title":"Phys. Rev. Lett."},{"issue":"6","key":"2324_CR20","doi-asserted-by":"publisher","first-page":"060502","DOI":"10.1103\/PhysRevLett.96.060502","volume":"96","author":"Y Yeo","year":"2006","unstructured":"Yeo, Y., Chua, W.K.: Teleportation and dense coding with genuine multipartite entanglement. Phys. Rev. Lett. 96(6), 060502 (2006)","journal-title":"Phys. Rev. Lett."},{"issue":"10","key":"2324_CR21","doi-asserted-by":"publisher","first-page":"100501","DOI":"10.1103\/PhysRevLett.113.100501","volume":"113","author":"M Huber","year":"2014","unstructured":"Huber, M., Sengupta, R.: Witnessing genuine multipartite entanglement with positive maps. Phys. Rev. Lett. 113(10), 100501 (2014)","journal-title":"Phys. Rev. Lett."},{"issue":"4","key":"2324_CR22","doi-asserted-by":"publisher","first-page":"040401","DOI":"10.1103\/PhysRevLett.102.040401","volume":"102","author":"A Cabello","year":"2009","unstructured":"Cabello, A., Rossi, A., Vallone, G., De Martini, F., Mataloni, P.: Proposed bell experiment with genuine energy-time entanglement. Phys. Rev. Lett. 102(4), 040401 (2009)","journal-title":"Phys. Rev. Lett."},{"issue":"6","key":"2324_CR23","doi-asserted-by":"publisher","first-page":"060502","DOI":"10.1103\/PhysRevLett.120.060502","volume":"120","author":"T Kraft","year":"2018","unstructured":"Kraft, T., Ritz, C., Brunner, N., Huber, M., G\u00fchne, O.: Characterizing genuine multilevel entanglement. Phys. Rev. Lett. 120(6), 060502 (2018)","journal-title":"Phys. Rev. Lett."},{"issue":"6","key":"2324_CR24","doi-asserted-by":"publisher","first-page":"060501","DOI":"10.1103\/PhysRevLett.94.060501","volume":"94","author":"G T\u00f3th","year":"2005","unstructured":"T\u00f3th, G., G\u00fchne, O.: Detecting genuine multipartite entanglement with two local measurements. Phys. Rev. Lett. 94(6), 060501 (2005)","journal-title":"Phys. Rev. Lett."},{"issue":"21","key":"2324_CR25","doi-asserted-by":"publisher","first-page":"210501","DOI":"10.1103\/PhysRevLett.104.210501","volume":"104","author":"M Huber","year":"2010","unstructured":"Huber, M., Mintert, F., Gabriel, A., Hiesmayr, B.C.: Detection of high-dimensional genuine multipartite entanglement of mixed states. Phys. Rev. Lett. 104(21), 210501 (2010)","journal-title":"Phys. Rev. Lett."},{"key":"2324_CR26","doi-asserted-by":"publisher","first-page":"032335","DOI":"10.1103\/PhysRevA.99.032335","volume":"99","author":"S Agrawal","year":"2019","unstructured":"Agrawal, S., Halder, S., Banik, M.: Genuinely entangled subspace with all-encompassing distillable entanglement across every bipartition. Phys. Rev. A 99, 032335 (2019)","journal-title":"Phys. Rev. A"},{"key":"2324_CR27","doi-asserted-by":"publisher","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. (2019). \n                    https:\/\/doi.org\/10.1080\/03081087.2019.1588849","DOI":"10.1080\/03081087.2019.1588849"},{"issue":"26","key":"2324_CR28","doi-asserted-by":"publisher","first-page":"265302","DOI":"10.1088\/1751-8121\/aac53b","volume":"51","author":"L Chen","year":"2018","unstructured":"Chen, L., Djokovic, D.Z.: Multiqubit UPB: the method of formally orthogonal matrices. J. Phys. A Math. Theor. 51(26), 265302 (2018)","journal-title":"J. Phys. A Math. Theor."},{"key":"2324_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."},{"issue":"6","key":"2324_CR30","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1007\/s11128-004-7076-z","volume":"3","author":"SB Bravyi","year":"2004","unstructured":"Bravyi, S.B.: Unextendible product bases and locally unconvertible bound entangled states. Quantum Inf. Process. 3(6), 309\u2013329 (2004)","journal-title":"Quantum Inf. Process."},{"issue":"46","key":"2324_CR31","first-page":"1103","volume":"46","author":"L Chen","year":"2013","unstructured":"Chen, L., Dokovic, D.Z.: Separability problem for multipartite states of rank at most 4. J. Phys. A Math. Theor. 46(46), 1103\u20131114 (2013)","journal-title":"J. Phys. A Math. Theor."}],"container-title":["Quantum Information Processing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-019-2324-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11128-019-2324-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-019-2324-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,13]],"date-time":"2020-05-13T23:31:22Z","timestamp":1589412682000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11128-019-2324-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,15]]},"references-count":31,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2019,7]]}},"alternative-id":["2324"],"URL":"https:\/\/doi.org\/10.1007\/s11128-019-2324-4","relation":{},"ISSN":["1570-0755","1573-1332"],"issn-type":[{"value":"1570-0755","type":"print"},{"value":"1573-1332","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,5,15]]},"assertion":[{"value":"8 February 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 May 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 May 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"202"}}