{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,8]],"date-time":"2025-09-08T06:35:51Z","timestamp":1757313351194,"version":"3.37.3"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2021,4,1]],"date-time":"2021-04-01T00:00:00Z","timestamp":1617235200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,4,1]],"date-time":"2021-04-01T00:00:00Z","timestamp":1617235200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation (NNSF) of China","doi-asserted-by":"crossref","award":["11871089","11947241"],"award-info":[{"award-number":["11871089","11947241"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"crossref","award":["KG12080401","ZG216S1902"],"award-info":[{"award-number":["KG12080401","ZG216S1902"]}],"id":[{"id":"10.13039\/501100012226","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":[[2021,4]]},"DOI":"10.1007\/s11128-021-03098-w","type":"journal-article","created":{"date-parts":[[2021,4,26]],"date-time":"2021-04-26T08:13:10Z","timestamp":1619424790000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Proving the distillability problem of two-copy $$4\\times 4$$ Werner states for monomial matrices"],"prefix":"10.1007","volume":"20","author":[{"given":"Lin","family":"Chen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huixia","family":"He","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xian","family":"Shi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Li-Jun","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,4,26]]},"reference":[{"key":"3098_CR1","doi-asserted-by":"publisher","first-page":"062312","DOI":"10.1103\/PhysRevA.61.062312","volume":"61","author":"DP DiVincenzo","year":"2000","unstructured":"DiVincenzo, D.P., Shor, P.W., Smolin, J.A., Terhal, B.M., Thapliyal, A.V.: Evidence for bound entangled states with negative partial transpose. Phys. Rev. A 61, 062312 (2000)","journal-title":"Phys. Rev. A"},{"issue":"1","key":"3098_CR2","first-page":"61","volume":"323","author":"BM Terhal","year":"2000","unstructured":"Terhal, B.M.: A family of indecomposable positive linear maps based on entangled quantum states. Linear Algebra Appl. 323(1), 61\u201373 (2000)","journal-title":"Linear Algebra Appl."},{"issue":"468","key":"3098_CR3","doi-asserted-by":"publisher","first-page":"122","DOI":"10.1016\/j.laa.2014.03.009","volume":"468","author":"E Poon","year":"2015","unstructured":"Poon, E.: Preservers of maximally entangled states. Linear Algebra Appl. 468(468), 122\u2013144 (2015)","journal-title":"Linear Algebra Appl."},{"key":"3098_CR4","first-page":"1","volume":"529","author":"D Cariello","year":"2016","unstructured":"Cariello, D.: A gap for ppt entanglement. Linear Algebra Appl. 529, 1 (2016)","journal-title":"Linear Algebra Appl."},{"issue":"5","key":"3098_CR5","doi-asserted-by":"publisher","first-page":"1245","DOI":"10.1016\/j.laa.2013.04.007","volume":"439","author":"J Hou","year":"2013","unstructured":"Hou, J., Qi, X.: Linear maps preserving separability of pure states. Linear Algebra Appl. 439(5), 1245\u20131257 (2013)","journal-title":"Linear Algebra Appl."},{"issue":"10","key":"3098_CR6","doi-asserted-by":"publisher","first-page":"2613","DOI":"10.1016\/j.laa.2012.06.018","volume":"437","author":"E Alfsen","year":"2012","unstructured":"Alfsen, E., Shultz, F.: Finding decompositions of a class of separable states. Linear Algebra Appl. 437(10), 2613\u20132629 (2012)","journal-title":"Linear Algebra Appl."},{"issue":"1","key":"3098_CR7","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1016\/j.laa.2003.01.001","volume":"376","author":"HJ Woerdeman","year":"2004","unstructured":"Woerdeman, H.J.: The separability problem and normal completions. Linear Algebra Appl. 376(1), 85\u201395 (2004)","journal-title":"Linear Algebra Appl."},{"key":"3098_CR8","unstructured":"Horodecki, P., Rudnicki, L., Zyczkowski, K.: Five open problems in quantum information (2020). arXiv:2002.03233v2"},{"issue":"8","key":"3098_CR9","doi-asserted-by":"publisher","first-page":"4085","DOI":"10.1109\/TIT.2010.2050810","volume":"56","author":"L Pankowski","year":"2010","unstructured":"Pankowski, L., Piani, M., Horodecki, M., Horodecki, P.: A few steps more towards NPT bound entanglement. IEEE Trans. Inf. Theory 56(8), 4085\u20134100 (2010)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"3098_CR10","doi-asserted-by":"publisher","first-page":"062313","DOI":"10.1103\/PhysRevA.61.062313","volume":"61","author":"W D\u00fcr","year":"2000","unstructured":"D\u00fcr, W., Cirac, J.I., Lewenstein, M., Bru\u00df, D.: Distillability and partial transposition in bipartite systems. Phys. Rev. A 61, 062313 (2000)","journal-title":"Phys. Rev. A"},{"key":"3098_CR11","doi-asserted-by":"publisher","first-page":"052306","DOI":"10.1103\/PhysRevA.74.052306","volume":"74","author":"RO Vianna","year":"2006","unstructured":"Vianna, R.O., Doherty, A.C.: Distillability of Werner states using entanglement witnesses and robust semidefinite programs. Phys. Rev. A 74, 052306 (2006)","journal-title":"Phys. Rev. A"},{"key":"3098_CR12","doi-asserted-by":"crossref","first-page":"022319","DOI":"10.1103\/PhysRevA.68.022319","volume":"68","author":"S Bandyopadhyay","year":"2003","unstructured":"Bandyopadhyay, S.: Classes of $$n$$-copy undistillable quantum states with negative partial transposition. Phys. Rev. A 68, 022319 (2003)","journal-title":"Phys. Rev. A"},{"key":"3098_CR13","doi-asserted-by":"publisher","first-page":"022318","DOI":"10.1103\/PhysRevA.78.022318","volume":"78","author":"L Chen","year":"2008","unstructured":"Chen, L., Chen, Y.X.: Rank-three bipartite entangled states are distillable. Phys. Rev. A 78, 022318 (2008)","journal-title":"Phys. Rev. A"},{"key":"3098_CR14","doi-asserted-by":"crossref","first-page":"574","DOI":"10.1103\/PhysRevLett.78.574","volume":"78","author":"M Horodecki","year":"1997","unstructured":"Horodecki, M., Horodecki, P., Horodecki, R.: Inseparable two spin- $$\\frac{1}{2}$$ density matrices can be distilled to a singlet form. Phys. Rev. Lett. 78, 574\u2013577 (1997)","journal-title":"Phys. Rev. Lett."},{"key":"3098_CR15","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1103\/PhysRevA.60.179","volume":"60","author":"EM Rains","year":"1999","unstructured":"Rains, E.M.: Bound on distillable entanglement. Phys. Rev. A 60, 179 (1999)","journal-title":"Phys. Rev. A"},{"issue":"28","key":"3098_CR16","doi-asserted-by":"publisher","first-page":"285303","DOI":"10.1088\/1751-8113\/44\/28\/285303","volume":"44","author":"L Chen","year":"2011","unstructured":"Chen, L., Djokovic\u00e1, D.Z.: Distillability and ppt entanglement of low-rank quantum states. J. Phys. A Math. Theor. 44(28), 285303 (2011)","journal-title":"J. Phys. A Math. Theor."},{"key":"3098_CR17","doi-asserted-by":"publisher","first-page":"052318","DOI":"10.1103\/PhysRevA.94.052318","volume":"94","author":"L Chen","year":"2016","unstructured":"Chen, L., Djokovic\u00e1, D.Z.: Distillability of non-positive-partial-transpose bipartite quantum states of rank four. Phys. Rev. A 94, 052318 (2016)","journal-title":"Phys. Rev. A"},{"issue":"8","key":"3098_CR18","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1109\/TIT.2010.2050810","volume":"56","author":"MHA Pankowski","year":"2010","unstructured":"Pankowski, M.H.A., Piani, M., Horodecki, P.: A few steps more towards NPT bound entanglement. IEEE. Trans. Inf. Theory 56(8), 1\u201319 (2010)","journal-title":"IEEE. Trans. Inf. Theory"},{"key":"3098_CR19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0375-9601(96)00706-2","volume":"223","author":"M Horodecki","year":"1996","unstructured":"Horodecki, M., Horodecki, P., Horodecki, R.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223, 1\u20138 (1996)","journal-title":"Phys. Lett. A"},{"key":"3098_CR20","doi-asserted-by":"publisher","first-page":"1413","DOI":"10.1103\/PhysRevLett.77.1413","volume":"77","author":"A Peres","year":"1996","unstructured":"Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413 (1996)","journal-title":"Phys. Rev. Lett."},{"key":"3098_CR21","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1016\/j.laa.2021.01.006","volume":"616","author":"L Qian","year":"2021","unstructured":"Qian, L., Chen, L., Chu, D., Shen, Y.: A matrix inequality for entanglement distillation problem. Linear Algebra Appl. 616, 139\u2013177 (2021)","journal-title":"Linear Algebra Appl."},{"key":"3098_CR22","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.: 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":"3098_CR23","first-page":"1","volume":"2019","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 2019, 1\u201316 (2019)","journal-title":"Linear Multilinear Algebra"},{"key":"3098_CR24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1080\/03081087.2019.1699497","volume":"2019","author":"M Liang","year":"2019","unstructured":"Liang, M., Mengyao, H., Sun, Y., Chen, L., Chen, X.: Real entries of complex hadamard matrices and mutually unbiased bases in dimension six. Linear Multilinear Algebra 2019, 1\u201318 (2019)","journal-title":"Linear Multilinear Algebra"},{"key":"3098_CR25","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., Yu, L.: The Birkhoff theorem for unitary matrices of arbitrary dimensions. Linear Algebra Appl. 514, 151\u2013164 (2017)","journal-title":"Linear Algebra Appl."},{"key":"3098_CR26","doi-asserted-by":"crossref","unstructured":"Ouyang, Y., Shen, Y., Chen, L.: Faster quantum computation with permutations and resonant couplings (2019). arXiv:1901.00252v4. Accepted by Linear Algebra and Its application","DOI":"10.1016\/j.laa.2020.01.020"},{"issue":"8","key":"3098_CR27","doi-asserted-by":"publisher","first-page":"588","DOI":"10.3390\/e20080588","volume":"20","author":"Y Shen","year":"2018","unstructured":"Shen, Y., Chen, L.: On a matrix inequality related to the distillability problem. Entropy 20(8), 588 (2018)","journal-title":"Entropy"}],"container-title":["Quantum Information Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-021-03098-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11128-021-03098-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-021-03098-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,5,8]],"date-time":"2021-05-08T12:13:46Z","timestamp":1620476026000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11128-021-03098-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4]]},"references-count":27,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,4]]}},"alternative-id":["3098"],"URL":"https:\/\/doi.org\/10.1007\/s11128-021-03098-w","relation":{},"ISSN":["1570-0755","1573-1332"],"issn-type":[{"type":"print","value":"1570-0755"},{"type":"electronic","value":"1573-1332"}],"subject":[],"published":{"date-parts":[[2021,4]]},"assertion":[{"value":"21 August 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 April 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 April 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"157"}}