{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T16:16:53Z","timestamp":1783700213299,"version":"3.55.0"},"reference-count":49,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T00:00:00Z","timestamp":1776211200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T00:00:00Z","timestamp":1776211200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Quantum Inf Process"],"DOI":"10.1007\/s11128-026-05156-7","type":"journal-article","created":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T07:29:25Z","timestamp":1776238165000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Entanglement-assisted local discrimination of strongly nonlocal tripartite orthogonal product states"],"prefix":"10.1007","volume":"25","author":[{"given":"Tian-Qing","family":"Cao","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shu-Na","family":"Peng","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qiao-Ling","family":"Xin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,4,15]]},"reference":[{"key":"5156_CR1","doi-asserted-by":"publisher","first-page":"419","DOI":"10.1103\/RevModPhys.86.419","volume":"86","author":"N Brunner","year":"2014","unstructured":"Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V., Wehner, S.: Bell nonlocality. Rev. Mod. Phys. 86, 419 (2014)","journal-title":"Rev. Mod. Phys."},{"key":"5156_CR2","doi-asserted-by":"publisher","first-page":"880","DOI":"10.1103\/PhysRevLett.23.880","volume":"23","author":"JF Clauser","year":"1969","unstructured":"Clauser, J.F., Horne, M.A., Shimony, A., Holt, R.A.: Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23, 880 (1969)","journal-title":"Phys. Rev. Lett."},{"key":"5156_CR3","doi-asserted-by":"publisher","first-page":"1804","DOI":"10.1103\/PhysRevLett.49.1804","volume":"49","author":"A Aspect","year":"1982","unstructured":"Aspect, A., Dalibard, J., Roger, G.: Experimental test of Bell\u2019s inequalities using time-varying analyzers. Phys. Rev. Lett. 49, 1804 (1982)","journal-title":"Phys. Rev. Lett."},{"key":"5156_CR4","doi-asserted-by":"publisher","first-page":"865","DOI":"10.1103\/RevModPhys.81.865","volume":"81","author":"R Horodecki","year":"2009","unstructured":"Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)","journal-title":"Rev. Mod. Phys."},{"key":"5156_CR5","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.119.010402","volume":"119","author":"W Rosenfeld","year":"2017","unstructured":"Rosenfeld, W., Burchardt, D., Garthoff, R., Redeker, K., Ortegel, N., Rau, M., Weinfurter, H.: Event-ready Bell test using entangled atoms simultaneously closing detection and locality loopholes. Phys. Rev. Lett. 119, 010402 (2017)","journal-title":"Phys. Rev. Lett."},{"key":"5156_CR6","doi-asserted-by":"crossref","unstructured":"BIG Bell Test Collaboration: Challenging local realism with human choices. Nature 557, 212 (2018)","DOI":"10.1038\/s41586-018-0085-3"},{"key":"5156_CR7","doi-asserted-by":"publisher","first-page":"1070","DOI":"10.1103\/PhysRevA.59.1070","volume":"59","author":"CH Bennett","year":"1999","unstructured":"Bennett, C.H., DiVincenzo, D.P., Fuchs, C.A., Mor, T., Rains, E., Shor, P.W., Smolin, J.A., Wootters, W.K.: Quantum nonlocality without entanglement. Phys. Rev. A 59, 1070 (1999)","journal-title":"Phys. Rev. A"},{"key":"5156_CR8","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.90.022313","volume":"90","author":"ZC Zhang","year":"2014","unstructured":"Zhang, Z.C., Gao, F., Tian, G.J., Cao, T.Q., Wen, Q.Y.: Nonlocality of orthogonal product basis quantum states. Phys. Rev. A 90, 022313 (2014)","journal-title":"Phys. Rev. A"},{"key":"5156_CR9","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2023.128956","volume":"624","author":"YY Zhu","year":"2023","unstructured":"Zhu, Y.Y., Jiang, D.H., Xu, G.B., Yang, Y.G.: Completable sets of orthogonal product states with minimal nonlocality. Physica A 624, 128956 (2023)","journal-title":"Physica A"},{"key":"5156_CR10","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.92.032313","volume":"92","author":"YL Wang","year":"2015","unstructured":"Wang, Y.L., Li, M.S., Zheng, Z.J., Fei, S.M.: Nonlocality of orthogonal product-basis quantum states. Phys. Rev. A 92, 032313 (2015)","journal-title":"Phys. Rev. A"},{"key":"5156_CR11","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.92.012332","volume":"92","author":"ZC Zhang","year":"2015","unstructured":"Zhang, Z.C., Gao, F., Qin, S.J., Yang, Y.H., Wen, Q.Y.: Nonlocality of orthogonal product states. Phys. Rev. A 92, 012332 (2015)","journal-title":"Phys. Rev. A"},{"key":"5156_CR12","doi-asserted-by":"publisher","first-page":"382","DOI":"10.1007\/s11128-021-03320-9","volume":"20","author":"HJ Zuo","year":"2021","unstructured":"Zuo, H.J., Liu, J.H., Zhen, X.F., Fei, S.M.: Nonlocal sets of orthogonal multipartite product states with less members. Quantum Inf. Process. 20, 382 (2021)","journal-title":"Quantum Inf. Process."},{"key":"5156_CR13","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.74.052103","volume":"74","author":"J Niset","year":"2006","unstructured":"Niset, J., Cerf, N.J.: Multipartite nonlocality without entanglement in many dimensions. Phys. Rev. A 74, 052103 (2006)","journal-title":"Phys. Rev. A"},{"key":"5156_CR14","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8121\/ac28cd","volume":"54","author":"MS Li","year":"2021","unstructured":"Li, M.S., Wang, Y.L., Shi, F., Yung, M.H.: Local distinguishability based genuinely quantum nonlocality without entanglement. J. Phys. A: Math. Theor. 54, 445301 (2021)","journal-title":"J. Phys. A: Math. Theor."},{"key":"5156_CR15","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.102.032211","volume":"102","author":"DH Jiang","year":"2020","unstructured":"Jiang, D.H., Xu, G.B.: Nonlocal sets of orthogonal product states in an arbitrary multipartite quantum system. Phys. Rev. A 102, 032211 (2020)","journal-title":"Phys. Rev. A"},{"key":"5156_CR16","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1007\/s11128-024-04591-8","volume":"23","author":"YQ Zhang","year":"2024","unstructured":"Zhang, Y.Q., Jiang, D.H., Yang, Y.G., Xu, G.B.: Nonlocal sets of orthogonal product states with less members in multipartite quantum systems. Quantum Inf. Process. 23, 391 (2024)","journal-title":"Quantum Inf. Process."},{"key":"5156_CR17","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.106.062432","volume":"106","author":"XF Zhen","year":"2022","unstructured":"Zhen, X.F., Fei, S.M., Zuo, H.J.: Nonlocality without entanglement in general multipartite quantum systems. Phys. Rev. A 106, 062432 (2022)","journal-title":"Phys. Rev. A"},{"key":"5156_CR18","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.111.022416","volume":"111","author":"GB Xu","year":"2025","unstructured":"Xu, G.B., Zhang, Y.Q., Jiang, D.H.: Minimal nonlocality in multipartite quantum systems. Phys. Rev. A 111, 022416 (2025)","journal-title":"Phys. Rev. A"},{"key":"5156_CR19","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.122.040403","volume":"122","author":"S Halder","year":"2019","unstructured":"Halder, S., Banik, M., Agrawal, S., Bandyopadhyay, S.: Strong quantum nonlocality without entanglement. Phys. Rev. Lett. 122, 040403 (2019)","journal-title":"Phys. Rev. Lett."},{"key":"5156_CR20","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.100.032321","volume":"100","author":"S Rout","year":"2019","unstructured":"Rout, S., Maity, A.G., Mukherjee, A., Halder, S., Banik, M.: Genuinely nonlocal product bases: classification and entanglement assisted discrimination. Phys. Rev. A 100, 032321 (2019)","journal-title":"Phys. Rev. A"},{"key":"5156_CR21","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.102.042202","volume":"102","author":"F Shi","year":"2020","unstructured":"Shi, F., Hu, M.Y., Chen, L., Zhang, X.D.: Strong quantum nonlocality with entanglement. Phys. Rev. A 102, 042202 (2020)","journal-title":"Phys. Rev. A"},{"key":"5156_CR22","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1007\/s11128-022-03506-9","volume":"21","author":"A Bhunia","year":"2022","unstructured":"Bhunia, A., Chattopadhyay, I., Sarkar, D.: Nonlocality without entanglement: an acyclic configuration. Quantum Inf. Process. 21, 169 (2022)","journal-title":"Quantum Inf. Process."},{"key":"5156_CR23","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.105.022209","volume":"105","author":"F Shi","year":"2022","unstructured":"Shi, F., Ye, Z., Chen, L., Zhang, X.D.: Strong quantum nonlocality in $$N$$-partite systems. Phys. Rev. A 105, 022209 (2022)","journal-title":"Phys. Rev. A"},{"key":"5156_CR24","doi-asserted-by":"publisher","DOI":"10.1016\/j.isci.2024.111528","volume":"28","author":"HQ Zhou","year":"2025","unstructured":"Zhou, H.Q., Gao, T., Yan, F.L.: Strong quantum nonlocality without entanglement in every $$(n-1)$$-partition. IScience 28, 111528 (2025)","journal-title":"IScience"},{"key":"5156_CR25","doi-asserted-by":"publisher","first-page":"619","DOI":"10.22331\/q-2022-01-05-619","volume":"6","author":"F Shi","year":"2022","unstructured":"Shi, F., Li, M.S., Hu, M.Y., Chen, L., Yung, M.H., Wang, Y.L., Zhang, X.D.: Strongly nonlocal unextendible product bases do exist. Quantum 6, 619 (2022)","journal-title":"Quantum"},{"key":"5156_CR26","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.109.052211","volume":"109","author":"A Bhunia","year":"2024","unstructured":"Bhunia, A., Bera, S., Biswas, I., Chattopadhyay, I., Sarkar, D.: Strong quantum nonlocality: unextendible biseparability beyond unextendible product basis. Phys. Rev. A 109, 052211 (2024)","journal-title":"Phys. Rev. A"},{"key":"5156_CR27","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8121\/ac3bea","volume":"55","author":"F Shi","year":"2022","unstructured":"Shi, F., Li, M.S., Chen, L., Zhang, X.D.: Strong quantum nonlocality for unextendible product bases in heterogeneous systems. J. Phys. A: Math. Theor. 55, 015305 (2022)","journal-title":"J. Phys. A: Math. Theor."},{"key":"5156_CR28","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.107.042214","volume":"107","author":"HQ Zhou","year":"2023","unstructured":"Zhou, H.Q., Gao, T., Yan, F.L.: Strong quantum nonlocality without entanglement in $$n$$-partite system with even $$n$$. Phys. Rev. A 107, 042214 (2023)","journal-title":"Phys. Rev. A"},{"key":"5156_CR29","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.99.062108","volume":"99","author":"ZC Zhang","year":"2019","unstructured":"Zhang, Z.C., Zhang, X.D.: Strong quantum nonlocality in multipartite quantum systems. Phys. Rev. A 99, 062108 (2019)","journal-title":"Phys. Rev. A"},{"key":"5156_CR30","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.109.022220","volume":"109","author":"MY Hu","year":"2024","unstructured":"Hu, M.Y., Gao, T., Yan, F.L.: Strong quantum nonlocality with genuine entanglement in an $$N$$-qutrit system. Phys. Rev. A 109, 022220 (2024)","journal-title":"Phys. Rev. A"},{"key":"5156_CR31","doi-asserted-by":"publisher","first-page":"1349","DOI":"10.22331\/q-2024-05-16-1349","volume":"8","author":"YY He","year":"2024","unstructured":"He, Y.Y., Shi, F., Zhang, X.D.: Strong quantum nonlocality and unextendibility without entanglement in $$N$$-partite systems with odd $$N$$. Quantum 8, 1349 (2024)","journal-title":"Quantum"},{"key":"5156_CR32","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.106.052209","volume":"106","author":"HQ Zhou","year":"2022","unstructured":"Zhou, H.Q., Gao, T., Yan, F.L.: Orthogonal product sets with strong quantum nonlocality on plane structure. Phys. Rev. A 106, 052209 (2022)","journal-title":"Phys. Rev. A"},{"key":"5156_CR33","doi-asserted-by":"publisher","first-page":"1895","DOI":"10.1103\/PhysRevLett.70.1895","volume":"70","author":"CH Bennett","year":"1993","unstructured":"Bennett, C.H., Brassard, G., Cr\u00e9peau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70, 1895 (1993)","journal-title":"Phys. Rev. Lett."},{"key":"5156_CR34","doi-asserted-by":"publisher","first-page":"813","DOI":"10.1007\/s00220-009-0890-5","volume":"291","author":"W Matthews","year":"2009","unstructured":"Matthews, W., Wehner, S., Winter, A.: Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding. Commun. Math. Phys. 291, 813 (2009)","journal-title":"Commun. Math. Phys."},{"key":"5156_CR35","doi-asserted-by":"publisher","DOI":"10.1088\/1402-4896\/acd3bd","volume":"98","author":"WY Ke","year":"2023","unstructured":"Ke, W.Y., Shi, R.H., Yu, H., Xu, X.T.: A receipt-free quantum voting protocol based on quantum public key encryption and quantum key agreement. Phys. Scr. 98, 065112 (2023)","journal-title":"Phys. Scr."},{"key":"5156_CR36","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.77.012304","volume":"77","author":"SM Cohen","year":"2008","unstructured":"Cohen, S.M.: Understanding entanglement as resource: locally distinguishing unextendible product bases. Phys. Rev. A 77, 012304 (2008)","journal-title":"Phys. Rev. A"},{"key":"5156_CR37","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8113\/42\/7\/072002","volume":"42","author":"S Bandyopadhyay","year":"2009","unstructured":"Bandyopadhyay, S., Walgate, J.: Local distinguishability of any three quantum states. J. Phys. A: Math. Theor. 42, 072002 (2009)","journal-title":"J. Phys. A: Math. Theor."},{"key":"5156_CR38","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.94.022311","volume":"94","author":"S Bandyopadhyay","year":"2016","unstructured":"Bandyopadhyay, S., Halder, S., Nathanson, M.: Entanglement as a resource for local state discrimination in multipartite systems. Phys. Rev. A 94, 022311 (2016)","journal-title":"Phys. Rev. A"},{"key":"5156_CR39","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2023.128852","volume":"623","author":"HQ Cao","year":"2023","unstructured":"Cao, H.Q., Zuo, H.J.: Locally distinguishing nonlocal sets with entanglement resource. Physica A 623, 128852 (2023)","journal-title":"Physica A"},{"key":"5156_CR40","doi-asserted-by":"publisher","first-page":"362","DOI":"10.1007\/s11128-021-03313-8","volume":"20","author":"TQ Cao","year":"2021","unstructured":"Cao, T.Q., Xin, Q.L., Zhang, Z.C.: Quantum entanglement as a resource to locally distinguish orthogonal product states. Quantum Inf. Process. 20, 362 (2021)","journal-title":"Quantum Inf. Process."},{"key":"5156_CR41","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8113\/43\/45\/455303","volume":"43","author":"S Bandyopadhyay","year":"2010","unstructured":"Bandyopadhyay, S., Rahaman, R., Wootters, W.K.: Entanglement cost of two-qubit orthogonal measurements. J. Phys. A: Math. Theor. 43, 455303 (2010)","journal-title":"J. Phys. A: Math. Theor."},{"key":"5156_CR42","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.97.022334","volume":"97","author":"ZC Zhang","year":"2018","unstructured":"Zhang, Z.C., Song, Y.Q., Song, T.T., Gao, F., Qin, S.J., Wen, Q.Y.: Local distinguishability of orthogonal quantum states with multiple copies of $$2\\otimes 2$$ maximally entangled states. Phys. Rev. A 97, 022334 (2018)","journal-title":"Phys. Rev. A"},{"key":"5156_CR43","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1007\/s11128-025-04718-5","volume":"24","author":"SJ Zhang","year":"2025","unstructured":"Zhang, S.J., Qiao, Q., Bai, C.M.: Locally distinguishing genuinely nonlocal sets with one GHZ state. Quantum Inf. Process. 24, 101 (2025)","journal-title":"Quantum Inf. Process."},{"key":"5156_CR44","doi-asserted-by":"publisher","first-page":"342","DOI":"10.1007\/s11128-022-03696-2","volume":"21","author":"ZC Zhang","year":"2022","unstructured":"Zhang, Z.C., Wei, X.J., Wang, A.L.: Entanglement as a resource to locally distinguish tripartite quantum states. Quantum Inf. Process. 21, 342 (2022)","journal-title":"Quantum Inf. Process."},{"key":"5156_CR45","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1007\/s11128-018-1962-2","volume":"17","author":"HQ Li","year":"2018","unstructured":"Li, H.Q., Jing, N.H., Tang, X.L.: Distinguishing multipartite orthogonal product states by LOCC with entanglement as a resource. Quantum Inf. Process. 17, 195 (2018)","journal-title":"Quantum Inf. Process."},{"key":"5156_CR46","doi-asserted-by":"publisher","DOI":"10.1088\/1572-9494\/ad6de6","volume":"76","author":"Q Qiao","year":"2024","unstructured":"Qiao, Q., Zhang, S.J., Bai, C.M., Liu, L.: Locally distinguishing genuinely nonlocal sets with entanglement resource. Commun. Theor. Phys. 76, 125101 (2024)","journal-title":"Commun. Theor. Phys."},{"key":"5156_CR47","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.102.042228","volume":"102","author":"P Yuan","year":"2020","unstructured":"Yuan, P., Tian, G.J., Sun, X.M.: Strong quantum nonlocality without entanglement in multipartite quantum systems. Phys. Rev. A 102, 042228 (2020)","journal-title":"Phys. Rev. A"},{"key":"5156_CR48","doi-asserted-by":"publisher","first-page":"361","DOI":"10.1007\/s11128-024-04558-9","volume":"23","author":"XJ Wei","year":"2024","unstructured":"Wei, X.J., Xie, Z.S., Li, Y.L., Zhang, Z.C.: Locally distinguishing tripartite strongly nonlocal quantum states with entanglement resource. Quantum Inf. Process. 23, 361 (2024)","journal-title":"Quantum Inf. Process."},{"key":"5156_CR49","doi-asserted-by":"publisher","first-page":"4972","DOI":"10.1103\/PhysRevLett.85.4972","volume":"85","author":"J Walgate","year":"2000","unstructured":"Walgate, J., Short, A.J., Hardy, L., Vedral, V.: Local distinguishability of multipartite orthogonal quantum states. Phys. Rev. Lett. 85, 4972 (2000)","journal-title":"Phys. Rev. Lett."}],"container-title":["Quantum Information Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-026-05156-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11128-026-05156-7","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-026-05156-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T16:01:51Z","timestamp":1783699311000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11128-026-05156-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,15]]},"references-count":49,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2026,5]]}},"alternative-id":["5156"],"URL":"https:\/\/doi.org\/10.1007\/s11128-026-05156-7","relation":{},"ISSN":["1573-1332"],"issn-type":[{"value":"1573-1332","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,15]]},"assertion":[{"value":"5 December 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 March 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 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":"144"}}