{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T23:42:42Z","timestamp":1770334962734,"version":"3.49.0"},"reference-count":61,"publisher":"MDPI AG","issue":"18","license":[{"start":{"date-parts":[[2022,9,7]],"date-time":"2022-09-07T00:00:00Z","timestamp":1662508800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea (NRF)","doi-asserted-by":"publisher","award":["2019R1A2C2007037"],"award-info":[{"award-number":["2019R1A2C2007037"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea (NRF)","doi-asserted-by":"publisher","award":["2022R1A4A3033401"],"award-info":[{"award-number":["2022R1A4A3033401"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea (NRF)","doi-asserted-by":"publisher","award":["IITP-2021-0-02046"],"award-info":[{"award-number":["IITP-2021-0-02046"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Ministry of Science and ICT","award":["2019R1A2C2007037"],"award-info":[{"award-number":["2019R1A2C2007037"]}]},{"name":"Ministry of Science and ICT","award":["2022R1A4A3033401"],"award-info":[{"award-number":["2022R1A4A3033401"]}]},{"name":"Ministry of Science and ICT","award":["IITP-2021-0-02046"],"award-info":[{"award-number":["IITP-2021-0-02046"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Quantum entanglement is a unique phenomenon of quantum mechanics, which has no classical counterpart and gives quantum systems their advantage in computing, communication, sensing, and metrology. In quantum sensing and metrology, utilizing an entangled probe state enhances the achievable precision more than its classical counterpart. Noise in the probe state preparation step can cause the system to output unentangled states, which might not be resourceful. Hence, an effective method for the detection and classification of tripartite entanglement is required at that step. However, current mathematical methods cannot robustly classify multiclass entanglement in tripartite quantum systems, especially in the case of mixed states. In this paper, we explore the utility of artificial neural networks for classifying the entanglement of tripartite quantum states into fully separable, biseparable, and fully entangled states. We employed Bell\u2019s inequality for the dataset of tripartite quantum states and train the deep neural network for multiclass classification. This entanglement classification method is computationally efficient due to using a small number of measurements. At the same time, it also maintains generalization by covering a large Hilbert space of tripartite quantum states.<\/jats:p>","DOI":"10.3390\/s22186767","type":"journal-article","created":{"date-parts":[[2022,9,8]],"date-time":"2022-09-08T04:18:32Z","timestamp":1662610712000},"page":"6767","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Multiclass Classification of Metrologically Resourceful Tripartite Quantum States with Deep Neural Networks"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1871-4058","authenticated-orcid":false,"given":"Syed Muhammad Abuzar","family":"Rizvi","sequence":"first","affiliation":[{"name":"Department of Electronics and Information Convergence Engineering, Kyung Hee University, Yongin 17104, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0007-5876","authenticated-orcid":false,"given":"Naema","family":"Asif","sequence":"additional","affiliation":[{"name":"Department of Electronics and Information Convergence Engineering, Kyung Hee University, Yongin 17104, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0769-1599","authenticated-orcid":false,"given":"Muhammad Shohibul","family":"Ulum","sequence":"additional","affiliation":[{"name":"Department of Electronics and Information Convergence Engineering, Kyung Hee University, Yongin 17104, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4703-4836","authenticated-orcid":false,"given":"Trung Q.","family":"Duong","sequence":"additional","affiliation":[{"name":"School of Electronics, Electrical Engineering and Computer Science, Queen\u2019s University Belfast, Belfast BT7 1NN, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3364-8084","authenticated-orcid":false,"given":"Hyundong","family":"Shin","sequence":"additional","affiliation":[{"name":"Department of Electronics and Information Convergence Engineering, Kyung Hee University, Yongin 17104, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,7]]},"reference":[{"key":"ref_1","unstructured":"Wilde, M.M. (2017). Quantum Information Theory, Cambridge University Press. [2nd ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1038\/35005001","article-title":"Quantum information and computation","volume":"404","author":"Bennett","year":"2000","journal-title":"Nature"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1038\/416238a","article-title":"Quantum information processing with atoms and photons","volume":"416","author":"Monroe","year":"2002","journal-title":"Nature"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"035002","DOI":"10.1103\/RevModPhys.89.035002","article-title":"Quantum sensing","volume":"89","author":"Degen","year":"2017","journal-title":"Rev. Mod. Phys."},{"key":"ref_5","first-page":"195","article-title":"On the Einstein Podolsky Rosen paradox","volume":"1","author":"Bell","year":"1964","journal-title":"Phys. Phys. Fiz."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1038\/nphys2904","article-title":"Quantum entanglement","volume":"10","author":"Vedral","year":"2014","journal-title":"Nat. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2015.12.002","article-title":"Quantum metrology and its application in biology","volume":"615","author":"Taylor","year":"2016","journal-title":"Phys. Rep."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"022322","DOI":"10.1103\/PhysRevA.85.022322","article-title":"Multipartite entanglement and high-precision metrology","volume":"85","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"250801","DOI":"10.1103\/PhysRevLett.113.250801","article-title":"Using Entanglement Against Noise in Quantum Metrology","volume":"113","author":"Maccone","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/S0304-3975(02)00139-1","article-title":"Detecting quantum entanglement","volume":"287","author":"Terhal","year":"2002","journal-title":"Theor. Comput. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"424006","DOI":"10.1088\/1751-8113\/47\/42\/424006","article-title":"Quantum metrology from a quantum information science perspective","volume":"47","author":"Apellaniz","year":"2014","journal-title":"J. Phys. A-Math. Theor."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1016\/j.optcom.2006.01.061","article-title":"Sudden death of entanglement: Classical noise effects","volume":"264","author":"Yu","year":"2006","journal-title":"Opt. Commun."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11128-018-2110-8","article-title":"Measurement-based quantum correlation in mixed-state quantum metrology","volume":"17","author":"Khalid","year":"2018","journal-title":"Quantum Inf. Process."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1038\/nature23474","article-title":"Quantum machine learning","volume":"549","author":"Biamonte","year":"2017","journal-title":"Nature"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"4778","DOI":"10.1038\/s41598-019-41324-9","article-title":"Hybrid quantum linear equation algorithm and its experimental test on IBM Quantum Experience","volume":"9","author":"Lee","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1080\/00107514.2014.964942","article-title":"An introduction to quantum machine learning","volume":"56","author":"Schuld","year":"2014","journal-title":"Contemp. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"240501","DOI":"10.1103\/PhysRevLett.120.240501","article-title":"Experimental Machine Learning of Quantum States","volume":"120","author":"Gao","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1058","DOI":"10.1002\/qua.24954","article-title":"Machine learning for quantum mechanics in a nutshell","volume":"115","author":"Rupp","year":"2015","journal-title":"Int. J. Quantum Chem."},{"key":"ref_19","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_20","doi-asserted-by":"crossref","first-page":"044305","DOI":"10.1103\/PhysRevA.71.044305","article-title":"Quantum secure direct communication with high-dimension quantum superdense coding","volume":"71","author":"Wang","year":"2005","journal-title":"Phys. Rev. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1038\/37539","article-title":"Experimental quantum teleportation","volume":"390","author":"Bouwmeester","year":"1997","journal-title":"Nature"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1103\/RevModPhys.74.145","article-title":"Quantum cryptography","volume":"74","author":"Gisin","year":"2002","journal-title":"Rev. Mod. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1038\/s41586-019-1427-5","article-title":"Parallel entangling operations on a universal ion-trap quantum computer","volume":"572","author":"Figgatt","year":"2019","journal-title":"Nature"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Cunha, M.M., Fonseca, A., and Silva, E.O. (2019). Tripartite entanglement: Foundations and applications. Universe, 5.","DOI":"10.3390\/universe5100209"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1330","DOI":"10.1126\/science.1104149","article-title":"Quantum-Enhanced Measurements: Beating the Standard Quantum Limit","volume":"306","author":"Giovannetti","year":"2004","journal-title":"Science"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"013602","DOI":"10.1103\/PhysRevLett.87.013602","article-title":"Two-Photon Diffraction and Quantum Lithography","volume":"87","author":"Chekhova","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2426","DOI":"10.1038\/ncomms3426","article-title":"An entanglement-enhanced microscope","volume":"4","author":"Ono","year":"2013","journal-title":"Nat. Commun."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"022309","DOI":"10.1103\/PhysRevA.65.022309","article-title":"Positioning and clock synchronization through entanglement","volume":"65","author":"Giovannetti","year":"2002","journal-title":"Phys. Rev. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"3775","DOI":"10.1038\/s41467-020-17559-w","article-title":"Quantum advantage in postselected metrology","volume":"11","author":"Halpern","year":"2020","journal-title":"Nat. Commun."},{"key":"ref_30","first-page":"222","article-title":"Advances in quantum metrology","volume":"5","author":"Giovannetti","year":"2011","journal-title":"Nat. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"022302","DOI":"10.1103\/PhysRevA.84.022302","article-title":"Quantum Fisher information of the Greenberger-Horne-Zeilinger state in decoherence channels","volume":"84","author":"Ma","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"120405","DOI":"10.1103\/PhysRevLett.112.120405","article-title":"Quantum Metrology in Open Systems: Dissipative Cram\u00e9r-Rao Bound","volume":"112","author":"Alipour","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"010401","DOI":"10.1103\/PhysRevLett.96.010401","article-title":"Quantum Metrology","volume":"96","author":"Giovannetti","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"5870","DOI":"10.1103\/PhysRevLett.86.5870","article-title":"Experimental Demonstration of Entanglement-Enhanced Rotation Angle Estimation Using Trapped Ions","volume":"86","author":"Meyer","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3865","DOI":"10.1103\/PhysRevLett.79.3865","article-title":"Improvement of frequency standards with quantum entanglement","volume":"79","author":"Huelga","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3439","DOI":"10.1103\/PhysRevLett.72.3439","article-title":"Statistical distance and the geometry of quantum states","volume":"72","author":"Braunstein","year":"1994","journal-title":"Phys. Rev. Lett."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"014301","DOI":"10.1103\/PhysRevA.88.014301","article-title":"Entanglement detection via quantum Fisher information","volume":"88","author":"Li","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"100401","DOI":"10.1103\/PhysRevLett.102.100401","article-title":"Entanglement, Nonlinear Dynamics, and the Heisenberg Limit","volume":"102","author":"Smerzi","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"025007","DOI":"10.1088\/2058-9565\/abd893","article-title":"Metrologically resourceful multipartite entanglement under quantum many-body effects","volume":"6","author":"Khalid","year":"2021","journal-title":"Quantum Sci. Technol."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"200501","DOI":"10.1103\/PhysRevLett.114.200501","article-title":"Machine Learning for Discriminating Quantum Measurement Trajectories and Improving Readout","volume":"114","author":"Magesan","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"042113","DOI":"10.1103\/PhysRevA.96.042113","article-title":"Deep learning and the Schr\u00f6dinger equation","volume":"96","author":"Mills","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1126\/science.aag2302","article-title":"Solving the quantum many-body problem with artificial neural network","volume":"355","author":"Carleo","year":"2017","journal-title":"Science"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"035105","DOI":"10.1103\/PhysRevB.95.035105","article-title":"Accelerated Monte Carlo simulations with restricted Boltzmann machines","volume":"95","author":"Huang","year":"2017","journal-title":"Phys. Rev. B."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1038\/nphys4035","article-title":"Machine learning phases of matter","volume":"13","author":"Carrasquilla","year":"2017","journal-title":"Nat. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1038\/nphys4037","article-title":"Learning phase transitions by confusion","volume":"13","author":"Liu","year":"2017","journal-title":"Nat. Phys."},{"key":"ref_46","first-page":"021021","article-title":"Quantum entanglement in neural network states","volume":"7","author":"Deng","year":"2017","journal-title":"Phys. Rev. X"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"063603","DOI":"10.1103\/PhysRevLett.104.063603","article-title":"Machine learning for precise quantum measurement","volume":"104","author":"Hentschel","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"123032","DOI":"10.1088\/1367-2630\/12\/12\/123032","article-title":"Quantum learning: Asymptotically optimal classification of qubit states","volume":"12","year":"2010","journal-title":"New J. Phys."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"880","DOI":"10.1103\/PhysRevLett.23.880","article-title":"Proposed Experiment to Test Local Hidden-Variable Theories","volume":"23","author":"Clauser","year":"1969","journal-title":"Phys. Rev. Lett."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/BF00417500","article-title":"Quantum generalizations of Bell\u2019s inequality","volume":"4","year":"1980","journal-title":"Lett. Math. Phys."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/S0375-9601(03)01115-0","article-title":"Symmetries of the Bell correlation inequalities","volume":"317","year":"2003","journal-title":"Phys. Lett. A"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1838","DOI":"10.1103\/PhysRevLett.65.1838","article-title":"Extreme quantum entanglement in a superposition of macroscopically distinct states","volume":"65","author":"Mermin","year":"1990","journal-title":"Phys. Rev. Lett."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"120402","DOI":"10.1103\/PhysRevLett.101.120402","article-title":"Necessary and Sufficient Detection Efficiency for the Mermin Inequalities","volume":"101","author":"Cabello","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"3066","DOI":"10.1103\/PhysRevD.35.3066","article-title":"Distinguishing three-body from two-body nonseparability by a Bell-type inequality","volume":"35","author":"Svetlichny","year":"1987","journal-title":"Phys. Rev. D"},{"key":"ref_55","unstructured":"MATLAB (2019). Version 9.6.0 1472908 (R2019a), The MathWorks Inc."},{"key":"ref_56","unstructured":"Johnston, N. (2016). QETLAB: A MATLAB Toolbox for Quantum Entanglement, Version 0.9, CVX Research, Inc."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"032343","DOI":"10.1103\/PhysRevA.101.032343","article-title":"Verifying multipartite entangled Greenberger-Horne-Zeilinger states via multiple quantum coherences","volume":"101","author":"Wei","year":"2020","journal-title":"Phys. Rev. A"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"054006","DOI":"10.1103\/PhysRevApplied.15.054006","article-title":"Accuracy of Entanglement Detection via Artificial Neural Networks and Human-Designed Entanglement Witnesses","volume":"15","author":"Roik","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjd\/e2016-60729-1","article-title":"Entangled graphs: A classification of four-qubit entanglement","volume":"70","author":"Ghahi","year":"2016","journal-title":"Eur. Phys. J. D"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"045001","DOI":"10.1088\/1367-2630\/ab783d","article-title":"Entanglement classification via neural network quantum states","volume":"22","author":"Harney","year":"2020","journal-title":"New J. Phys."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"063033","DOI":"10.1088\/1367-2630\/ac0388","article-title":"Mixed state entanglement classification using artificial neural networks","volume":"23","author":"Harney","year":"2021","journal-title":"New J. Phys."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/18\/6767\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:25:05Z","timestamp":1760142305000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/18\/6767"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,7]]},"references-count":61,"journal-issue":{"issue":"18","published-online":{"date-parts":[[2022,9]]}},"alternative-id":["s22186767"],"URL":"https:\/\/doi.org\/10.3390\/s22186767","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,7]]}}}