{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,27]],"date-time":"2026-07-27T15:01:53Z","timestamp":1785164513520,"version":"3.55.0"},"reference-count":68,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,11,15]],"date-time":"2021-11-15T00:00:00Z","timestamp":1636934400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11974213"],"award-info":[{"award-number":["11974213"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["92065112"],"award-info":[{"award-number":["92065112"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2019YFA0308200"],"award-info":[{"award-number":["2019YFA0308200"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"publisher","award":["ZR2019MA001"],"award-info":[{"award-number":["ZR2019MA001"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"publisher","award":["ZR2020JQ05"],"award-info":[{"award-number":["ZR2020JQ05"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Taishan Scholar of Shandong Province","award":["tsqn202103013"],"award-info":[{"award-number":["tsqn202103013"]}]},{"name":"Shandong University Multidisciplinary Research and Innovation Team of Young Scholars","award":["2020QNQT"],"award-info":[{"award-number":["2020QNQT"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Detecting multipartite quantum coherence usually requires quantum state reconstruction, which is quite inefficient for large-scale quantum systems. Along this line of research, several efficient procedures have been proposed to detect multipartite quantum coherence without quantum state reconstruction, among which the spectrum-estimation-based method is suitable for various coherence measures. Here, we first generalize the spectrum-estimation-based method for the geometric measure of coherence. Then, we investigate the tightness of the estimated lower bound of various coherence measures, including the geometric measure of coherence, the l1-norm of coherence, the robustness of coherence, and some convex roof quantifiers of coherence multiqubit GHZ states and linear cluster states. Finally, we demonstrate the spectrum-estimation-based method as well as the other two efficient methods. We observe that the spectrum-estimation-based method outperforms other methods in various coherence measures, which significantly enhances the accuracy of estimation.<\/jats:p>","DOI":"10.3390\/e23111519","type":"journal-article","created":{"date-parts":[[2021,11,15]],"date-time":"2021-11-15T20:46:47Z","timestamp":1637009207000},"page":"1519","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["The Tightness of Multipartite Coherence from Spectrum Estimation"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0548-6519","authenticated-orcid":false,"given":"Qi-Ming","family":"Ding","sequence":"first","affiliation":[{"name":"School of Physics, Shandong University, Jinan 250100, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6077-6925","authenticated-orcid":false,"given":"Xiao-Xu","family":"Fang","sequence":"additional","affiliation":[{"name":"School of Physics, Shandong University, Jinan 250100, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6501-5447","authenticated-orcid":false,"given":"He","family":"Lu","sequence":"additional","affiliation":[{"name":"School of Physics, Shandong University, Jinan 250100, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,15]]},"reference":[{"key":"ref_1","unstructured":"Nielsen, M.A., and Chuang, I.L. (2010). Quantum Computation and Quantum Information, Cambridge University Press. [10th anniversary ed.]."},{"key":"ref_2","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_3","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1038\/nphoton.2011.35","article-title":"Advances in Quantum Metrology","volume":"5","author":"Giovannetti","year":"2011","journal-title":"Nat. Photonics"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"6383","DOI":"10.1038\/ncomms7383","article-title":"Description of Quantum Coherence in Thermodynamic Processes Requires Constraints beyond Free Energy","volume":"6","author":"Lostaglio","year":"2015","journal-title":"Nat. Commun."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"7689","DOI":"10.1038\/ncomms8689","article-title":"Low-Temperature Thermodynamics with Quantum Coherence","volume":"6","author":"Narasimhachar","year":"2015","journal-title":"Nat. Commun."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"150402","DOI":"10.1103\/PhysRevLett.113.150402","article-title":"Catalytic Coherence","volume":"113","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2015.04.003","article-title":"The Resource Theory of Informational Nonequilibrium in Thermodynamics","volume":"583","author":"Gour","year":"2015","journal-title":"Phys. Rep."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1080\/00405000.2013.829687","article-title":"Vibrations, Quanta and Biology","volume":"54","author":"Huelga","year":"2013","journal-title":"Contemp. Phys."},{"key":"ref_9","first-page":"012037","article-title":"Quantum Coherence in Biological Systems","volume":"302","author":"Lloyd","year":"2011","journal-title":"J. Phys.: Conf. Ser."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1038\/nphys2474","article-title":"Quantum Biology","volume":"9","author":"Lambert","year":"2013","journal-title":"Nat. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"676","DOI":"10.1038\/nphys3017","article-title":"Quantum Coherence in Photosynthesis for Efficient Solar-Energy Conversion","volume":"10","author":"Romero","year":"2014","journal-title":"Nat. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"140401","DOI":"10.1103\/PhysRevLett.113.140401","article-title":"Quantifying Coherence","volume":"113","author":"Baumgratz","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"041003","DOI":"10.1103\/RevModPhys.89.041003","article-title":"Colloquium","volume":"89","author":"Streltsov","year":"2017","journal-title":"Rev. Mod. Phys."},{"key":"ref_14","first-page":"1","article-title":"Quantum Coherence and Geometric Quantum Discord","volume":"762\u2013764","author":"Hu","year":"2018","journal-title":"Phys. Rep."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"210401","DOI":"10.1103\/PhysRevLett.114.210401","article-title":"Frozen Quantum Coherence","volume":"114","author":"Bromley","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"150504","DOI":"10.1103\/PhysRevLett.116.150504","article-title":"Distribution of Quantum Coherence in Multipartite Systems","volume":"116","author":"Radhakrishnan","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"020403","DOI":"10.1103\/PhysRevLett.115.020403","article-title":"Measuring Quantum Coherence with Entanglement","volume":"115","author":"Streltsov","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"022112","DOI":"10.1103\/PhysRevA.92.022112","article-title":"Quantum Coherence in Multipartite Systems","volume":"92","author":"Yao","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"150502","DOI":"10.1103\/PhysRevLett.116.150502","article-title":"Robustness of Coherence: An Operational and Observable Measure of Quantum Coherence","volume":"116","author":"Napoli","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"042107","DOI":"10.1103\/PhysRevA.93.042107","article-title":"Robustness of Asymmetry and Coherence of Quantum States","volume":"93","author":"Piani","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"022124","DOI":"10.1103\/PhysRevA.92.022124","article-title":"Intrinsic Randomness as a Measure of Quantum Coherence","volume":"92","author":"Yuan","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"120404","DOI":"10.1103\/PhysRevLett.116.120404","article-title":"Operational Resource Theory of Coherence","volume":"116","author":"Winter","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"032316","DOI":"10.1103\/PhysRevA.96.032316","article-title":"Operational One-to-One Mapping between Coherence and Entanglement Measures","volume":"96","author":"Zhu","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1007\/s11128-017-1650-7","article-title":"A New Coherence Measure Based on Fidelity","volume":"16","author":"Liu","year":"2017","journal-title":"Quantum Inf. Process"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"285301","DOI":"10.1088\/1751-8121\/aa7638","article-title":"Measuring Coherence with Entanglement Concurrence","volume":"50","author":"Qi","year":"2017","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"042120","DOI":"10.1103\/PhysRevA.91.042120","article-title":"Fidelity and Trace-Norm Distances for Quantifying Coherence","volume":"91","author":"Shao","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"042336","DOI":"10.1103\/PhysRevA.96.042336","article-title":"Coherence Number as a Discrete Quantum Resource","volume":"96","author":"Chin","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"012110","DOI":"10.1103\/PhysRevA.93.012110","article-title":"Trace-Distance Measure of Coherence","volume":"93","author":"Rana","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"123033","DOI":"10.1088\/1367-2630\/aa91fa","article-title":"Polynomial Measure of Coherence","volume":"19","author":"Zhou","year":"2017","journal-title":"New J. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"022340","DOI":"10.1103\/PhysRevA.99.022340","article-title":"Coherence Measure: Logarithmic Coherence Number","volume":"99","author":"Xi","year":"2019","journal-title":"Phys. Rev. A"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"012308","DOI":"10.1103\/PhysRevA.99.012308","article-title":"Epsilon-Smooth Measure of Coherence","volume":"99","author":"Xi","year":"2019","journal-title":"Phys. Rev. A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"022420","DOI":"10.1103\/PhysRevA.102.022420","article-title":"Examining the Validity of Schatten-$p$-Norm-Based Functionals as Coherence Measures","volume":"102","author":"Cui","year":"2020","journal-title":"Phys. Rev. A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"240405","DOI":"10.1103\/PhysRevLett.116.240405","article-title":"Entanglement and Coherence in Quantum State Merging","volume":"116","author":"Streltsov","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"070402","DOI":"10.1103\/PhysRevLett.116.070402","article-title":"Assisted Distillation of Quantum Coherence","volume":"116","author":"Chitambar","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_35","first-page":"011024","article-title":"Towards Resource Theory of Coherence in Distributed Scenarios","volume":"7","author":"Streltsov","year":"2017","journal-title":"Phys. Rev. X"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"224204","DOI":"10.1103\/PhysRevB.100.224204","article-title":"Quantum Coherence and the Localization Transition","volume":"100","author":"Styliaris","year":"2019","journal-title":"Phys. Rev. B"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"180504","DOI":"10.1103\/PhysRevLett.123.180504","article-title":"Demonstrating Quantum Coherence and Metrology That Is Resilient to Transversal Noise","volume":"123","author":"Zhang","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_38","unstructured":"Smith, G., Smolin, J.A., Yuan, X., Zhao, Q., Girolami, D., and Ma, X. (2017). Quantifying Coherence and Entanglement via Simple Measurements. arXiv."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"020403","DOI":"10.1103\/PhysRevLett.118.020403","article-title":"Directly Measuring the Degree of Quantum Coherence Using Interference Fringes","volume":"118","author":"Wang","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41534-020-0280-6","article-title":"Direct Estimation of Quantum Coherence by Collective Measurements","volume":"6","author":"Yuan","year":"2020","journal-title":"Npj Quantum Inf."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"170501","DOI":"10.1103\/PhysRevLett.120.170501","article-title":"Estimating Coherence Measures from Limited Experimental Data Available","volume":"120","author":"Zhang","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"062310","DOI":"10.1103\/PhysRevA.99.062310","article-title":"Detecting Coherence via Spectrum Estimation","volume":"99","author":"Yu","year":"2019","journal-title":"Phys. Rev. A"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"023228","DOI":"10.1103\/PhysRevResearch.3.023228","article-title":"Efficient Estimation of Multipartite Quantum Coherence","volume":"3","author":"Ding","year":"2021","journal-title":"Phys. Rev. Res."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"054022","DOI":"10.1103\/PhysRevApplied.13.054022","article-title":"Experimentally Accessible Lower Bounds for Genuine Multipartite Entanglement and Coherence Measures","volume":"13","author":"Dai","year":"2020","journal-title":"Phys. Rev. Appl."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"013157","DOI":"10.1103\/PhysRevResearch.2.013157","article-title":"Characterizing coherence with quantum observables","volume":"2","author":"Mandal","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"012409","DOI":"10.1103\/PhysRevA.103.012409","article-title":"Detecting and Estimating Coherence Based on Coherence Witnesses","volume":"103","author":"Ma","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"060303","DOI":"10.1103\/PhysRevA.93.060303","article-title":"Measure-Independent Freezing of Quantum Coherence","volume":"93","author":"Yu","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"062325","DOI":"10.1103\/PhysRevA.99.062325","article-title":"Operational Interpretation of Coherence in Quantum Key Distribution","volume":"99","author":"Ma","year":"2019","journal-title":"Phys. Rev. A"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"012118","DOI":"10.1103\/PhysRevA.92.012118","article-title":"Duality of Quantum Coherence and Path Distinguishability","volume":"92","author":"Bera","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"012111","DOI":"10.1103\/PhysRevA.93.012111","article-title":"Coherence as a Resource in Decision Problems: The Deutsch-Jozsa Algorithm and a Variation","volume":"93","author":"Hillery","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"032307","DOI":"10.1103\/PhysRevA.95.032307","article-title":"Coherence Depletion in the Grover Quantum Search Algorithm","volume":"95","author":"Shi","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Liu, Y.C., Shang, J., and Zhang, X. (2019). Coherence Depletion in Quantum Algorithms. Entropy, 21.","DOI":"10.3390\/e21030260"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"140402","DOI":"10.1103\/PhysRevLett.122.140402","article-title":"Operational Advantage of Quantum Resources in Subchannel Discrimination","volume":"122","author":"Takagi","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"057902","DOI":"10.1103\/PhysRevLett.91.057902","article-title":"Majorization Criterion for Distillability of a Bipartite Quantum State","volume":"91","author":"Hiroshima","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"5184","DOI":"10.1103\/PhysRevLett.86.5184","article-title":"Separable States Are More Disordered Globally than Locally","volume":"86","author":"Nielsen","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1103\/PhysRevLett.83.436","article-title":"Conditions for a Class of Entanglement Transformations","volume":"83","author":"Nielsen","year":"1999","journal-title":"Phys. Rev. Lett."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1088\/0253-6102\/67\/2\/166","article-title":"Estimation on Geometric Measure of Quantum Coherence","volume":"67","author":"Zhang","year":"2017","journal-title":"Commun. Theor. Phys."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"910","DOI":"10.1103\/PhysRevLett.86.910","article-title":"Persistent Entanglement in Arrays of Interacting Particles","volume":"86","author":"Briegel","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"12122","DOI":"10.1038\/s41598-020-68979-z","article-title":"Numerical and Analytical Results for Geometric Measure of Coherence and Geometric Measure of Entanglement","volume":"10","author":"Zhang","year":"2020","journal-title":"Sci. Rep."},{"key":"ref_60","doi-asserted-by":"crossref","unstructured":"Boyd, S.P., and Vandenberghe, L. (2004). Convex Optimization, Cambridge University Press.","DOI":"10.1017\/CBO9780511804441"},{"key":"ref_61","unstructured":"Grant, M., and Boyd, S. (2021, November 12). CVX: MATLAB Software for Disciplined Convex Programming, Version 2.1. Available online: http:\/\/cvxr.com\/cvx."},{"key":"ref_62","unstructured":"Blondel, V., Boyd, S., and Kimura, H. (2008). Graph implementations for nonsmooth convex programs. Recent Advances in Learning and Control, Springer-Verlag Limited. Available online: https:\/\/link.springer.com\/content\/pdf\/10.1007%2F978-1-84800-155-8.pdf."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"010201","DOI":"10.1103\/PRXQuantum.2.010201","article-title":"Theory of Quantum System Certification","volume":"2","author":"Kliesch","year":"2021","journal-title":"PRX Quantum"},{"key":"ref_64","doi-asserted-by":"crossref","unstructured":"Aaronson, S. (2018). Shadow Tomography of Quantum States. arXiv.","DOI":"10.1145\/3188745.3188802"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"1050","DOI":"10.1038\/s41567-020-0932-7","article-title":"Predicting many properties of a quantum system from very few measurements","volume":"16","author":"Huang","year":"2020","journal-title":"Nat. Phys."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"200501","DOI":"10.1103\/PhysRevLett.127.200501","article-title":"Experimental Quantum State Measurement with Classical Shadows","volume":"127","author":"Zhang","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"010307","DOI":"10.1103\/PRXQuantum.2.010307","article-title":"Experimental Estimation of Quantum State Properties from Classical Shadows","volume":"2","author":"Struchalin","year":"2021","journal-title":"PRX Quantum"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"100401","DOI":"10.1103\/PhysRevLett.124.100401","article-title":"Quantum Overlapping Tomography","volume":"124","author":"Cotler","year":"2020","journal-title":"Phys. Rev. Lett."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/11\/1519\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:30:40Z","timestamp":1760167840000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/11\/1519"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,15]]},"references-count":68,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2021,11]]}},"alternative-id":["e23111519"],"URL":"https:\/\/doi.org\/10.3390\/e23111519","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,11,15]]}}}