{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,12]],"date-time":"2026-03-12T13:34:55Z","timestamp":1773322495987,"version":"3.50.1"},"reference-count":85,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2010,5,7]],"date-time":"2010-05-07T00:00:00Z","timestamp":1273190400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Quantum entropy is a fundamental concept for quantum information recently developed in various directions. We will review the mathematical aspects of quantum entropy (entropies) and discuss some applications to quantum communication, statistical physics. All topics taken here are somehow related to the quantum entropy that the present authors have been studied. Many other fields recently developed in quantum information theory, such as quantum algorithm, quantum teleportation, quantum cryptography, etc., are totally discussed in the book (reference number 60).<\/jats:p>","DOI":"10.3390\/e12051194","type":"journal-article","created":{"date-parts":[[2010,5,7]],"date-time":"2010-05-07T10:41:03Z","timestamp":1273228863000},"page":"1194-1245","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":33,"title":["Quantum Entropy and Its Applications to Quantum Communication and Statistical Physics"],"prefix":"10.3390","volume":"12","author":[{"given":"Masanori","family":"Ohya","sequence":"first","affiliation":[{"name":"Department of Information Sciences, Tokyo University of Science, Noda City, Chiba 278-8510, Japan"}]},{"given":"Noboru","family":"Watanabe","sequence":"additional","affiliation":[{"name":"Department of Information Sciences, Tokyo University of Science, Noda City, Chiba 278-8510, Japan"}]}],"member":"1968","published-online":{"date-parts":[[2010,5,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1214\/aoms\/1177729694","article-title":"On information and sufficiency","volume":"22","author":"Kullback","year":"1951","journal-title":"Ann. Math. Stat."},{"key":"ref_3","first-page":"199","article-title":"Calculation of the amount of information about a random function contained in another such function","volume":"12","author":"Gelfand","year":"1959","journal-title":"Amer. Math. Soc. Transl."},{"key":"ref_4","first-page":"291","article-title":"Theory of transmission of information","volume":"33","author":"Kolmogorov","year":"1963","journal-title":"Amer. Math. Soc. Transl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1016\/0022-247X(81)90170-0","article-title":"Quantum ergodic channels in operator algebras","volume":"84","author":"Ohya","year":"1981","journal-title":"J. Math. Anal. Appl."},{"key":"ref_6","unstructured":"Von Neumann, J. (1932). Die Mathematischen Grundlagen der Quantenmechanik, Springer."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"59","DOI":"10.2996\/kmj\/1138844604","article-title":"Conditional expectations in an operator algebra IV(entropy and information)","volume":"14","author":"Umegaki","year":"1962","journal-title":"Kodai Math. Sem. Rep."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"809","DOI":"10.2977\/prims\/1195191148","article-title":"Relative entropy of states of von Neumann Algebras","volume":"11","author":"Araki","year":"1976","journal-title":"Publ.RIMS, Kyoto Univ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"173","DOI":"10.2977\/prims\/1195190105","article-title":"Relative entropy for states of von Neumann algebras II","volume":"13","author":"Araki","year":"1977","journal-title":"Publ.RIMS, Kyoto Univ."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/BF01609834","article-title":"Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in interpolation theory","volume":"54","author":"Uhlmann","year":"1977","journal-title":"Commun. Math. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1007\/BF01212339","article-title":"On the relative entropy","volume":"105","author":"Donald","year":"1985","journal-title":"Commun. Math. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"317","DOI":"10.4064\/sm-21-1-117-133","article-title":"Joint probability distribution of observables in quantum mechanics","volume":"21","author":"Urbanik","year":"1961","journal-title":"Stud. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"770","DOI":"10.1109\/TIT.1983.1056719","article-title":"On compound state and mutual information in quantum information theory","volume":"29","author":"Ohya","year":"1983","journal-title":"IEEE Trans. Infor. Theo."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"402","DOI":"10.1007\/BF02789599","article-title":"Note on quantum probability","volume":"38","author":"Ohya","year":"1983","journal-title":"L. Nuo. Cimen."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Schatten, R. (1970). Norm Ideals of Completely Continuous Operators, Springer Verlag.","DOI":"10.1007\/978-3-662-35155-0"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/0034-4877(89)90034-7","article-title":"Some aspects of quantum information theory and their applications to irreversible processes","volume":"27","author":"Ohya","year":"1989","journal-title":"Rep. Math. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Ohya, M., and Petz, D. (1993). Quantum Entropy and its Use, Springer Verlag.","DOI":"10.1007\/978-3-642-57997-4"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"99","DOI":"10.2140\/pjm.1981.96.99","article-title":"Sufficiency, KMS condition and relative entropy in von Neumann algebras","volume":"96","author":"Hiai","year":"1981","journal-title":"Pacif. J. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"117","DOI":"10.2140\/pjm.1983.107.117","article-title":"Sufficiency and relative entropy in *-algebras with applications to quantum systems","volume":"107","author":"Hiai","year":"1983","journal-title":"Pacif. J. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/BF01212345","article-title":"Sufficient subalgebras and the relative entropy of states of a von Neumann algebra","volume":"105","author":"Petz","year":"1986","journal-title":"Commun. Math. Phys."},{"key":"ref_21","first-page":"3","article-title":"Some estimates for the amount of information transmittable by a quantum communication channel (in Russian)","volume":"9","author":"Holevo","year":"1973","journal-title":"Prob. Pered. Infor."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/0034-4877(76)90005-7","article-title":"Quantum information theory","volume":"10","author":"Ingarden","year":"1976","journal-title":"Rep. Math. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1016\/0022-247X(84)90076-3","article-title":"Entropy Transmission in C*-dynamical systems","volume":"100","author":"Ohya","year":"1984","journal-title":"J. Math. Anal. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1007\/BF02228859","article-title":"Computation of mutual entropy in quantum Markov chains","volume":"2","author":"Accardi","year":"1994","journal-title":"Open Sys. Infor. Dyn."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1023\/A:1009605501218","article-title":"Superposition representability problems of quantum information channels","volume":"4","author":"Akashi","year":"1997","journal-title":"Open Sys. Infor. Dyn."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1007\/BF02228935","article-title":"Entropies of general quantum systems","volume":"1","author":"Muraki","year":"1992","journal-title":"Open Sys. Infor. Dyn."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1007\/BF00943285","article-title":"Entropy functionals of Kolmogorov\u2014Sinai type and their limit theorems","volume":"36","author":"Muraki","year":"1996","journal-title":"Lett. in Math. Phys."},{"key":"ref_28","first-page":"29","article-title":"Construction and analysis of a mathematical model in quantum communication processes","volume":"68","author":"Ohya","year":"1985","journal-title":"Scripta Thechnica, Inc., Elect. Commun. Japan"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1007\/BFb0074488","article-title":"State change and entropies in quantum dynamical systems","volume":"1136","author":"Ohya","year":"1985","journal-title":"Springer Lect. Not. in Math."},{"key":"ref_30","first-page":"179","article-title":"On capacities of quantum channels","volume":"17","author":"Ohya","year":"1997","journal-title":"Prob. and Math. Stat."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1023\/A:1026672330321","article-title":"Numerical computation of quantum capacity","volume":"38","author":"Ohya","year":"1998","journal-title":"Inter. J. Theor. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1007\/978-1-4899-1391-3_36","article-title":"A mathematical study of information transmission in quantum communication processes","volume":"2","author":"Ohya","year":"1995","journal-title":"Quant. Commun. Measur."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1007\/BF03167097","article-title":"A new treatment of communication processes with Gaussian Channels","volume":"3","author":"Ohya","year":"1986","journal-title":"Japan J. Appl. Math."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"959","DOI":"10.1007\/BF01206171","article-title":"Point processes and the position, distribution of infinite boson systems","volume":"47","author":"Fichtner","year":"1987","journal-title":"J. Stat. Phys."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1007\/BF02431883","article-title":"Characterization of states of infinite Boson systems I. On the construction of states","volume":"137","author":"Fichtner","year":"1991","journal-title":"Commun. Math. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"97","DOI":"10.2977\/prims\/1195184017","article-title":"Quantum stochastic processes","volume":"18","author":"Accardi","year":"1982","journal-title":"Publ. Res. Inst. Math. Sci."},{"key":"ref_37","first-page":"251","article-title":"Markov cocycles","volume":"83A","author":"Accardi","year":"1983","journal-title":"Proc. R. Ir. Acad."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2124","DOI":"10.1103\/PhysRevLett.62.2124","article-title":"Quantum optical Fredkin gate","volume":"63","author":"Milburn","year":"1989","journal-title":"Phys. Rev. Lett."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1103\/PhysRevLett.70.363","article-title":"Ultimate information carrying limit of quantum systems","volume":"70","author":"Yuen","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1023\/A:1009676318267","article-title":"Fundamentals of quantum mutual entropy and capacity","volume":"6","author":"Ohya","year":"1999","journal-title":"Open Sys. Infor. Dyn."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1142\/S0219025703001171","article-title":"On quantum entropy and its bound","volume":"6","author":"Ohya","year":"2003","journal-title":"Infi. Dimen. Anal., Quant. Prob. and Rela. Topics"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1109\/18.651037","article-title":"The capacity of quantum channel with general signal states","volume":"44","author":"Holevo","year":"1998","journal-title":"IEEE Trans. Info. Theor."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2614","DOI":"10.1103\/PhysRevA.54.2614","article-title":"Sending entanglement through noisy quantum channels","volume":"54","author":"Schumacher","year":"1996","journal-title":"Phys. Rev. A"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1142\/S0219025701000425","article-title":"Quantum entropy and information in discrete entangled states","volume":"4","author":"Belavkin","year":"2001","journal-title":"Infi. Dimen. Anal., Quant. Prob. Rela. Topics"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1098\/rspa.2001.0867","article-title":"Quantum entanglements and entangled mutual entropy","volume":"458","author":"Belavkin","year":"2002","journal-title":"Proc. Roy. Soc. Lond. A."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Ingarden, R.S., Kossakowski, A., and Ohya, M. (1997). Information Dynamics and Open Systems, Kluwer.","DOI":"10.1007\/978-94-017-1882-0"},{"key":"ref_47","unstructured":"Kolmogorov, A.N. (1958 and 1959). Dokl. Akad. Nauk SSSR, 119 and 124, 861 and 754."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1007\/BF02392105","article-title":"Entropy for automorphisms of II von Neumann algebras","volume":"134","author":"Connes","year":"1975","journal-title":"Acta Math."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/BF00536283","article-title":"Positivity of the K-entropy on non-abelian K-flows","volume":"29","author":"Emch","year":"1974","journal-title":"Z. Wahrscheinlichkeitstheory verw. Gebiete"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1007\/BF01225381","article-title":"Dynamical entropy of C*-algebras and von Neumann algebras","volume":"112","author":"Connes","year":"1987","journal-title":"Commun.Math.Phys."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/BF00761125","article-title":"Defining quantum dynamical entropy","volume":"32","author":"Alicki","year":"1994","journal-title":"Lett. Math. Phys."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Benatti, F. (1993). Deterministic Chaos in Infinite Quantum Systems, Series: Trieste Notes in Physics, Springer-Verlag.","DOI":"10.1007\/978-3-642-84999-2"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/BF00761124","article-title":"Dynamical entropy of generalized quantum Markov chains","volume":"32","author":"Park","year":"1994","journal-title":"Lett. Math. Phys."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"4303","DOI":"10.1063\/1.530855","article-title":"Topological entropy for appropriately approximated C*-algebras","volume":"35","author":"Hudetz","year":"1994","journal-title":"J. Math. Phys."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1007\/BF02108329","article-title":"Dynamical approximation entropies and topological entropy in operator algebras","volume":"170","author":"Voiculescu","year":"1995","journal-title":"Commun. Math. Phys."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"1197","DOI":"10.1017\/S0143385700009986","article-title":"Entropy for extensions of Bernoulli shifts","volume":"16","author":"Choda","year":"1996","journal-title":"Ergod. Theo. Dyn. Sys."},{"key":"ref_57","first-page":"81","article-title":"Information dynamics and its application to optical communication processes","volume":"378","author":"Ohya","year":"1991","journal-title":"Springer Lect. Not. Math."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1023\/A:1009609602126","article-title":"Dynamical entropy through quantum Markov chain","volume":"4","author":"Accardi","year":"1997","journal-title":"Open Sys. Infor. Dyn."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1142\/S021902579900014X","article-title":"Quantum dynamical entropy for completely positive maps","volume":"2","author":"Kossakowski","year":"1999","journal-title":"Infi. Dimen. Anal., Quant. Prob. Rela. Topics"},{"key":"ref_60","unstructured":"Ohya, M., and Volovich, I.V. Mathematical Foundation of Quantum Information and Computation, (in preparation)."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/BF01646743","article-title":"Entropy, information and quantum measurements","volume":"33","author":"Lindblad","year":"1973","journal-title":"Commun. Math. Phys."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/BF01609396","article-title":"Completely positive maps and entropy inequalities","volume":"40","author":"Lindblad","year":"1975","journal-title":"Commun. Math. Phys."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"9","DOI":"10.2140\/pjm.1989.138.9","article-title":"State extensions and a radon-Nikodym theorem for conditional expectations on von Neumann algebras","volume":"138","author":"Cecchini","year":"1989","journal-title":"Pacif. J. Math."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1007\/s002200100497","article-title":"Quantum teleportation with entangled states given by beam splittings","volume":"222","author":"Fichtner","year":"2001","journal-title":"Commun. Math. Phys."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/s002201000571","article-title":"Quantum teleportation and beam splitting","volume":"225","author":"Fichtner","year":"2002","journal-title":"Commun. Math. Phys."},{"key":"ref_66","unstructured":"Bilingsley, L. (1965). Ergodic Theory and Information, Wiley."},{"key":"ref_67","first-page":"50","article-title":"A formulation of noncommutative McMillan theorem","volume":"63, Ser.A","author":"Ohya","year":"1987","journal-title":"Proc. Japan Acad."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1007\/BF01196936","article-title":"Stationary states of quantum dynamical semigroups","volume":"63","author":"Frigerio","year":"1978","journal-title":"Commun. Math. Phys."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1103\/RevModPhys.50.221","article-title":"General properties of entropy","volume":"50","author":"Wehrl","year":"1978","journal-title":"Rev. Mod. Phys."},{"key":"ref_70","unstructured":"Shor, P. (2002, January 21\u201323). The Quantum Channel Capacity and Coherent Information, San Francisco, CA, USA. (unpublished)."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"4153","DOI":"10.1103\/PhysRevA.57.4153","article-title":"Information transmission through a noisy quantum channel","volume":"57","author":"Barnum","year":"1998","journal-title":"Phys. Rev. A"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"2637","DOI":"10.1109\/TIT.2002.802612","article-title":"Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem","volume":"48","author":"Bennett","year":"2002","journal-title":"IEEE Trans. Info. Theory"},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"2629","DOI":"10.1103\/PhysRevA.54.2629","article-title":"Quantum data processing and error correction","volume":"54","author":"Schumacher","year":"1996","journal-title":"Phys. Rev. A"},{"key":"ref_74","unstructured":"Ohya, M., and Watanabe, N. (2003). Comparison of mutual entropy-type measures, TUS preprint."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1007\/3-540-53862-3_196","article-title":"Efficiency of optical modulations with coherent states","volume":"378","author":"Watanabe","year":"1991","journal-title":"Springer Lect. Note. Phys."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1023\/A:1026620313483","article-title":"Complexities and their applications to characterization of chaos","volume":"37","author":"Ohya","year":"1998","journal-title":"Inter. J. Theor. Phys."},{"key":"ref_77","first-page":"423","article-title":"Notes on quantum entropy","volume":"31","author":"Ohya","year":"1996","journal-title":"Stud. Scien. Math. Hungar."},{"key":"ref_78","first-page":"22","article-title":"Capacity of memoryless quantum communication channels","volume":"94","author":"Fujiwara","year":"1994","journal-title":"Math. Eng. Tech. Rep., Univ. Tokyo"},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/978-1-4615-5923-8_23","article-title":"Quantum capacity of noisy quantum channel","volume":"3","author":"Ohya","year":"1997","journal-title":"Quant. Commun. Measur."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1016\/S0034-4877(97)84895-1","article-title":"Note on quantum dynamical entropies","volume":"38","author":"Accardi","year":"1996","journal-title":"Rep. Math. Phys."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1007\/978-1-4899-1391-3_29","article-title":"State change, complexity and fractal in quantum systems","volume":"2","author":"Ohya","year":"1995","journal-title":"Quant. Commun. Measur."},{"key":"ref_82","unstructured":"Accardi, L. (1974). Noncommutative Markov chains. Inter. Sch. Math. Phys., Camerino, 268."},{"key":"ref_83","unstructured":"Ohya, M., and Heyde, C.C. (1998). Probability towards 2000. Lecture Notes in Statistics, Springer-Verlag."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s002459900097","article-title":"Compound channels, transition expectations, and liftings","volume":"39","author":"Accardi","year":"1999","journal-title":"Appl. Math. Optim."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"411","DOI":"10.4064\/-43-1-411-420","article-title":"Comparing quantum dynamical entropies","volume":"43","author":"Tuyls","year":"1998","journal-title":"Banach Cent. Pub."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/12\/5\/1194\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:02:23Z","timestamp":1760220143000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/12\/5\/1194"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,5,7]]},"references-count":85,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2010,5]]}},"alternative-id":["e12051194"],"URL":"https:\/\/doi.org\/10.3390\/e12051194","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,5,7]]}}}